(* A mixed bag of examples of triangular and rectangular control polygons, and calls to the various programs to draw them. (11/25/2002) Remember, all points must have a weight. If poly surface, then weight = 1 *) (* Pattern of calls for the various rendering procedures *) (* if light = 1, light triangulation trianglight, *) (* else ltriang *) (* *) (* if fcnet = 1, displays control net, otherwise skip it *) (* *) (* sqrender[{net__}, mm_, a_, b_, c_, d_, n_, light_, fcnet_, *) (* lx_,mx_,ly_,my_,lz_,mz_,debug_] *) (* *) (* This is wrt the square [c, a] x [d, b] *) (* *) (* rendersq[{net__}, mm_, a_, b_, c_, d_, n_, light_, fcnet_, *) (* lx_,mx_,ly_,my_,lz_,mz_,debug_] := *) (* Same as sqrender, but wrt [a, b] x [c, d], more intuitive order *) (* *) (* trender[{net__}, mm_, {newtrig_}, n_, light_, fcnet_, *) (* lx_,mx_,ly_,my_,lz_,mz_,debug_] *) (* *) (* render1[{net__}, mm_, n_, light_, fcnet_, *) (* lx_,mx_,ly_,my_,lz_,mz_,debug_] *) (* *) (* render3[{net__}, mm_, n_, light_, fcnet_, *) (* lx_,mx_,ly_,my_,lz_,mz_,debug_] *) (* *) (* render4[{net__}, mm_, n_, light_, fcnet_, *) (* lx_,mx_,ly_,my_,lz_,mz,debug_] *) (* *) (* render5[{net__}, mm_, n_, light_, fcnet_, *) (* lx_,mx_,ly_,my_,lz_,mz_,debug_] *) (* *) (* render6[{net__}, mm_, n_, light_, fcnet_, *) (* lx_,mx_,ly_,my_,lz_,mz_,debug_] *) (* *) (* render34[{net__}, mm_, n_, light_, fcnet_, *) (* lx_,mx_,ly_,my_,lz_,mz_,debug_] *) (* *) (* render56[{net__}, mm_, n_, light_, fcnet_, *) (* lx_,mx_,ly_,my_,lz_,mz_,debug_] *) (* *) (* render3to6[{net__}, mm_, n_, light_, fcnet_, *) (* lx_,mx_,ly_,my_,lz_,mz_,debug] *) (* *) (* lrender renders the surface by u-curves and v-curves *) (* wrt square [-1, 1] x [-1, 1] *) (* *) (* lrender[{net__},mm_,n_,ni_,lx_,mx_,ly_,my_,lz_,mz_,debug_] *) (* *) (* *) (* lfrender renders a closed surface by u-curves and v-curves *) (* *) (* lfrender[{net__},mm_,n_,ni_,lx_,mx_,ly_,my_,lz_,mz_,debug_] *) (* *) (* render[{net__}, mm_, n_, light_, fcnet_, *) (* lx_,mx_,ly_,my_,lz_,mz_,debug_] *) (* *) (* new2net[{net__}, mm_, a_, b_, n_, *) (* lx_,mx_,ly_,my_,lz_,mz_, debug_] *) (* *) (* subdiv4[{net__}, mm_, n_, lx_,mx_,ly_,my_,lz_,mz_,debug_] *) (* *) (* To render a square patch *) (* recrender[{net__}, p_, q_, n_, light_, fcnet_, *) (* lx_,mx_,ly_,my_,lz_,mz_,debug_] *) (* *) (* rfrac6[{net__}, mm_, n_, light_, fcnet_, *) (* lx_,mx_,ly_,my_,lz_,mz_] *) (* *) (* sqfractal[{net__}, mm_, a_, b_, c_, d_, n_, light_, fcnet_, *) (* lx_,mx_,ly_,my_,lz_,mz_] *) (* *) (* tfractal[{net__}, mm_, {newtrig__}, n_, light_, fcnet_, *) (* lx_,mx_,ly_,my_,lz_,mz_] *) (* *) (* fractal4[{net__}, mm_, n_, lx_,mx_,ly_,my_,lz_,mz_,light_] *) (* *) (* fractal4 is the same as tfractal for (r,s,t) *) (* *) (* fractal2[{net__}, mm_, n_, lx_,mx_,ly_,my_,lz_,mz_,light_] *) (* *) (* fractal2 is the same as sqfractal for standard square *) (* *) (* fastdiv6 is the "spider-style" subdivision *) (* fastdiv6[{net__}, mm_, n_, lx_,mx_,ly_,my_,lz_,mz_] *) (* *) (* fastdiv4 is the "diamond-style" subdivision *) (* fastdiv4[{net__}, mm_, n_, lx_,mx_,ly_,my_,lz_,mz_,light_] *) (* *) (* clocurv3D renders a closed curve on a surface *) (* specified by points a, b in the parameter plane *) (* clocurv3D[{net__},mm_,n_,a_,b_,lx_,mx_,ly_,my_,lz_,mz_,debug_] *) (* *) (* scurv3D renders a curve segment on a surface, specified by *) (* points a, b in the parameter plane *) (* scurv3D[{net__},mm_,n_,a_,b_,lx_,mx_,ly_,my_,lz_,mz_,debug_] *) (* *) (* Some useful reference triangles *) reftrig1 = {{-1, 1, 1}, {-1, -1, 3}, {1, 1, -1}}; reftrig2 = {{1, -1, 1}, {1, 1, -1}, {-1, -1, 3}}; reftrig3 = {{-1, 1, 1}, {1, 1, -1}, {1, -1, 1}}; reftrig = {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}; (* a parabolic cylinder *) xnet = {{0, 0, 0, 1}, {3, 0, 3, 1}, {6, 0, 0, 1}, {0, 4, 0, 1}, {3, 4, 3, 1}, {6, 4, 0, 1}} ynet = makerecnet[xnet, 1, 2] recrender[xnet,1,2,0,0,1,0,6,0,4,0,3,0] recrender[xnet,1,2,1,0,1,0,6,0,4,0,3,0] recrender[xnet,1,2,3,0,1,0,6,0,4,0,3,0] (* solid *) recrender[xnet,1,2,3,3,1,0,6,0,4,0,3,0] (* solid *) recrender[xnet,1,2,1,4,1,0,6,0,4,0,3,0] (* four colors *) recrender[xnet,1,2,2,4,1,0,6,0,4,0,3,0] (* four colors *) recrender[xnet,1,2,3,4,1,0,6,0,4,0,3,0] (* four colors *) recrender[xnet,1,2,2,0,1,0,6,0,4,0,3,0] recrender[xnet,1,2,2,2,1,0,6,0,4,0,3,0] (* 2 colors *) recrender[xnet,1,2,2,-1,1,0,6,0,4,0,3,0] (* edges only *) (* portion of sphere *) net = {{0, 0, -1, 1}, {0, 1, -1, 1}, {0, 1, 0, 2}, {1, 0, -1, 1}, {1, 1, -1, 1}, {1, 0, 0, 2}}; subdiv3[net,2,3,0,1,0,1,-1,0] subdiv4[net,2,3,0,1,0,1,-1,0] fastdiv4[net,2,3,0,1,0,1,-1,0] new6net[net,2,3,-1,1,-1,1,-1,1] frac6net[net,2,3,-1,1,-1,1,-1,1] mixsubdiv[net,2,1,0,1,0,1,-1,0] newtrig = {{-1, -1, 3}, {3, -1, -1}, {-1, 3, -1}}; subd4[net,2,newtrig,3,-1.5,1.5,-1.5,1.5,-1.5,1.5] subd4[net,2,newtrig,2,-5,3,-5,3,-2,3] newt = {{0, -1, 2}, {-1, 0 ,2}, {1, 1, -1}}; subd4[net,2,newt,2,-1,1,-1,1,-1,1] subd4[net,2,reftrig1,2,-1,1,-1,1,-1,1] subd2[net,2,2,-1,1,-1,1,-1,1] subd2[net,2,3,-1,1,-1,1,-1,1] fastdiv4[net,2,3,0,1,0,1,-1,0] (* portion of ellipsoid *) net = {{0, 0, -2, 1}, {0, 3, -2, 1}, {0, 3, 0, 2}, {4, 0, -2, 1}, {4, 3, -2, 1}, {4, 0, 0, 2}}; subdiv4[net,2,3,0,4,0,3,-2,0,0] subd2[net,2,3,-4,4,-3,3,-2,2,0] fastdiv4[net,2,3,0,4,0,3,-2,0] fastdiv6[net,2,3,0,4,0,3,-2,0] render1[net,2,1,0,1,0,4,0,3,-2,0,0] render1[net,2,2,0,1,0,4,0,3,-2,0,0] render1[net,2,3,0,1,0,4,0,3,-2,0,0] sqrender[net,2,1,1,-1,-1,3,0,0,-4,4,-3,3,-2,2,0] sqrender[net,2,1,1,-1,-1,3,2,0,-4,4,-3,3,-2,2,0] (* also ViewPoint -> {-2, -10, 2} *) (* patches of ellipsoid and the full ellipsoid ! *) render34[net,2,3,0,0,-4,4,-3,3,-2,2,0] render3[net,2,3,2,0,-4,4,-3,3,-2,2,0] render4[net,2,3,2,0,-4,4,-3,3,-2,2,0] render34[net,2,3,2,0,-4,4,-3,3,-2,2,0] render56[net,2,3,0,0,-4,4,-3,3,-2,2,0] render[net,2,3,0,0,-4,4,-3,3,-2,2,0] render[net,2,3,2,0,-4,4,-3,3,-2,2,0] (* 2 colors *) new3anet[net,2,3,-4,4,-3,3,-2,2] new4anet[net,2,3,-4,4,-3,3,-2,2] new34net[net,2,3,-4,4,-3,3,-2,2] new56net[net,2,3,-4,4,-3,3,-2,2] new4net[net,2,3,-4,4,-3,3,-2,2] new6net[net,2,3,-4,4,-3,3,-2,2] new2net[net,2,2,2,3,-4,4,-3,3,-2,2] srender3[net,2,3,0,0,-4,4,-3,3,-2,2,0,0,0] srender34[net,2,3,0,0,-4,4,-3,3,-2,2,0,0,0] fractal4[net,2,4,0,4,0,3,-2,0] (* very cute *) tfractal[net,2,reftrig,4,0,0,0,4,0,3,-2,0] (* very cute *) frac6net[net,2,3,-4,4,-3,3,-2,2] fractal2[net,2,3,-4,4,-3,3,-2,2] (* full ellipsoid, polar coord, degree 8 *) poly1 = 4*aa*v*(u^4 - 6*u^2 + 1)*(1 - v^2); poly2 = 16*bb*u*v*(1 - u^2)*(1 - v^2); poly3 = cc*(v^4 - 6*v^2 + 1)*(1 + u^2)^2; poly4 = (1 + u^2)^2*(1 + v^2)^2; p1 = Expand[poly1]; p1 = InputForm[p1]; p1 = 4*aa*v - 24*aa*u^2*v + 4*aa*u^4*v - 4*aa*v^3 + 24*aa*u^2*v^3 - 4*aa*u^4*v^3 p2 = Expand[poly2]; p2 = InputForm[p2]; p2 = 16*bb*u*v - 16*bb*u^3*v - 16*bb*u*v^3 + 16*bb*u^3*v^3 p3 = Expand[poly3]; p3 = InputForm[p3]; p3 = cc + 2*cc*u^2 + cc*u^4 - 6*cc*v^2 - 12*cc*u^2*v^2 - 6*cc*u^4*v^2 + cc*v^4 + 2*cc*u^2*v^4 + cc*u^4*v^4 p4 = Expand[poly4]; p4 = InputForm[p4]; p4 = 1 + 2*u^2 + u^4 + 2*v^2 + 4*u^2*v^2 + 2*u^4*v^2 + v^4 + 2*u^2*v^4 + u^4*v^4 poly = {{{4*aa, 0, 1}, {-24*aa, 2, 1}, {4*aa, 4, 1}, {-4*aa, 0, 3}, {24*aa, 2, 3}, {-4*aa, 4, 3}}, {{16*bb, 1, 1}, {-16*bb, 3, 1}, {-16*bb, 1, 3}, {16*bb, 3, 3}}, {{cc, 0, 0}, {2*cc, 2, 0}, {cc, 4, 0}, {-6*cc, 0, 2}, {-12*cc, 2, 2}, {-6*cc, 4, 2}, {cc, 0, 4}, {2*cc, 2, 4}, {cc, 4, 4}}, {{1, 0, 0}, {2, 2, 0}, {1, 4, 0}, {2, 0, 2}, {4, 2, 2}, {2, 4, 2}, {1, 0, 4}, {2, 2, 4}, {1, 4, 4}}}; net = contnet[poly, 8]; ellipnet = InputForm[net]; (* Master net *) ellipnet = {{0, 0, cc, 1}, {aa/2, 0, cc, 1}, {(14*aa)/15, 0, (11*cc)/15, 15/14}, {(20*aa)/17, 0, (5*cc)/17, 17/14}, {(120*aa)/101, 0, (-19*cc)/101, 101/70}, {aa, 0, (-3*cc)/5, 25/14}, {(11*aa)/16, 0, (-7*cc)/8, 16/7}, {aa/3, 0, -cc, 3}, {0, 0, -cc, 4}, {0, 0, cc, 1}, {aa/2, (2*bb)/7, cc, 1}, {(14*aa)/15, (8*bb)/15, (11*cc)/15, 15/14}, {(20*aa)/17, (56*bb)/85, (5*cc)/17, 17/14}, {(120*aa)/101, (64*bb)/101, (-19*cc)/101, 101/70}, {aa, (12*bb)/25, (-3*cc)/5, 25/14}, {(11*aa)/16, bb/4, (-7*cc)/8, 16/7}, {aa/3, 0, -cc, 3}, {0, 0, cc, 15/14}, {aa/3, (8*bb)/15, cc, 15/14}, {(75*aa)/121, (120*bb)/121, (87*cc)/121, 121/105}, {(73*aa)/92, (28*bb)/23, (6*cc)/23, 46/35}, {(276*aa)/331, (384*bb)/331, (-77*cc)/331, 331/210}, {(63*aa)/83, (72*bb)/83, (-53*cc)/83, 83/42}, {(11*aa)/18, (4*bb)/9, (-8*cc)/9, 18/7}, {0, 0, cc, 17/14}, {aa/17, (56*bb)/85, cc, 17/14}, {(5*aa)/46, (28*bb)/23, (16*cc)/23, 46/35}, {(19*aa)/106, (79*bb)/53, (11*cc)/53, 53/35}, {(12*aa)/43, (184*bb)/129, (-13*cc)/43, 129/70}, {(13*aa)/33, (12*bb)/11, (-23*cc)/33, 33/14}, {0, 0, cc, 101/70}, {(-24*aa)/101, (64*bb)/101, cc, 101/70}, {(-144*aa)/331, (384*bb)/331, (219*cc)/331, 331/210}, {(-20*aa)/43, (184*bb)/129, (17*cc)/129, 129/70}, {(-48*aa)/161, (32*bb)/23, (-9*cc)/23, 23/10}, {0, 0, cc, 25/14}, {(-12*aa)/25, (12*bb)/25, cc, 25/14}, {(-72*aa)/83, (72*bb)/83, (51*cc)/83, 83/42}, {(-32*aa)/33, (12*bb)/11, cc/33, 33/14}, {0, 0, cc, 16/7}, {(-5*aa)/8, bb/4, cc, 16/7}, {(-10*aa)/9, (4*bb)/9, (5*cc)/9, 18/7}, {0, 0, cc, 3}, {(-2*aa)/3, 0, cc, 3}, {0, 0, cc, 4}}; (* Case aa = 4, bb = 3, cc = 2 *) aa= 4; bb = 3; cc = 2; elnet = {{0, 0, 2, 1}, {2, 0, 2, 1}, {56/15, 0, 22/15, 15/14}, {80/17, 0, 10/17, 17/14}, {480/101, 0, -38/101, 101/70}, {4, 0, -6/5, 25/14}, {11/4, 0, -7/4, 16/7}, {4/3, 0, -2, 3}, {0, 0, -2, 4}, {0, 0, 2, 1}, {2, 6/7, 2, 1}, {56/15, 8/5, 22/15, 15/14}, {80/17, 168/85, 10/17, 17/14}, {480/101, 192/101, -38/101, 101/70}, {4, 36/25, -6/5, 25/14}, {11/4, 3/4, -7/4, 16/7}, {4/3, 0, -2, 3}, {0, 0, 2, 15/14}, {4/3, 8/5, 2, 15/14}, {300/121, 360/121, 174/121, 121/105}, {73/23, 84/23, 12/23, 46/35}, {1104/331, 1152/331, -154/331, 331/210}, {252/83, 216/83, -106/83, 83/42}, {22/9, 4/3, -16/9, 18/7}, {0, 0, 2, 17/14}, {4/17, 168/85, 2, 17/14}, {10/23, 84/23, 32/23, 46/35}, {38/53, 237/53, 22/53, 53/35}, {48/43, 184/43, -26/43, 129/70}, {52/33, 36/11, -46/33, 33/14}, {0, 0, 2, 101/70}, {-96/101, 192/101, 2, 101/70}, {-576/331, 1152/331, 438/331, 331/210}, {-80/43, 184/43, 34/129, 129/70}, {-192/161, 96/23, -18/23, 23/10}, {0, 0, 2, 25/14}, {-48/25, 36/25, 2, 25/14}, {-288/83, 216/83, 102/83, 83/42}, {-128/33, 36/11, 2/33, 33/14}, {0, 0, 2, 16/7}, {-5/2, 3/4, 2, 16/7}, {-40/9, 4/3, 10/9, 18/7}, {0, 0, 2, 3}, {-8/3, 0, 2, 3}, {0, 0, 2, 4}}; sqrender[elnet,8,1,1,-1,-1,2,0,0,-4.2,4.2,-3.2,3.2,-2.2,2.2,0] (* fine *) (* the pattern of curves is interesting *) sqrender[elnet,8,1,1,-1,-1,2,1,0,-4.2,4.2,-3.2,3.2,-2.2,2.2,0] (* lite *) trender[elnet,8,reftrig1,2,1,0,-4.2,4.2,-3.2,3.2,-2.2,2.2,0] (* lite *) trender[elnet,8,reftrig1,1,1,1,-4.2,4.2,-3.2,3.2,-2.2,2.2,0] (* lite *) render1[elnet,8,1,1,1,-4.2,4.2,-3.2,3.2,-2.2,2.2,0] (* lite *) render1[elnet,8,2,1,1,-4.2,4.2,-3.2,3.2,-2.2,2.2,0] (* lite *) render1[elnet,8,2,1,0,-4.2,4.2,-3.2,3.2,-2.2,2.2,0] (* lite *) render1[elnet,8,2,0,0,-4.2,4.2,-3.2,3.2,-2.2,2.2,0] lrender[elnet,8,5,2,-4.2,4.2,-3.2,3.2,-2.2,2.2,0] lfrender[elnet,8,5,2,-4.2,4.2,-3.2,3.2,-2.2,2.2,0] a = {-1, -1, 3}; b = {1, 1, -1}; a = {-1, 1, 1}; b = {1, -1, 1}; scurv3D[elnet,8,5,a,b,-4.2,4.2,-3.2,3.2,-2.2,2.2,0] clocurv3D[elnet,8,5,a,b,-4.2,4.2,-3.2,3.2,-2.2,2.2,0] (* ellipsoid, degree 4 *) poly = {{{aa, 2, 2}, {-aa, 2, 0}, {-aa, 0, 2}, {aa, 0, 0}}, {{2*bb, 1, 0}, {-2*bb, 1, 2}}, {{2*cc, 0, 1}, {2*cc, 2, 1}}, {{1, 2, 2}, {1, 2, 0}, {1, 0, 2}, {1, 0, 0}}}; net = contnet[poly, 4]; ellipnet1 = InputForm[net]; (* Master net *) ellipnet1 = {{aa, 0, 0, 1}, {aa, 0, cc/2, 1}, {(5*aa)/7, 0, (6*cc)/7, 7/6}, {aa/3, 0, cc, 3/2}, {0, 0, cc, 2}, {aa, bb/2, 0, 1}, {aa, bb/2, cc/2, 1}, {(5*aa)/7, (2*bb)/7, (6*cc)/7, 7/6}, {aa/3, 0, cc, 3/2}, {(5*aa)/7, (6*bb)/7, 0, 7/6}, {(5*aa)/7, (6*bb)/7, (4*cc)/7, 7/6}, {(5*aa)/9, (4*bb)/9, (8*cc)/9, 3/2}, {aa/3, bb, 0, 3/2}, {aa/3, bb, (2*cc)/3, 3/2}, {0, bb, 0, 2}}; (* Case aa = 4, bb = 3, cc = 2 *) elnet2 = {{4, 0, 0, 1}, {4, 0, 1, 1}, {20/7, 0, 12/7, 7/6}, {4/3, 0, 2, 3/2}, {0, 0, 2, 2}, {4, 3/2, 0, 1}, {4, 3/2, 1, 1}, {20/7, 6/7, 12/7, 7/6}, {4/3, 0, 2, 3/2}, {20/7, 18/7, 0, 7/6}, {20/7, 18/7, 8/7, 7/6}, {20/9, 4/3, 16/9, 3/2}, {4/3, 3, 0, 3/2}, {4/3, 3, 4/3, 3/2}, {0, 3, 0, 2}}; sqrender[elnet2,4,1,1,-1,-1,2,0,0,-4.2,4.2,-3.2,3.2,-2.2,2.2,0] (* fine *) render1[elnet2,4,2,0,0,-4.2,4.2,-3.2,3.2,-2.2,2.2,0] render1[elnet2,4,2,1,0,-4.2,4.2,-3.2,3.2,-2.2,2.2,0] (* lite * render3[elnet2,4,2,0,0,-4.2,4.2,-3.2,3.2,-2.2,2.2,0] render4[elnet2,4,2,0,0,-4.2,4.2,-3.2,3.2,-2.2,2.2,0] render34[elnet2,4,2,0,0,-4.2,4.2,-3.2,3.2,-2.2,2.2,0] render56[elnet2,4,3,0,0,-4.2,4.2,-3.2,3.2,-2.2,2.2,0] render3to6[elnet2,4,2,0,0,-4.2,4.2,-3.2,3.2,-2.2,2.2,0] trender[elnet2,4,reftrig1,2,0,0,-4.2,4.2,-3.2,3.2,-2.2,2.2,0] trender[elnet2,4,reftrig2,2,0,0,-4.2,4.2,-3.2,3.2,-2.2,2.2,0] trender[elnet2,4,reftrig2,2,1,0,-4.2,4.2,-3.2,3.2,-2.2,2.2,0] (* lite *) render[elnet2,4,2,0,0,-4.2,4.2,-3.2,3.2,-2.2,2.2,0] lrender[elnet2,4,5,2,-4.2,4.2,-3.2,3.2,-2.2,2.2,0] lfrender[elnet2,4,5,2,-4.2,4.2,-3.2,3.2,-2.2,2.2,0] (* very nice *) scurv3D[elnet2,4,5,a,b,-4.2,4.2,-3.2,3.2,-2.2,2.2,0] clocurv3D[elnet2,4,5,a,b,-4.2,4.2,-3.2,3.2,-2.2,2.2,0] (* Hyperboloid of 1 sheet, degree 6 *) poly = {{{aa, 4, 2}, {-6*aa, 2, 2}, {aa, 4, 0}, {-6*aa, 2, 0}, {aa, 0, 2}, {aa, 0, 0}}, {{-4*bb, 3, 2}, {-4*bb, 3, 0}, {4*bb, 1, 2}, {4*bb, 1, 0}}, {{2*cc, 4, 1}, {4*cc, 2, 1}, {2*cc, 0, 1}}, {{-1, 4, 2}, {1, 4, 0}, {-2, 2, 2}, {2, 2, 0}, {-1, 0, 2}, {1, 0, 0}}}; net = contnet[poly, 6]; hypnet1 = InputForm[net]; (* Master Net *) hypnet = {{aa, 0, 0, 1}, {aa, 0, cc/3, 1}, {(8*aa)/7, 0, (5*cc)/7, 14/15}, {(3*aa)/2, 0, (5*cc)/4, 4/5}, {(7*aa)/3, 0, (20*cc)/9, 3/5}, {5*aa, 0, 5*cc, 1/3}, {2*aa, 0, 2*cc, 0}, {aa, (2*bb)/3, 0, 1}, {aa, (2*bb)/3, cc/3, 1}, {(8*aa)/7, (11*bb)/14, (5*cc)/7, 14/15}, {(3*aa)/2, (13*bb)/12, (5*cc)/4, 4/5}, {(7*aa)/3, (16*bb)/9, (20*cc)/9, 3/5}, {5*aa, 4*bb, 5*cc, 1/3}, {(9*aa)/17, (20*bb)/17, 0, 17/15}, {(9*aa)/17, (20*bb)/17, (6*cc)/17, 17/15}, {(27*aa)/47, (66*bb)/47, (36*cc)/47, 47/45}, {(9*aa)/13, 2*bb, (18*cc)/13, 13/15}, {aa, (32*bb)/9, (8*cc)/3, 3/5}, {-aa/7, (9*bb)/7, 0, 7/5}, {-aa/7, (9*bb)/7, (8*cc)/21, 7/5}, {(-5*aa)/19, (29*bb)/19, (16*cc)/19, 19/15}, {(-3*aa)/5, (11*bb)/5, (8*cc)/5, 1}, {(-5*aa)/7, bb, 0, 28/15}, {(-5*aa)/7, bb, (3*cc)/7, 28/15}, {-aa, (7*bb)/6, cc, 8/5}, {-aa, bb/2, 0, 8/3}, {-aa, bb/2, cc/2, 8/3}, {-aa, 0, 0, 4}}; (* Case aa = 2, bb = 1, cc = 2 *) aa= 2; bb = 1; cc = 2; hypnet = {{2, 0, 0, 1}, {2, 0, 2/3, 1}, {16/7, 0, 10/7, 14/15}, {3, 0, 5/2, 4/5}, {14/3, 0, 40/9, 3/5}, {10, 0, 10, 1/3}, {4, 0, 4, 0}, {2, 2/3, 0, 1}, {2, 2/3, 2/3, 1}, {16/7, 11/14, 10/7, 14/15}, {3, 13/12, 5/2, 4/5}, {14/3, 16/9, 40/9, 3/5}, {10, 4, 10, 1/3}, {18/17, 20/17, 0, 17/15}, {18/17, 20/17, 12/17, 17/15}, {54/47, 66/47, 72/47, 47/45}, {18/13, 2, 36/13, 13/15}, {2, 32/9, 16/3, 3/5}, {-2/7, 9/7, 0, 7/5}, {-2/7, 9/7, 16/21, 7/5}, {-10/19, 29/19, 32/19, 19/15}, {-6/5, 11/5, 16/5, 1}, {-10/7, 1, 0, 28/15}, {-10/7, 1, 6/7, 28/15}, {-2, 7/6, 2, 8/5}, {-2, 1/2, 0, 8/3}, {-2, 1/2, 1, 8/3}, {-2, 0, 0, 4}}; reftrig = {{-1, -0.5, 2.5}, {-1, 0.5, 1.5}, {1, 0.5, -0.5}}; sqrender[hypnet,6,1,0.4,-1,-0.4,2,0,0,-4.2,4.2,-3.2,3.2,-2.1,2.1,0] sqrender[hypnet,6,1,0.5,-1,-0.5,2,0,0,-4.2,4.2,-3.2,3.2,-2.7,2.7,0] (* nice *) sqrender[hypnet,6,1,0.5,-1,-0.5,3,0,0,-4.2,4.2,-3.2,3.2,-2.7,2.7,0] (* great *) trender[hypnet,6,reftrig,2,0,0,-4.2,4.2,-3.2,3.2,-2.7,2.7,0] a = {0, -1, 2}; b = {0, 1, 0}; a = {-1, 0, 2}; b = {1, 0, 0}; scurv3D[hypnet,6,5,a,b,-4.2,4.2,-3.2,3.2,-2.2,2.2,0] clocurv3D[hypnet,6,5,a,b,-4.2,4.2,-3.2,3.2,-2.2,2.2,0] (* Hyperboloid of 1 sheet, degree 4 *) poly = {{{-aa, 2, 2}, {-aa, 2, 0}, {aa, 0, 2}, {aa, 0, 0}}, {{2*bb, 1, 0}, {2*bb, 1, 2}}, {{2*cc, 0, 1}, {2*cc, 2, 1}}, {{-1, 2, 2}, {1, 2, 0}, {-1, 0, 2}, {1, 0, 0}}}; net = contnet[poly, 4]; hypnet2 = InputForm[net]; (* Master net *) hypnet2 = {{aa, 0, 0, 1}, {aa, 0, cc/2, 1}, {(7*aa)/5, 0, (6*cc)/5, 5/6}, {3*aa, 0, 3*cc, 1/2}, {2*aa, 0, 2*cc, 0}, {aa, bb/2, 0, 1}, {aa, bb/2, cc/2, 1}, {(7*aa)/5, (4*bb)/5, (6*cc)/5, 5/6}, {3*aa, 2*bb, 3*cc, 1/2}, {(5*aa)/7, (6*bb)/7, 0, 7/6}, {(5*aa)/7, (6*bb)/7, (4*cc)/7, 7/6}, {aa, (8*bb)/5, (8*cc)/5, 5/6}, {aa/3, bb, 0, 3/2}, {aa/3, bb, (2*cc)/3, 3/2}, {0, bb, 0, 2}}; (* Case aa = 2, bb = 1, cc = 2 *) aa= 2; bb = 1; cc = 2; hypnet = {{2, 0, 0, 1}, {2, 0, 1, 1}, {14/5, 0, 12/5, 5/6}, {6, 0, 6, 1/2}, {4, 0, 4, 0}, {2, 1/2, 0, 1}, {2, 1/2, 1, 1}, {14/5, 4/5, 12/5, 5/6}, {6, 2, 6, 1/2}, {10/7, 6/7, 0, 7/6}, {10/7, 6/7, 8/7, 7/6}, {2, 8/5, 16/5, 5/6}, {2/3, 1, 0, 3/2}, {2/3, 1, 4/3, 3/2}, {0, 1, 0, 2}}; sqrender[hypnet,4,1,0.4,-1,-0.4,2,0,0,-4.2,4.2,-3.2,3.2,-2.1,2.1,0] sqrender[hypnet,4,1,0.5,-1,-0.5,3,0,0,-4.2,4.2,-3.2,3.2,-2.7,2.7,0] (* nice *) trender[hypnet,4,reftrig,2,0,0,-4.2,4.2,-3.2,3.2,-2.7,2.7,0] (* Hyperboloid of 2 sheets, degree 6 *) poly = {{{2*aa, 4, 1}, {-12*aa, 2, 1}, {2*aa, 0, 1}}, {{-8*bb, 3, 1}, {8*bb, 1, 1}}, {{cc, 4, 2}, {2*cc, 2, 2}, {cc, 4, 0}, {2*cc, 2, 0}, {cc, 0, 2}, {cc, 0, 0}}, {{-1, 4, 2}, {1, 4, 0}, {-2, 2, 2}, {2, 2, 0}, {-1, 0, 2}, {1, 0, 0}}}; net = contnet[poly, 6]; hypnet3 = InputForm[net]; (* Master net *) hypnet3 = {{0, 0, cc, 1}, {aa/3, 0, cc, 1}, {(5*aa)/7, 0, (8*cc)/7, 14/15}, {(5*aa)/4, 0, (3*cc)/2, 4/5}, {(20*aa)/9, 0, (7*cc)/3, 3/5}, {5*aa, 0, 5*cc, 1/3}, {2*aa, 0, 2*cc, 0}, {0, 0, cc, 1}, {aa/3, (4*bb)/15, cc, 1}, {(5*aa)/7, (4*bb)/7, (8*cc)/7, 14/15}, {(5*aa)/4, bb, (3*cc)/2, 4/5}, {(20*aa)/9, (16*bb)/9, (7*cc)/3, 3/5}, {5*aa, 4*bb, 5*cc, 1/3}, {0, 0, cc, 17/15}, {(2*aa)/17, (8*bb)/17, cc, 17/15}, {(12*aa)/47, (48*bb)/47, (55*cc)/47, 47/45}, {(6*aa)/13, (24*bb)/13, (21*cc)/13, 13/15}, {(8*aa)/9, (32*bb)/9, (25*cc)/9, 3/5}, {0, 0, cc, 7/5}, {(-4*aa)/21, (10*bb)/21, cc, 7/5}, {(-8*aa)/19, (20*bb)/19, (23*cc)/19, 19/15}, {(-4*aa)/5, 2*bb, (9*cc)/5, 1}, {0, 0, cc, 28/15}, {(-3*aa)/7, (2*bb)/7, cc, 28/15}, {-aa, (2*bb)/3, (4*cc)/3, 8/5}, {0, 0, cc, 8/3}, {-aa/2, 0, cc, 8/3}, {0, 0, cc, 4}}; (* Case aa = 2, bb = 1, cc = 2 *) aa= 2; bb = 1; cc = 2; hypnet = {{0, 0, 2, 1}, {2/3, 0, 2, 1}, {10/7, 0, 16/7, 14/15}, {5/2, 0, 3, 4/5}, {40/9, 0, 14/3, 3/5}, {10, 0, 10, 1/3}, {4, 0, 4, 0}, {0, 0, 2, 1}, {2/3, 4/15, 2, 1}, {10/7, 4/7, 16/7, 14/15}, {5/2, 1, 3, 4/5}, {40/9, 16/9, 14/3, 3/5}, {10, 4, 10, 1/3}, {0, 0, 2, 17/15}, {4/17, 8/17, 2, 17/15}, {24/47, 48/47, 110/47, 47/45}, {12/13, 24/13, 42/13, 13/15}, {16/9, 32/9, 50/9, 3/5}, {0, 0, 2, 7/5}, {-8/21, 10/21, 2, 7/5}, {-16/19, 20/19, 46/19, 19/15}, {-8/5, 2, 18/5, 1}, {0, 0, 2, 28/15}, {-6/7, 2/7, 2, 28/15}, {-2, 2/3, 8/3, 8/5}, {0, 0, 2, 8/3}, {-1, 0, 2, 8/3}, {0, 0, 2, 4}}; reftrig = {{-1, 0.6, 1.4}, {-1, 0, 2}, {1, 0.6, -0.6}}; sqrender[hypnet,6,1,0.6,-1,0,2,0,0,-5,5,-3,3,-4.5,4.5,0] (* nice *) sqrender[hypnet,6,1,0.6,-1,0,2,1,0,-5,5,-3,3,-4.5,4.5,0] (* lite *) trender[hypnet,6,reftrig,2,0,0,-5,5,-3,3,-4.5,4.5,0] (* Hyperboloid of 2 sheets, degree 4 *) poly = {{{2*aa, 0, 1}, {-2*aa, 2, 1}}, {{4*bb, 1, 1}}, {{cc, 2, 2}, {cc, 2, 0}, {cc, 0, 2}, {cc, 0, 0}}, {{-1, 2, 2}, {1, 2, 0}, {-1, 0, 2}, {1, 0, 0}}}; net = contnet[poly, 4]; hypnet4 = InputForm[net]; (* Master net *) hypnet4 = {{0, 0, cc, 1}, {aa/2, 0, cc, 1}, {(6*aa)/5, 0, (7*cc)/5, 5/6}, {3*aa, 0, 3*cc, 1/2}, {2*aa, 0, 2*cc, 0}, {0, 0, cc, 1}, {aa/2, bb/3, cc, 1}, {(6*aa)/5, (4*bb)/5, (7*cc)/5, 5/6}, {3*aa, 2*bb, 3*cc, 1/2}, {0, 0, cc, 7/6}, {(2*aa)/7, (4*bb)/7, cc, 7/6}, {(4*aa)/5, (8*bb)/5, (9*cc)/5, 5/6}, {0, 0, cc, 3/2}, {0, (2*bb)/3, cc, 3/2}, {0, 0, cc, 2}}; (* Case aa = 2, bb = 1, cc = 2 *) aa= 2; bb = 1; cc = 2; hypnet = {{0, 0, 2, 1}, {1, 0, 2, 1}, {12/5, 0, 14/5, 5/6}, {6, 0, 6, 1/2}, {4, 0, 4, 0}, {0, 0, 2, 1}, {1, 1/3, 2, 1}, {12/5, 4/5, 14/5, 5/6}, {6, 2, 6, 1/2}, {0, 0, 2, 7/6}, {4/7, 4/7, 2, 7/6}, {8/5, 8/5, 18/5, 5/6}, {0, 0, 2, 3/2}, {0, 2/3, 2, 3/2}, {0, 0, 2, 2}}; sqrender[hypnet,4,1,0.6,-1,0,2,0,0,-5,5,-3,3,-4.5,4.5,0] (* nice *) trender[hypnet,4,reftrig,2,0,0,-5,5,-3,3,-4.5,4.5,0] (* Steiner's Roman surface *) net = {{0, 0, 0, 1}, {1, 0, 0, 1}, {1, 0, 0, 2}, {0, 1, 0, 1}, {1, 1, 1, 1}, {0, 1, 0, 2}}; subdiv4[net,2,3,0,1,0,1,0,1,0] subd2[net,2,3,-1,1,-1,1,-1,1,0] new2net[net,2,2,2,3,-1,1,-1,1,-1,1,0] new4net[net,2,3,-1,1,-1,1,-1,1] new3anet[net,2,3,-1,1,-1,1,-1,1] new4anet[net,2,3,-1,1,-1,1,-1,1] new34net[net,2,3,-1,1,-1,1,-1,1] new56net[net,2,3,-1,1,-1,1,-1,1] new6net[net,2,3,-1,1,-1,1,-1,1] fastdiv4[net,2,3,0,1,0,1,0,1] render[net,2,3,0,0,-1,1,-1,1,-1,1,0] (* very good *) render[net,2,4,0,0,-1,1,-1,1,-1,1,0] (* great *) render[net,2,3,2,0,-1,1,-1,1,-1,1,0] (* in/outside colors *) sqrender[net,2,1,1,-1,-1,4,0,0,-1,1,-1,1,-1,1,0] (* very nice *) sqrender[net,2,1.5,1.5,-1.5,-1.5,4,0,0,-1,1,-1,1,-1,1,0] render1[net,2,4,0,0,-1,1,-1,1,-1,1,0] render1[net,2,4,0,0,0,1,0,1,0,1,0] render34[net,2,3,0,0,-1,1,-1,1,-1,1,0] render56[net,2,3,0,0,-1,1,-1,1,-1,1,0] (* Also very good viewpoint: ViewPoint -> {2, -15, -2}, great *) (* Also very good viewpoint: ViewPoint -> {2, -15, 2} *) (* Show[%136, ViewPoint -> {2, -20, -2}] *) fractal4[net,2,4,0,1,0,1,0,1] fractal2[net,2,4,-1,1,-1,1,-1,1] frac6net[net,2,3,-1,1,-1,1,-1,1] lrender[net,2,5,3,-1,1,-1,1,-1,1,0] lfrender[net,2,5,3,-1,1,-1,1,-1,1,0] (* Steiner's Roman surface, polar coordinates, degree 6 *) (* u = angle, v = polar radius *) poly1 = {{{4, 3, 1}, {4, 1, 1}}, {{2, 0, 1}, {-2, 4, 1}}, {{4, 1, 2}, {-4, 3, 2}}, {{1, 4, 2}, {1, 4, 0}, {2, 2, 2}, {2, 2, 0}, {1, 0, 2}, {1, 0, 0}}}; stnet1 = contnet[poly1, 6]; steinet1 = InputForm[stnet1] steinet1 = {{0, 0, 0, 1}, {0, 1/3, 0, 1}, {0, 5/8, 0, 16/15}, {0, 5/6, 0, 6/5}, {0, 20/21, 0, 7/5}, {0, 1, 0, 5/3}, {0, 1, 0, 2}, {0, 0, 0, 1}, {2/15, 1/3, 0, 1}, {1/4, 5/8, 1/16, 16/15}, {1/3, 5/6, 1/6, 6/5}, {8/21, 20/21, 2/7, 7/5}, {2/5, 1, 2/5, 5/3}, {0, 0, 0, 17/15}, {4/17, 5/17, 0, 17/15}, {24/55, 6/11, 6/55, 11/9}, {4/7, 5/7, 2/7, 7/5}, {16/25, 4/5, 12/25, 5/3}, {0, 0, 0, 7/5}, {1/3, 5/21, 0, 7/5}, {14/23, 10/23, 2/23, 23/15}, {7/9, 5/9, 2/9, 9/5}, {0, 0, 0, 28/15}, {3/7, 1/7, 0, 28/15}, {3/4, 1/4, 0, 32/15}, {0, 0, 0, 8/3}, {1/2, 0, 0, 8/3}, {0, 0, 0, 4}}; (* Complementary patch (v -> 1/v) *) poly2 = {{{4, 3, 1}, {4, 1, 1}}, {{2, 0, 1}, {-2, 4, 1}}, {{4, 1, 0}, {-4, 3, 0}}, {{1, 4, 2}, {1, 4, 0}, {2, 2, 2}, {2, 2, 0}, {1, 0, 2}, {1, 0, 0}}}; stnet2 = contnet[poly2, 6]; steinet2 = InputForm[stnet2] steinet2 = {{0, 0, 0, 1}, {0, 1/3, 0, 1}, {0, 5/8, 0, 16/15}, {0, 5/6, 0, 6/5}, {0, 20/21, 0, 7/5}, {0, 1, 0, 5/3}, {0, 1, 0, 2}, {0, 0, 2/3, 1}, {2/15, 1/3, 2/3, 1}, {1/4, 5/8, 5/8, 16/15}, {1/3, 5/6, 5/9, 6/5}, {8/21, 20/21, 10/21, 7/5}, {2/5, 1, 2/5, 5/3}, {0, 0, 20/17, 17/15}, {4/17, 5/17, 20/17, 17/15}, {24/55, 6/11, 12/11, 11/9}, {4/7, 5/7, 20/21, 7/5}, {16/25, 4/5, 4/5, 5/3}, {0, 0, 9/7, 7/5}, {1/3, 5/21, 9/7, 7/5}, {14/23, 10/23, 27/23, 23/15}, {7/9, 5/9, 1, 9/5}, {0, 0, 1, 28/15}, {3/7, 1/7, 1, 28/15}, {3/4, 1/4, 7/8, 32/15}, {0, 0, 1/2, 8/3}, {1/2, 0, 1/2, 8/3}, {0, 0, 0, 4}}; rendersq[steinet1,6,-1,1,-1,0,3,0,0,-1.1,1.1,-1.1,1.1,-1.1,1.1,0] rendersq[steinet1,6,-1,1,0,1,3,0,0,-1.1,1.1,-1.1,1.1,-1.1,1.1,0] rendersq[steinet1,6,-1,1,-1,1,3,0,0,-1.1,1.1,-1.1,1.1,-1.1,1.1,0] rendersq[steinet1,6,-1,1,-1,1,2,4,0,-1.1,1.1,-1.1,1.1,-1.1,1.1,0] render34[steinet1,6,3,0,0,-1,1,-1,1,-1,1,0] rendersq[steinet2,6,-1,1,-1,0,3,0,0,-1.1,1.1,-1.1,1.1,-1.1,1.1,0] (* Show[%270, ViewPoint -> {4, -4, 3}] *) rendersq[steinet2,6,-1,1,-1,1,3,0,0,-1.1,1.1,-1.1,1.1,-1.1,1.1,0] (* Show[%1426, ViewPoint -> {-4, 4, 1}] *) rendersq[steinet2,6,-1,1,-1,1,3,0,0,-1.1,1.1,-1.1,1.1,-1.1,0.6,0] (* nice *) rendersq[steinet2,6,-1,1,-1,0,3,0,0,-1.1,1.1,-1.1,1.1,-1.1,1.1,0] rendersq[steinet2,6,-1,1,0,1,3,0,0,-1.1,1.1,-1.1,1.1,-1.1,1.1,0] rendersq[steinet2,6,-1,1,-1,1,3,4,0,-1.1,1.1,-1.1,1.1,-1.1,1.1,0] (* other closed Steiner type surface *) net = {{-1, 0, 0, 1}, {0, 1, 0, 1}, {0, 2, 0, 2}, {1, 0, 0, 1}, {1, 1, -2, 1}, {1, 0, 0, 2}}; subdiv4[net,2,3,-1,1,0,2,-2,1,0] subd2[net,2,3,-5,3,-2,4,-4,4,0] new6net[net,2,3,-4,2,-2,2,-2,3] frac6net[net,2,3,-4,2,-2,2,-2,3] (* cross-cap surface (image of projective plane in 3D) *) poly = {{{2, 2, 1}, {2, 0, 3}, {-2, 0, 1}}, {{8, 1, 1}}, {{4, 2, 0}, {-4, 0, 2}}, {{1, 4, 0}, {1, 0, 4}, {2, 2, 2}, {2, 2, 0}, {2, 0, 2}, {1, 0, 0}}}; net = contnet[poly, 4] projnet2 = {{0, 0, 0, 1}, {-1/2, 0, 0, 1}, {-3/4, 0, -1/2, 4/3}, {-1/2, 0, -1, 2}, {0, 0, -1, 4}, {0, 0, 0, 1}, {-1/2, 2/3, 0, 1}, {-3/4, 1, -1/2, 4/3}, {-1/2, 1, -1, 2}, {0, 0, 1/2, 4/3}, {-1/4, 1, 1/2, 4/3}, {-1/3, 4/3, 0, 2}, {0, 0, 1, 2}, {0, 1, 1, 2}, {0, 0, 1, 4}}; subdiv4[projnet2,4,2,-0.8,0.6,-0.1,1,-1.1,1.1,0] sqrender[projnet2,4,1,1,-1,-1,3,0,0,-0.8,0.8,-1.1,1.1,-1.1,1.1,0] sqrender[projnet2,4,1,1,-1,-1,3,0,0,-0.8,0.8,-1.1,1.1,-1.1,0.5,0] (* top cut, great *) sqrender[projnet2,4,1,1,-1,-1,3,1,0,-0.8,0.8,-1.1,1.1,-1.1,1.1,0] (* lite *) trender[projnet2,4,reftrig1,3,0,0,-0.8,0.8,-1.1,1.1,-1.1,1.1,0] trender[projnet2,4,reftrig1,4,0,0,-0.8,0.8,-1.1,1.1,-1.1,1.1,0] (* 4 iterations *) trender[projnet2,4,reftrig2,3,0,0,-0.8,0.8,-1.1,1.1,-1.1,1.1,0] trender[projnet2,4,reftrig2,4,0,0,-0.8,0.8,-1.1,1.1,-1.1,1.1,0] (* 4 iterations *) sqrender[projnet2,4,1,1,-1,-1,4,0,0,-0.8,0.8,-1.1,1.1,-1.1,1.1,0] (* great *) (* good Show[%260, ViewPoint -> {0, 8,10}] *) (* Show[%260, ViewPoint -> {2, 8, 10}] *) (* Show[%260, ViewPoint -> {0, 0.2, 10}] from above *) (* Show[%260, ViewPoint -> {0, 6, 10}] *) render1[projnet2,4,3,0,0,-0.8,0.8,-1.1,1.1,-1.1,1.1,0] render3[projnet2,4,3,0,0,-0.8,0.8,-1.1,1.1,-1.1,1.1,0] render4[projnet2,4,3,0,0,-0.8,0.8,-1.1,1.1,-1.1,1.1,0] render34[projnet2,4,3,0,0,-0.8,0.8,-1.1,1.1,-1.1,1.1,0] render5[projnet2,4,3,0,0,-0.8,0.8,-1.1,1.1,-1.1,1.1,0] render6[projnet2,4,3,0,0,-0.8,0.8,-1.1,1.1,-1.1,1.1,0] render56[projnet2,4,3,0,0,-0.8,0.8,-1.1,1.1,-1.1,1.1,0] render3to6[projnet2,4,3,1,0,-0.8,0.8,-1.1,1.1,-1.1,1.1,0] (* lite *) render[projnet2,4,3,0,0,-0.8,0.8,-1.1,1.1,-1.1,1.1,0] (* very good *) lrender[projnet2,4,5,8,-0.8,0.8,-1.1,1.1,-1.1,1.1,0] (* nice *) lrender[projnet2,4,5,2,-0.8,0.8,-1.1,1.1,-1.1,1.1,0] (* interesting *) lrender[projnet2,4,5,4,-0.8,0.8,-1.1,1.1,-1.1,1.1,0] lrender[projnet2,4,5,0,-0.8,0.8,-1.1,1.1,-1.1,1.1,0] lfrender[projnet2,4,5,2,-0.8,0.8,-1.1,1.1,-1.1,1.1,0] (* nice *) pta = {0, -1, 2}; ptb = {0, 1, 0}; pta = {-1, 0, 2}; ptb = {1, 0, 0}; pta = {-1, 1/2, 3/2}; ptb = {1, 1/2, -1/2}; pta = {-1, -1, 3}; ptb = {-1, 1, 1}; pta = {-1, -1, 3}; ptb = {1, -1, 1}; pta = {1, -1, 1}; ptb = {1, 1, -1}; pta = {-1, 1, 1}; ptb = {1, 1, -1}; clocurv3D[projnet2,4,5,pta,ptb,-0.8,0.8,-1.1,1.1,-1.1,1.1,0] (* other image of the projective plane in 3D *) (* twisted double cross cap *) poly = {{{2, 2, 1}, {2, 0, 3}, {-2, 0, 1}}, {{2, 3, 0}, {2, 1, 2}, {-2, 1, 0}}, {{4, 2, 0}, {-4, 0, 2}}, {{1, 4, 0}, {1, 0, 4}, {2, 2, 2}, {2, 2, 0}, {2, 0, 2}, {1, 0, 0}}} prjnet3 = contnet[poly, 4] projnet3 = {{0, 0, 0, 1}, {-1/2, 0, 0, 1}, {-3/4, 0, -1/2, 4/3}, {-1/2, 0, -1, 2}, {0, 0, -1, 4}, {0, -1/2, 0, 1}, {-1/2, -1/2, 0, 1}, {-3/4, -1/4, -1/2, 4/3}, {-1/2, 0, -1, 2}, {0, -3/4, 1/2, 4/3}, {-1/4, -3/4, 1/2, 4/3}, {-1/3, -1/3, 0, 2}, {0, -1/2, 1, 2}, {0, -1/2, 1, 2}, {0, 0, 1, 4}} sqrender[projnet3,4,1,1,-1,-1,3,0,0,-0.55,0.55,-0.55,0.55,-1.1,1.1,0] (* great *) (* with top cut off *) sqrender[projnet3,4,1,1,-1,-1,3,0,0,-0.55,0.55,-0.55,0.55,-1.1,0.5,0] sqrender[projnet3,4,1,1,-1,-1,3,2,0,-0.55,0.55,-0.55,0.55,-1.1,1.1,0] (* 2 colors *) sqrender[projnet3,4,1,1,-1,-1,3,4,0,-0.55,0.55,-0.55,0.55,-1.1,1.1,0] (* 4 colors *) rendersq[projnet3,4,-1,1,-1,0,3,0,0,-0.55,0.55,-0.55,0.55,-1.1,1.1,0] rendersq[projnet3,4,-1,1,0,1,3,0,0,-0.55,0.55,-0.55,0.55,-1.1,1.1,0] rendersq[projnet3,4,-1,0,-1,1,3,0,0,-0.55,0.55,-0.55,0.55,-1.1,1.1,0] rendersq[projnet3,4,0,1,-1,1,3,0,0,-0.55,0.55,-0.55,0.55,-1.1,1.1,0] rendersq[projnet3,4,-1,0,-1,0,3,0,0,-0.55,0.55,-0.55,0.55,-1.1,1.1,0] rendersq[projnet3,4,-1,0,0,1,3,0,0,-0.55,0.55,-0.55,0.55,-1.1,1.1,0] sqrender[projnet3,4,1,1,-1,-1,4,0,0,-0.55,0.55,-0.55,0.55,-1.1,1.1,0] sqrender[projnet3,4,1,1,-1,-1,4,1,0,-0.55,0.55,-0.55,0.55,-1.1,1.1,0] (* lite *) trender[projnet3,4,reftrig1,3,0,0,-0.55,0.55,-0.55,0.55,-1.1,1.1,0] trender[projnet3,4,reftrig1,4,0,0,-0.55,0.55,-0.55,0.55,-1.1,1.1,0] (* 4 iterations *) trender[projnet3,4,reftrig2,3,0,0,-0.55,0.55,-0.55,0.55,-1.1,1.1,0] trender[projnet3,4,reftrig2,4,0,0,-0.55,0.55,-0.55,0.55,-1.1,1.1,0] (* 4 iterations *) render1[projnet3,4,3,0,0,-0.55,0.55,-0.55,0.55,-1.1,1.1,0] render3[projnet3,4,3,0,0,-0.55,0.55,-0.55,0.55,-1.1,1.1,0] render4[projnet3,4,3,0,0,-0.55,0.55,-0.55,0.55,-1.1,1.1,0] render34[projnet3,4,3,0,0,-0.55,0.55,-0.55,0.55,-1.1,1.1,0] render5[projnet3,4,3,0,0,-0.55,0.55,-0.55,0.55,-1.1,1.1,0] render6[projnet3,4,3,0,0,-0.55,0.55,-0.55,0.55,-1.1,1.1,0] render56[projnet3,4,3,0,0,-0.55,0.55,-0.55,0.55,-1.1,1.1,0] render3to6[projnet3,4,3,1,0,-0.55,0.55,-0.55,0.55,-1.1,1.1,0] (* lite *) render[projnet3,4,3,0,0,-0.55,0.55,-0.55,0.55,-1.1,1.1,0] (* Also Show[%152, ViewPoint -> {0, -6, 5}] *) (* Show[%152, ViewPoint -> {0, -6, 4}] *) lrender[projnet3,4,5,2,-0.55,0.55,-0.55,0.55,-1.1,1.1,0] (* nice *) lfrender[projnet3,4,5,2,-0.55,0.55,-0.55,0.55,-1.1,1.1,0] (* cross-cap algebraic surface from Hilbert and Cohn-Vossen *) (* Only get the bottom part *) poly = {{{2, 0, 1}, {-2, 4, 1}}, {{4, 1, 1}, {4, 3, 1}}, {{2, 4, 2}, {4, 2, 2}, {2, 0, 2}}, {{1, 4, 2}, {1, 4, 0}, {1, 0, 2}, {1, 0, 0}}}; net = contnet[poly, 6] hilnetb1 = {{0, 0, 0, 1}, {1/3, 0, 0, 1}, {5/8, 0, 1/8, 16/15}, {5/6, 0, 1/3, 6/5}, {20/21, 0, 4/7, 7/5}, {1, 0, 4/5, 5/3}, {1, 0, 1, 2}, {0, 0, 0, 1}, {1/3, 2/15, 0, 1}, {5/8, 1/4, 1/8, 16/15}, {5/6, 1/3, 1/3, 6/5}, {20/21, 8/21, 4/7, 7/5}, {1, 2/5, 4/5, 5/3}, {0, 0, 0, 1}, {1/3, 4/15, 0, 1}, {5/8, 1/2, 1/6, 16/15}, {5/6, 2/3, 4/9, 6/5}, {20/21, 16/21, 16/21, 7/5}, {0, 0, 0, 1}, {1/3, 7/15, 0, 1}, {5/8, 7/8, 1/4, 16/15}, {5/6, 7/6, 2/3, 6/5}, {0, 0, 0, 16/15}, {1/4, 3/4, 0, 16/15}, {4/9, 4/3, 4/9, 6/5}, {0, 0, 0, 4/3}, {0, 1, 0, 4/3}, {0, 0, 0, 2}}; render1[hilnetb1,6,2,0,0,-1.1,1.1,-2.1,2.1,0,4.1,0] rendersq[hilnetb1,6,-1,1,-1,1,2,0,0,-1.1,1.1,-2.1,2.1,0,4.1,0] rendersq[hilnetb1,6,-1,1,-1,1,2,2,0,-1.1,1.1,-2.1,2.1,0,4.1,0] (* 2 colors *) (* algebraic cross-cap surface, second parameterization, degree 6 *) (* Only get the top part *) poly = {{{2, 0, 1}, {-2, 4, 1}}, {{4, 1, 1}, {4, 3, 1}}, {{2, 4, 0}, {4, 2, 0}, {2, 0, 0}}, {{1, 4, 2}, {1, 4, 0}, {1, 0, 2}, {1, 0, 0}}}; net = contnet[poly, 6] hilnetb2 = {{0, 0, 2, 1}, {1/3, 0, 2, 1}, {5/8, 0, 15/8, 16/15}, {5/6, 0, 5/3, 6/5}, {20/21, 0, 10/7, 7/5}, {1, 0, 6/5, 5/3}, {1, 0, 1, 2}, {0, 0, 2, 1}, {1/3, 2/15, 2, 1}, {5/8, 1/4, 15/8, 16/15}, {5/6, 1/3, 5/3, 6/5}, {20/21, 8/21, 10/7, 7/5}, {1, 2/5, 6/5, 5/3}, {0, 0, 34/15, 1}, {1/3, 4/15, 34/15, 1}, {5/8, 1/2, 17/8, 16/15}, {5/6, 2/3, 17/9, 6/5}, {20/21, 16/21, 34/21, 7/5}, {0, 0, 14/5, 1}, {1/3, 7/15, 14/5, 1}, {5/8, 7/8, 21/8, 16/15}, {5/6, 7/6, 7/3, 6/5}, {0, 0, 7/2, 16/15}, {1/4, 3/4, 7/2, 16/15}, {4/9, 4/3, 28/9, 6/5}, {0, 0, 4, 4/3}, {0, 1, 4, 4/3}, {0, 0, 4, 2}}; render1[hilnetb2,6,2,0,0,-1.1,1.1,-2.1,2.1,0,4.1,0] rendersq[hilnetb2,6,-1,1,-1,1,2,0,0,-1.1,1.1,-2.1,2.1,0,4.1,0] rendersq[hilnetb2,6,-1,1,-1,1,2,2,0,-1.1,1.1,-2.1,2.1,0,4.1,0] (* 2 colors *) rendersq[hilnetb2,6,2,2,-2,-2,3,0,0,-1.1,1.1,-2.1,2.1,0,4.1,0] (* algebraic cross-cap surface, third parameterization, degree 8 *) (* general parameterization depending on a, b *) pol4 = (u^4 + 2*u^2 + 1) * (aa*v^4 + 2*(2*bb - aa)*v^2 + aa); pp4 = Expand[pol4]; pp4 = aa + 2*aa*u^2 + aa*u^4 - 2*aa*v^2 + 4*bb*v^2 - 4*aa*u^2*v^2 + 8*bb*u^2*v^2 - 2*aa*u^4*v^2 + 4*bb*u^4*v^2 + aa*v^4 + 2*aa*u^2*v^4 + aa*u^4*v^4; poly = {{{-4, 3, 4}, {4, 1, 4}, {4, 3, 0}, {-4, 1, 0}}, {{8, 3, 3}, {8, 3, 1}, {-8, 1, 3}, {-8, 1, 1}}, {{2, 4, 4}, {4, 4, 2}, {-4, 2, 4}, {2, 4, 0}, {2, 0, 4}, {-8, 2, 2}, {-4, 2, 0}, {4, 0, 2}, {2, 0, 0}}, {{aa, 4, 4}, {2*aa, 2, 4}, {2*(2*bb - aa), 4, 2}, {4*(2*bb -aa), 2, 2}, {aa, 4, 0}, {aa, 0, 4}, {2*aa, 2, 0}, {2*(2*bb - aa), 0, 2}, {aa, 0, 0}}}; hilbertn1 = contnet[poly, 8]; hilbertnet1 = InputForm[hilbertn1] hilbertnet = {{0, 0, 2/aa, aa}, {0, 0, 2/aa, aa}, {0, 0, 15/(7*(aa + (-aa + 2*bb)/14)), aa + (-aa + 2*bb)/14}, {0, 0, 17/(7*(aa + (3*(-aa + 2*bb))/14)), aa + (3*(-aa + 2*bb))/14}, {0, 0, 101/(35*((71*aa)/70 + (3*(-aa + 2*bb))/7)), (71*aa)/70 + (3*(-aa + 2*bb))/7}, {0, 0, 25/(7*((15*aa)/14 + (5*(-aa + 2*bb))/7)), (15*aa)/14 + (5*(-aa + 2*bb))/7}, {0, 0, 32/(7*((17*aa)/14 + (15*(-aa + 2*bb))/14)), (17*aa)/14 + (15*(-aa + 2*bb))/14}, {0, 0, 6/((3*aa)/2 + (3*(-aa + 2*bb))/2), (3*aa)/2 + (3*(-aa + 2*bb))/2}, {0, 0, 8/(2*aa + 2*(-aa + 2*bb)), 2*aa + 2*(-aa + 2*bb)}, {-1/(2*aa), 0, 2/aa, aa}, {-1/(2*aa), -1/(7*aa), 2/aa, aa}, {-1/(2*(aa + (-aa + 2*bb)/14)), -2/(7*(aa + (-aa + 2*bb)/14)), 15/(7*(aa + (-aa + 2*bb)/14)), aa + (-aa + 2*bb)/14}, {-1/(2*(aa + (3*(-aa + 2*bb))/14)), -16/(35*(aa + (3*(-aa + 2*bb))/14)), 17/(7*(aa + (3*(-aa + 2*bb))/14)), aa + (3*(-aa + 2*bb))/14}, {-17/(35*((71*aa)/70 + (3*(-aa + 2*bb))/7)), -24/(35*((71*aa)/70 + (3*(-aa + 2*bb))/7)), 101/(35*((71*aa)/70 + (3*(-aa + 2*bb))/7)), (71*aa)/70 + (3*(-aa + 2*bb))/7}, {-3/(7*((15*aa)/14 + (5*(-aa + 2*bb))/7)), -((15*aa)/14 + (5*(-aa + 2*bb))/7)^(-1), 25/(7*((15*aa)/14 + (5*(-aa + 2*bb))/7)), (15*aa)/14 + (5*(-aa + 2*bb))/7}, {-2/(7*((17*aa)/14 + (15*(-aa + 2*bb))/14)), -10/(7*((17*aa)/14 + (15*(-aa + 2*bb))/14)), 32/(7*((17*aa)/14 + (15*(-aa + 2*bb))/14)), (17*aa)/14 + (15*(-aa + 2*bb))/14}, {0, -2/((3*aa)/2 + (3*(-aa + 2*bb))/2), 6/((3*aa)/2 + (3*(-aa + 2*bb))/2), (3*aa)/2 + (3*(-aa + 2*bb))/2}, {-14/(15*aa), 0, 26/(15*aa), (15*aa)/14}, {-14/(15*aa), -4/(15*aa), 26/(15*aa), (15*aa)/14}, {-((15*aa)/14 + (17*(-aa + 2*bb))/210)^(-1), -4/(7*((15*aa)/14 + (17*(-aa + 2*bb))/210)), 208/(105*((15*aa)/14 + (17*(-aa + 2*bb))/210)), (15*aa)/14 + (17*(-aa + 2*bb))/210}, {-((15*aa)/14 + (17*(-aa + 2*bb))/70)^(-1), -32/(35*((15*aa)/14 + (17*(-aa + 2*bb))/70)), 78/(35*((15*aa)/14 + (17*(-aa + 2*bb))/70)), (15*aa)/14 + (17*(-aa + 2*bb))/70}, {-34/(35*((229*aa)/210 + (17*(-aa + 2*bb))/35)), -48/(35*((229*aa)/210 + (17*(-aa + 2*bb))/35)), 55/(21*((229*aa)/210 + (17*(-aa + 2*bb))/35)), (229*aa)/210 + (17*(-aa + 2*bb))/35}, {-6/(7*((7*aa)/6 + (17*(-aa + 2*bb))/21)), -2/((7*aa)/6 + (17*(-aa + 2*bb))/21), 67/(21*((7*aa)/6 + (17*(-aa + 2*bb))/21)), (7*aa)/6 + (17*(-aa + 2*bb))/21}, {-4/(7*((19*aa)/14 + (17*(-aa + 2*bb))/14)), -20/(7*((19*aa)/14 + (17*(-aa + 2*bb))/14)), 4/((19*aa)/14 + (17*(-aa + 2*bb))/14), (19*aa)/14 + (17*(-aa + 2*bb))/14}, {-20/(17*aa), 0, 22/(17*aa), (17*aa)/14}, {-20/(17*aa), -28/(85*aa), 22/(17*aa), (17*aa)/14}, {-10/(7*((17*aa)/14 + (-aa + 2*bb)/10)), -4/(5*((17*aa)/14 + (-aa + 2*bb)/10)), 58/(35*((17*aa)/14 + (-aa + 2*bb)/10)), (17*aa)/14 + (-aa + 2*bb)/10}, {-10/(7*((17*aa)/14 + (3*(-aa + 2*bb))/10)), -89/(70*((17*aa)/14 + (3*(-aa + 2*bb))/10)), 64/(35*((17*aa)/14 + (3*(-aa + 2*bb))/10)), (17*aa)/14 + (3*(-aa + 2*bb))/10}, {-7/(5*((87*aa)/70 + (3*(-aa + 2*bb))/5)), -66/(35*((87*aa)/70 + (3*(-aa + 2*bb))/5)), 73/(35*((87*aa)/70 + (3*(-aa + 2*bb))/5)), (87*aa)/70 + (3*(-aa + 2*bb))/5}, {-9/(7*((5*aa)/14 + 2*bb)), -19/(7*((5*aa)/14 + 2*bb)), 17/(7*((5*aa)/14 + 2*bb)), (5*aa)/14 + 2*bb}, {-120/(101*aa), 0, 82/(101*aa), (101*aa)/70}, {-120/(101*aa), -32/(101*aa), 82/(101*aa), (101*aa)/70}, {-12/(7*((101*aa)/70 + (2*(-aa + 2*bb))/15)), -32/(35*((101*aa)/70 + (2*(-aa + 2*bb))/15)), 127/(105*((101*aa)/70 + (2*(-aa + 2*bb))/15)), (101*aa)/70 + (2*(-aa + 2*bb))/15}, {-12/(7*((101*aa)/70 + (2*(-aa + 2*bb))/5)), -10/(7*((101*aa)/70 + (2*(-aa + 2*bb))/5)), 9/(7*((101*aa)/70 + (2*(-aa + 2*bb))/5)), (101*aa)/70 + (2*(-aa + 2*bb))/5}, {-12/(7*((3*aa)/2 + (4*(-aa + 2*bb))/5)), -72/(35*((3*aa)/2 + (4*(-aa + 2*bb))/5)), 7/(5*((3*aa)/2 + (4*(-aa + 2*bb))/5)), (3*aa)/2 + (4*(-aa + 2*bb))/5}, {-aa^(-1), 0, 2/(5*aa), (25*aa)/14}, {-aa^(-1), -6/(25*aa), 2/(5*aa), (25*aa)/14}, {-25/(14*((25*aa)/14 + (4*(-aa + 2*bb))/21)), -6/(7*((25*aa)/14 + (4*(-aa + 2*bb))/21)), 5/(7*((25*aa)/14 + (4*(-aa + 2*bb))/21)), (25*aa)/14 + (4*(-aa + 2*bb))/21}, {-25/(14*((25*aa)/14 + (4*(-aa + 2*bb))/7)), -9/(7*((25*aa)/14 + (4*(-aa + 2*bb))/7)), 5/(7*((25*aa)/14 + (4*(-aa + 2*bb))/7)), (25*aa)/14 + (4*(-aa + 2*bb))/7}, {-11/(16*aa), 0, 1/(8*aa), (16*aa)/7}, {-11/(16*aa), -1/(8*aa), 1/(8*aa), (16*aa)/7}, {-11/(7*((16*aa)/7 + (2*(-aa + 2*bb))/7)), -4/(7*((16*aa)/7 + (2*(-aa + 2*bb))/7)), 2/(7*((16*aa)/7 + (2*(-aa + 2*bb))/7)), (16*aa)/7 + (2*(-aa + 2*bb))/7}, {-1/(3*aa), 0, 0, 3*aa}, {-1/(3*aa), 0, 0, 3*aa}, {0, 0, 0, 4*aa}}; (* there was an error in the input polynomial, in z, the coeff of u^2v^2 was 4 instead of 8 *) hilbertfalse = {{0, 0, 2, aa}, {0, 0, 2, aa}, {0, 0, 15/7, aa + (-aa + 2*bb)/14}, {0, 0, 17/7, aa + (3*(-aa + 2*bb))/14}, {0, 0, 101/35, (71*aa)/70 + (3*(-aa + 2*bb))/7}, {0, 0, 25/7, (15*aa)/14 + (5*(-aa + 2*bb))/7}, {0, 0, 32/7, (17*aa)/14 + (15*(-aa + 2*bb))/14}, {0, 0, 6, (3*aa)/2 + (3*(-aa + 2*bb))/2}, {0, 0, 8, 2*aa + 2*(-aa + 2*bb)}, {-1/2, 0, 2, aa}, {-1/2, -1/7, 2, aa}, {-1/2, -2/7, 15/7, aa + (-aa + 2*bb)/14}, {-1/2, -16/35, 17/7, aa + (3*(-aa + 2*bb))/14}, {-17/35, -24/35, 101/35, (71*aa)/70 + (3*(-aa + 2*bb))/7}, {-3/7, -1, 25/7, (15*aa)/14 + (5*(-aa + 2*bb))/7}, {-2/7, -10/7, 32/7, (17*aa)/14 + (15*(-aa + 2*bb))/14}, {0, -2, 6, (3*aa)/2 + (3*(-aa + 2*bb))/2}, {-1, 0, 13/7, (15*aa)/14}, {-1, -2/7, 13/7, (15*aa)/14}, {-1, -4/7, 209/105, (15*aa)/14 + (17*(-aa + 2*bb))/210}, {-1, -32/35, 79/35, (15*aa)/14 + (17*(-aa + 2*bb))/70}, {-34/35, -48/35, 281/105, (229*aa)/210 + (17*(-aa + 2*bb))/35}, {-6/7, -2, 23/7, (7*aa)/6 + (17*(-aa + 2*bb))/21}, {-4/7, -20/7, 29/7, (19*aa)/14 + (17*(-aa + 2*bb))/14}, {-10/7, 0, 11/7, (17*aa)/14}, {-10/7, -2/5, 11/7, (17*aa)/14}, {-10/7, -4/5, 59/35, (17*aa)/14 + (-aa + 2*bb)/10}, {-10/7, -89/70, 67/35, (17*aa)/14 + (3*(-aa + 2*bb))/10}, {-7/5, -66/35, 79/35, (87*aa)/70 + (3*(-aa + 2*bb))/5}, {-9/7, -19/7, 19/7, (5*aa)/14 + 2*bb}, {-12/7, 0, 41/35, (101*aa)/70}, {-12/7, -16/35, 41/35, (101*aa)/70}, {-12/7, -32/35, 19/15, (101*aa)/70 + (2*(-aa + 2*bb))/15}, {-12/7, -10/7, 51/35, (101*aa)/70 + (2*(-aa + 2*bb))/5}, {-12/7, -72/35, 61/35, (3*aa)/2 + (4*(-aa + 2*bb))/5}, {-25/14, 0, 5/7, (25*aa)/14}, {-25/14, -3/7, 5/7, (25*aa)/14}, {-25/14, -6/7, 17/21, (25*aa)/14 + (4*(-aa + 2*bb))/21}, {-25/14, -9/7, 1, (25*aa)/14 + (4*(-aa + 2*bb))/7}, {-11/7, 0, 2/7, (16*aa)/7}, {-11/7, -2/7, 2/7, (16*aa)/7}, {-11/7, -4/7, 3/7, (16*aa)/7 + (2*(-aa + 2*bb))/7}, {-1, 0, 0, 3*aa}, {-1, 0, 0, 3*aa}, {0, 0, 0, 4*aa}}; (* Case aa = 1, bb = 1/2 *) poly = {{{-4, 3, 4}, {4, 1, 4}, {4, 3, 0}, {-4, 1, 0}}, {{8, 3, 3}, {8, 3, 1}, {-8, 1, 3}, {-8, 1, 1}}, {{2, 4, 4}, {4, 4, 2}, {-4, 2, 4}, {2, 4, 0}, {2, 0, 4}, {-4, 2, 2}, {-4, 2, 0}, {4, 0, 2}, {2, 0, 0}}, {{1, 4, 4}, {1, 4, 0}, {1, 0, 4}, {2, 2, 4}, {2, 2, 0}, {1, 0, 0}}}; (* Case aa = 1, bb = 1/2 *) hilnet = {{0, 0, 2, 1}, {0, 0, 2, 1}, {0, 0, 15/7, 1}, {0, 0, 17/7, 1}, {0, 0, 202/71, 71/70}, {0, 0, 10/3, 15/14}, {0, 0, 64/17, 17/14}, {0, 0, 4, 3/2}, {0, 0, 4, 2}, {-1/2, 0, 2, 1}, {-1/2, -1/7, 2, 1}, {-1/2, -2/7, 15/7, 1}, {-1/2, -16/35, 17/7, 1}, {-34/71, -48/71, 202/71, 71/70}, {-2/5, -14/15, 10/3, 15/14}, {-4/17, -20/17, 64/17, 17/14}, {0, -4/3, 4, 3/2}, {-14/15, 0, 26/15, 15/14}, {-14/15, -4/15, 26/15, 15/14}, {-14/15, -8/15, 416/225, 15/14}, {-14/15, -64/75, 52/25, 15/14}, {-204/229, -288/229, 550/229, 229/210}, {-36/49, -12/7, 134/49, 7/6}, {-8/19, -40/19, 56/19, 19/14}, {-20/17, 0, 22/17, 17/14}, {-20/17, -28/85, 22/17, 17/14}, {-20/17, -56/85, 116/85, 17/14}, {-20/17, -89/85, 128/85, 17/14}, {-98/87, -44/29, 146/87, 87/70}, {-18/19, -2, 34/19, 19/14}, {-120/101, 0, 82/101, 101/70}, {-120/101, -32/101, 82/101, 101/70}, {-120/101, -64/101, 254/303, 101/70}, {-120/101, -100/101, 90/101, 101/70}, {-8/7, -48/35, 14/15, 3/2}, {-1, 0, 2/5, 25/14}, {-1, -6/25, 2/5, 25/14}, {-1, -12/25, 2/5, 25/14}, {-1, -18/25, 2/5, 25/14}, {-11/16, 0, 1/8, 16/7}, {-11/16, -1/8, 1/8, 16/7}, {-11/16, -1/4, 1/8, 16/7}, {-1/3, 0, 0, 3}, {-1/3, 0, 0, 3}, {0, 0, 0, 4}}; falsehilnet = {{0, 0, 2, 1}, {0, 0, 2, 1}, {0, 0, 15/7, 1}, {0, 0, 17/7, 1}, {0, 0, 202/71, 71/70}, {0, 0, 10/3, 15/14}, {0, 0, 64/17, 17/14}, {0, 0, 4, 3/2}, {0, 0, 4, 2}, {-1/2, 0, 2, 1}, {-1/2, -1/7, 2, 1}, {-1/2, -2/7, 15/7, 1}, {-1/2, -16/35, 17/7, 1}, {-34/71, -48/71, 202/71, 71/70}, {-2/5, -14/15, 10/3, 15/14}, {-4/17, -20/17, 64/17, 17/14}, {0, -4/3, 4, 3/2}, {-14/15, 0, 26/15, 15/14}, {-14/15, -4/15, 26/15, 15/14}, {-14/15, -8/15, 418/225, 15/14}, {-14/15, -64/75, 158/75, 15/14}, {-204/229, -288/229, 562/229, 229/210}, {-36/49, -12/7, 138/49, 7/6}, {-8/19, -40/19, 58/19, 19/14}, {-20/17, 0, 22/17, 17/14}, {-20/17, -28/85, 22/17, 17/14}, {-20/17, -56/85, 118/85, 17/14}, {-20/17, -89/85, 134/85, 17/14}, {-98/87, -44/29, 158/87, 87/70}, {-18/19, -2, 2, 19/14}, {-120/101, 0, 82/101, 101/70}, {-120/101, -32/101, 82/101, 101/70}, {-120/101, -64/101, 266/303, 101/70}, {-120/101, -100/101, 102/101, 101/70}, {-8/7, -48/35, 122/105, 3/2}, {-1, 0, 2/5, 25/14}, {-1, -6/25, 2/5, 25/14}, {-1, -12/25, 34/75, 25/14}, {-1, -18/25, 14/25, 25/14}, {-11/16, 0, 1/8, 16/7}, {-11/16, -1/8, 1/8, 16/7}, {-11/16, -1/4, 3/16, 16/7}, {-1/3, 0, 0, 3}, {-1/3, 0, 0, 3}, {0, 0, 0, 4}}; render1[hilnet,8,2,0,0,-2.1,2.1,-2.1,2.1,0,4.1,0] sqrender[hilnet,8,1,1,-1,-1,2,1,1,-2.1,2.1,-2.1,2.1,0,4.1,0] (* light *) sqrender[hilnet,8,1,1,-1,-1,2,0,0,-2.1,2.1,-2.1,2.1,0,4.1,0] (* beautiful *) sqrender[hilnet,8,1,1,-1,-1,2,2,0,-2.1,2.1,-2.1,2.1,0,4.1,0] (* in/outside colors *) sqrender[hilnet,8,1,1,-1,-1,3,0,0,-2.1,2.1,-2.1,2.1,0,4.1,0] (* gorgeous *) lrender[hilnet,8,5,8,-2.1,2.1,-2.1,2.1,0,4.1,0] (* beautiful *) (* Monkey saddle (triangular net, degree 3 *) monknet = {{0, 0, 0, 1}, {0, 1/3, 0, 1}, {0, 2/3, 0, 1}, {0, 1, 0, 1}, {1/3, 0, 0, 1}, {1/3, 1/3, 0, 1}, {1/3, 2/3, -1, 1}, {2/3, 0, 0, 1}, {2/3, 1/3, 0, 1}, {1, 0, 1, 1}}; subdiv4[monknet,3,2,0,1.01,0,1.01,-1,1,0] subdiv4[monknet,3,2,0,1.01,0,1.01,-0.5,1,0] new2net[monknet,3,1,1,3,-1.01,1.01,-1.01,1.01,-2.01,2.01,0] fractal2[monknet,3,3,-1.01,1.01,-1.01,1.01,-2.01,2.01,-1] (* cute *) fractal2[monknet,3,4,-1.01,1.01,-1.01,1.01,-2.01,2.01,0] (* solid *) fractal2[monknet,3,3,-1.01,1.01,-1.01,1.01,-2.01,2.01,2] (* 2 sides *) fractal2[monknet,3,3,-1.01,1.01,-1.01,1.01,-2.01,2.01,4] (* 3 colors *) new2net[monknet,3,1.5,1.5,3,-1.51,1.51,-1.51,1.51,-6.8,6.8,0] new2net[monknet,3,1.5,1.5,3,-1.51,1.51,-1.51,1.51,-6.8,6.8,0] sqrender[monknet,3,1,1,-1,-1,3,0,0, -1.01,1.01,-1.01,1.01,-2.01,2.01, 0] (* very nice *) sqrender[monknet,3,1,1,-1,-1,3,2,0, -1.01,1.01,-1.01,1.01,-2.01,2.01, 0] (* 2 sides *) new2net[monknet,3,2,2,2,-2.01,2.01,-2.01,2.01,-16,16,0] render[monknet,3,3,0,0, -1.01,1.01,-1.01,1.01,-2.01,2.01, 0] subd2[net,3,3,-2,2,-2,3,-2,2,0] (* Monkey saddle (rectangular net, degree 3, 2 *) sqmonknet1 = {{0, 0, 0, 1}, {0, 1/2, 0, 1}, {0, 1, 0, 1}, {1/3, 0, 0, 1}, {1/3, 1/2, 0, 1}, {1/3, 1, -1, 1}, {2/3, 0, 0, 1}, {2/3, 1/2, 0, 1}, {2/3, 1, -2, 1}, {1, 0, 1, 1}, {1, 1/2, 1, 1}, {1, 1, -2, 1}} recrender[sqmonknet1,3,2,3,0,0,0,1.01,-1.01,1.01,-2.01,1.01, 0] (* good *) sqmonknet = newrecnet[sqmonknet1,3,2,-1,1,-1,1,0] sqmonknet = {{-1, -1, 2, 1}, {-1, 0, -4, 1}, {-1, 1, 2, 1}, {-1/3, -1, 2, 1}, {-1/3, 0, 0, 1}, {-1/3, 1, 2, 1}, {1/3, -1, -2, 1}, {1/3, 0, 0, 1}, {1/3, 1, -2, 1}, {1, -1, -2, 1}, {1, 0, 4, 1}, {1, 1, -2, 1}} recrender[sqmonknet,3,2,1,0,0,-1.01,1.01,-1.01,1.01,-2.01,2.01, 0] recrender[sqmonknet,3,2,2,0,0,-1.01,1.01,-1.01,1.01,-2.01,2.01, 0] recrender[sqmonknet,3,2,3,0,0,-1.01,1.01,-1.01,1.01,-2.01,2.01, 0] (* good *) (* Whitney's umbrella degree 2 *) net = {{0, 0, 0, 1}, {0, 0, 0, 1}, {0, 0, 1, 1}, {0, 1/2, 0, 1}, {1, 1/2, 0, 1}, {0, 1, 0, 1}}; subdiv4[net,2,3,0,1,0,1,-1,1,0] render1[net,2,3,0,0,0,1,0,1,-1,1,0] new2net[net,2,1,1,3,-2,2,-1,1,-1,1,0] sqrender[net,2,1,1,-1,-1,4,0,0,-2,2,-1,1,-1,1,0] sqrender[net,2,1,1,-1,-1,3,0,0,-2,2,-1,1,-1,1,0] (* great *) (* triangulation by line segments *) sqrender[net,2,1,1,-1,-1,3,0,0,-2,2,-1,1,-1,1,0] (* solid tiangulation *) sqrender[net,2,1,1,-1,-1,3,-1,0,-2,2,-1,1,-1,1,0] new2net[net,2,2,2,3,-8,8,-2,2,-4,4,0] (* Plucker's conoid, bad version *) plucknet = {{0, 0, 0, 0}, {0, 1/2, 0, 0}, {0, 1, 0, 1}, {1/2, 0, 0, 0}, {1/2, 1/2, 1, 0}, {1, 0, 0, 1}}; subdiv4[plucknet,2,3,0,4,0,6,0,4] subdiv4[plucknet,2,3,0,4,0,4,0,1] new2net[plucknet,2,2,2,3,-2,2,-2,2,-1.5,1.5] reftrigb = {{1, -1}, {1, 1}, {-1, -1}} recpluck = blowtrsq[plucknet, 2, 1, reftrigb, 0] pluck = testblow2[plucknet, 2, 1, 0] recrender[recpluck,1,2,3,0,0,-2,2,-2,2,-1.5,1.5, 0] sqrender[pluck,3,1, 1, -1, -1, 3,0,0,-3,3,-3,3,-3,3, 0] (* Plucker's conoid (degree 5) *) poly = {{{1, 0, 1}, {-1, 4, 1}}, {{2, 1, 1}, {2, 3, 1}}, {{4, 1, 0}, {-4, 3, 0}}, {{1, 4, 0}, {2, 2, 0}, {1, 0, 0}}}; net = contnet[poly, 5] plucknet = {{0, 0, 0, 1}, {1/5, 0, 0, 1}, {2/5, 0, 0, 1}, {3/5, 0, 0, 1}, {4/5, 0, 0, 1}, {1, 0, 0, 1}, {0, 0, 4/5, 1}, {1/5, 1/10, 4/5, 1}, {2/5, 1/5, 4/5, 1}, {3/5, 3/10, 4/5, 1}, {4/5, 2/5, 4/5, 1}, {0, 0, 4/3, 6/5}, {1/6, 1/6, 4/3, 6/5}, {1/3, 1/3, 4/3, 6/5}, {1/2, 1/2, 4/3, 6/5}, {0, 0, 5/4, 8/5}, {1/8, 1/4, 5/4, 8/5}, {1/4, 1/2, 5/4, 8/5}, {0, 0, 2/3, 12/5}, {0, 1/3, 2/3, 12/5}, {0, 0, 0, 4}}; render1[plucknet,5,3,0,0,-0.1,1.1,-0.1,1.1,-0.1,1.1,0] sqrender[plucknet,5,1,1,0,-1,3,0,1,-1.1,1.1,-1.1,1.1,-1.1,1.1,0] sqrender[plucknet,5,1,1,-1,-1,2,1,1,-1.1,1.1,-1.1,1.1,-1.1,1.1,0] (* light *) sqrender[plucknet,5,1,1,-1,-1,3,0,0,-1.1,1.1,-1.1,1.1,-1.1,1.1,0] (* very nice *) sqrender[plucknet,5,1,2,-1,-2,3,0,0,-2.1,2.1,-2.1,2.1,-1.1,1.1,0] (* great *) sqrender[plucknet,5,1,3/2,-1,-3/2,3,0,0,-2.1,2.1,-2.1,2.1,-1.1,1.1,0] render[plucknet,5,3,0,0,-4,4,-4,4,-1.1,1.1,0] (* Right conoid, degree 3 *) poly = {{{1, 0, 1}, {-1, 2, 1}}, {{2, 1, 1}}, {{4, 1, 0}}, {{1, 2, 0}, {1, 0, 0}}}; net = contnet[poly, 3] rconet0 = {{0, 0, 0, 1}, {1/3, 0, 0, 1}, {2/3, 0, 0, 1}, {1, 0, 0, 1}, {0, 0, 4/3, 1}, {1/3, 1/3, 4/3, 1}, {2/3, 2/3, 4/3, 1}, {0, 0, 2, 4/3}, {0, 1/2, 2, 4/3}, {0, 0, 2, 2}}; (* for -1 <= u <= 1, we only get half of the conoid *) render1[rconet0,3,3,1,0,-2.1,2.1,-2.1,2.1,-2.1,2.1,0] (* light *) render1[rconet0,3,3,0,0,-2.1,2.1,-2.1,2.1,-2.1,2.1,0] trender[rconet0,3,reftrig1,3,0,0,-2.1,2.1,-2.1,2.1,-2.1,2.1,0] trender[rconet0,3,reftrig2,3,0,0,-2.1,2.1,-2.1,2.1,-2.1,2.1,0] sqrender[rconet0,3,1,1,0,-1,3,0,0,-2.1,2.1,-2.1,2.1,-2.1,2.1,0] sqrender[rconet0,3,1,1,-1,-1,3,0,0,-2.1,2.1,-2.1,2.1,-2.1,2.1,0] sqrender[rconet0,3,1,3/2,-1,-3/2,3,0,0,-2.2,2.2,-2.2,2.2,-2.2,2.2,0] (* nice *) sqrender[rconet0,3,1,2,-1,-2,3,0,0,-2.2,2.2,-2.2,2.2,-2.1,2.1,0] (* very pretty *) sqrender[rconet0,3,1,2,-1,-2,3,0,1,-2.6,2.6,-2.6,2.6,-1.1,1.1,0] (* lite *) sqrender[rconet0,3,1,3,-1,-3,3,0,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0] (* pretty *) (* Right conoid, degree 5 *) poly = {{{1, 4, 1}, {-6, 2, 1}, {1, 0, 1}}, {{4, 1, 1}, {-4, 3, 1}}, {{8, 1, 0}, {-8, 3, 0}}, {{1, 4, 0}, {2, 2, 0}, {1, 0, 0}}}; net = contnet[poly, 5]; rconet = InputForm[net]; rconet = {{0, 0, 0, 1}, {1/5, 0, 0, 1}, {2/5, 0, 0, 1}, {3/5, 0, 0, 1}, {4/5, 0, 0, 1}, {1, 0, 0, 1}, {0, 0, 8/5, 1}, {1/5, 1/5, 8/5, 1}, {2/5, 2/5, 8/5, 1}, {3/5, 3/5, 8/5, 1}, {4/5, 4/5, 8/5, 1}, {0, 0, 8/3, 6/5}, {0, 1/3, 8/3, 6/5}, {0, 2/3, 8/3, 6/5}, {0, 1, 8/3, 6/5}, {0, 0, 5/2, 8/5}, {-1/4, 1/4, 5/2, 8/5}, {-1/2, 1/2, 5/2, 8/5}, {0, 0, 4/3, 12/5}, {-1/3, 0, 4/3, 12/5}, {0, 0, 0, 4}}; render1[rconet,5,3,0,0,-2.2,2.2,-2.2,2.2,-2.2,2.2,0] trender[rconet,5,reftrig1,3,0,0,-2.2,2.2,-2.2,2.2,-2.2,2.2,0] trender[rconet,5,reftrig2,3,0,0,-2.2,2.2,-2.2,2.2,-2.2,2.2,0] sqrender[rconet,5,1,3/2,-1,-3/2,3,0,0,-2.2,2.2,-2.2,2.2,-2.2,2.2,0] (* very nice *) sqrender[rconet,5,1,1,-1,-1,3,0,0,-1.2,1.2,-1.2,1.2,-2.2,2.2,0] sqrender[rconet,5,1,2,-1,-2,3,0,0,-2.2,2.2,-2.2,2.2,-2.2,2.2,0] (* nice *) (* Show[%295, ViewPoint -> {8, 1, 8}] *) (* Show[%7, ViewPoint -> {8, 0.2, 4}] *) (* Other conoid, from Cagnac, Vol III, page 425-426 *) x = v cos theta y = v sin theta z = aa cos theta - bb v x = u^4v - 6u^2v + v y = 4uv - 4u^3v z = aa u^4 - 6aa u^2 + aa - bb u^4v - 2bb u^2v - bb v w = u^4 + 2u^2 + 1 poly = {{{1, 4, 1}, {-6, 2, 1}, {1, 0, 1}}, {{4, 1, 1}, {-4, 3, 1}}, {{aa, 4, 0}, {-6aa, 2, 0}, {aa, 0, 0}, {-bb, 4 ,1}, {-2bb, 2, 1}, {-bb, 0, 1}}, {{1, 4, 0}, {2, 2, 0}, {1, 0, 0}}}; netl = contnet[poly, 4]; netlim = InputForm[netl]; netlim = {{0, 0, aa, 1}, {1/4, 0, aa - bb/4, 1}, {1/2, 0, aa - bb/2, 1}, {3/4, 0, aa - (3*bb)/4, 1}, {1, 0, aa - bb, 1}, {0, 0, aa, 1}, {1/4, 1/3, aa - bb/4, 1}, {1/2, 2/3, aa - bb/2, 1}, {3/4, 1, aa - (3*bb)/4, 1}, {0, 0, 0, 4/3}, {-3/16, 1/2, (-5*bb)/16, 4/3}, {-3/8, 1, (-5*bb)/8, 4/3}, {0, 0, -aa, 2}, {-5/8, 0, (-2*aa - (3*bb)/4)/2, 2}, {0, 0, -aa, 4}}; (* Case a = 1, b = 1 *) netlim = {{0, 0, 1, 1}, {1/4, 0, 3/4, 1}, {1/2, 0, 1/2, 1}, {3/4, 0, 1/4, 1}, {1, 0, 0, 1}, {0, 0, 1, 1}, {1/4, 1/3, 3/4, 1}, {1/2, 2/3, 1/2, 1}, {3/4, 1, 1/4, 1}, {0, 0, 0, 4/3}, {-3/16, 1/2, -5/16, 4/3}, {-3/8, 1, -5/8, 4/3}, {0, 0, -1, 2}, {-5/8, 0, -11/8, 2}, {0, 0, -1, 4}}; rendersq[netlim,4,-1,1,-2,2,3,0,0,-2.3,2.3,-2.3,2.3,-3.1,3.1,0] (* Torus , degree 4 *) poly = {{{-aa, 2, 2}, {2 * bb, 2, 1}, {-aa, 2, 0}, {aa, 0, 2}, {-2 * bb, 0, 1}, {aa, 0, 0}}, {{2 * aa, 1, 2}, {-4 * bb, 1, 1}, {2 * aa, 1, 0}}, {{-cc, 2, 2}, {-cc, 0, 2}, {cc, 2, 0}, {cc, 0, 0}}, {{1, 2, 2}, {1, 2, 0}, {1, 0, 2}, {1, 0, 0}}}; net = contnet[poly, 4]; tornets = InputForm[net]; (* Master net *) tornets = {{aa, 0, cc, 1}, {aa - bb/2, 0, cc, 1}, {(6*((7*aa)/6 - bb))/7, 0, (5*cc)/7, 7/6}, {(2*((3*aa)/2 - (3*bb)/2))/3, 0, cc/3, 3/2}, {(2*aa - 2*bb)/2, 0, 0, 2}, {aa, aa/2, cc, 1}, {aa - bb/2, aa/2 - bb/3, cc, 1}, {(6*((7*aa)/6 - bb))/7, (6*((2*aa)/3 - (2*bb)/3))/7, (5*cc)/7, 7/6}, {(2*((3*aa)/2 - (3*bb)/2))/3, (2*(aa - bb))/3, cc/3, 3/2}, {(5*aa)/7, (6*aa)/7, cc, 7/6}, {(6*((5*aa)/6 - bb/3))/7, (6*(aa - (2*bb)/3))/7, cc, 7/6}, {(2*((5*aa)/6 - (2*bb)/3))/3, (2*((4*aa)/3 - (4*bb)/3))/3, (5*cc)/9, 3/2}, {aa/3, aa, cc, 3/2}, {aa/3, (2*((3*aa)/2 - bb))/3, cc, 3/2}, {0, aa, cc, 2}}; (* aa = 2, bb = 1 cc = 1 *) tornet = {{2, 0, 1, 1}, {3/2, 0, 1, 1}, {8/7, 0, 5/7, 7/6}, {1, 0, 1/3, 3/2}, {1, 0, 0, 2}, {2, 1, 1, 1}, {3/2, 2/3, 1, 1}, {8/7, 4/7, 5/7, 7/6}, {1, 2/3, 1/3, 3/2}, {10/7, 12/7, 1, 7/6}, {8/7, 8/7, 1, 7/6}, {2/3, 8/9, 5/9, 3/2}, {2/3, 2, 1, 3/2}, {2/3, 4/3, 1, 3/2}, {0, 2, 1, 2}}; (* testing the net123 program blowup, and conversion programs *) znets = net123[tornet, 4, 0] zog = testnet[tornet, 4, 0] reftrig = {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}; bonet = maptohat[tornet]; zig1 = newcnet3old[bonet, 4, reftrig1]; zig2 = newcnet3[bonet, 4, reftrig1]; dnet = {{0, 0, 0, 0}, {3, 0, 4, 1}, {6, 0, 0, 1}, {2, 3, 4, 1}, {5, 3, 4, 1}, {3, 6, 0, 1}}; trender[dnet, 2, reftrig, 2, 0, 0, 0,6,0,6,0,4, 0] zut = testblow2[dnet, 2, 1, 0] ddnet = {{2, 3, 4, 1}, {7/3, 2, 4, 1}, {8/3, 1, 4, 1}, {3, 0, 4, 1}, {7/3, 4, 8/3, 1}, {19/6, 5/2, 4, 1}, {11/3, 1, 8/3, 1}, {8/3, 5, 4/3, 1}, {4, 3, 4, 1}, {3, 6, 0, 1}} ddnetz = {{6, 0, 0, 1}, {5, 0, 4/3, 1}, {4, 0, 8/3, 1}, {3, 0, 4, 1}, {16/3, 2, 8/3, 1}, {9/2, 3/2, 8/3, 1}, {11/3, 1, 8/3, 1}, {13/3, 4, 8/3, 1}, {4, 3, 4, 1}, {3, 6, 0, 1}} trender[ddnet, 3, reftrig, 2, 0, 0, 0,6,0,6,0,4, 0] sqrender[ddnet, 3, 1, 1, 0, 0, 2, 0, 0, 0,6,0,6,0,4, 0] sqrender[ddnetz, 3, 1, 1, 0, 0, 2, 0, 0, 0,6,0,6,0,4, 0] u = {1/2} v = {1/2, 1/2} popo = sqpoldecas[dnet, 1, 2, 0, 1, 0, 1, u, v, 2] net = {{0, 0, 0, 1}, {3, 0, 4, 1}, {6, 0, 0, 1}, {2, 3, 4, 1}, {5, 3, 4, 1}, {3, 6, 0, 1}}; blo = tconvert[net, 2, 2] reftrig = {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}; zog = tconvert[net, 2, 2] gnet1 = {{0, 0, 0, 1}, {3/2, 0, 2, 1}, {3, 0, 8/3, 1}, {9/2, 0, 2, 1}, {6, 0, 0, 1}, {1, 3/2, 2, 1}, {5/2, 3/2, 10/3, 1}, {4, 3/2, 10/3, 1}, {11/2, 3/2, 2, 1}, {11/6, 3, 8/3, 1}, {10/3, 3, 10/3, 1}, {29/6, 3, 8/3, 1}, {5/2, 9/2, 2, 1}, {4, 9/2, 2, 1}, {3, 6, 0, 1}} trender[gnet1, 4, reftrig, 2, 0, 0, 0,6,0,6,0,4, 0] sqrender[gnet1, 4, 1, 1, 0, 0, 2, 0, 0, 0,6,0,6,0,4, 0] net = {{0, 0, -2, 1}, {0, 3, -2, 1}, {0, 3, 0, 2}, {4, 0, -2, 1}, {4, 3, -2, 1}, {4, 0, 0, 2}}; subdiv4[net,2,3,0,4,0,3,-2,0,0] subd2[net,2,3,-4,4,-3,3,-2,2,0] zog = tconvert[net, 2, 0] gnet2 = {{0, 0, -2, 1}, {0, 3/2, -2, 1}, {0, 18/7, -10/7, 7/6}, {0, 3, -2/3, 3/2}, {0, 3, 0, 2}, {2, 0, -2, 1}, {2, 3/2, -2, 1}, {12/7, 18/7, -10/7, 7/6}, {4/3, 3, -2/3, 3/2}, {24/7, 0, -10/7, 7/6}, {24/7, 9/7, -10/7, 7/6}, {3, 9/4, -1, 4/3}, {4, 0, -2/3, 3/2}, {4, 1, -2/3, 3/2}, {4, 0, 0, 2}} trender[gnet2, 4, reftrig, 2, 0, 0, 0,4,0,3,-2,0,0] sqrender[gnet2, 4, 1, 1, -1, -1, 2, 0, 0, -4,4,-3,3,-2,2,0] net = {{0, 0, 0, 1}, {2, 0, 2, 1}, {4, 0, 2, 1}, {6, 0, 0, 1}, {1, 2, 2, 1}, {3, 2, 5, 1}, {5, 2, 2, 1}, {2, 4, 2, 1}, {4, 4, 2, 1}, {3, 6, 0, 1}}; netdd = {{0, 0, 0, 0}, {0, 0, 0, 0}, {4, 0, 2, 1}, {6, 0, 0, 1}, {0, 0, 0, 0}, {3, 2, 5, 1}, {5, 2, 2, 1}, {2, 4, 2, 1}, {4, 4, 2, 1}, {3, 6, 0, 1}}; netdd2 = testblow2[netdd, 3, 2, 0] netddd = {{6, 0, 0, 1}, {11/2, 0, 1/2, 1}, {5, 0, 1, 1}, {9/2, 0, 3/2, 1}, {4, 0, 2, 1}, {21/4, 3/2, 3/2, 1}, {29/6, 4/3, 2, 1}, {53/12, 7/6, 5/2, 1}, {4, 1, 3, 1}, {9/2, 3, 2, 1}, {25/6, 8/3, 5/2, 1}, {23/6, 7/3, 3, 1}, {15/4, 9/2, 3/2, 1}, {7/2, 4, 2, 1}, {3, 6, 0, 1}} netdd2 = {{2, 4, 2, 1}, {5/2, 3, 7/2, 1}, {3, 2, 4, 1}, {7/2, 1, 7/2, 1}, {4, 0, 2, 1}, {9/4, 9/2, 3/2, 1}, {17/6, 10/3, 3, 1}, {41/12, 13/6, 7/2, 1}, {4, 1, 3, 1}, {5/2, 5, 1, 1}, {19/6, 11/3, 5/2, 1}, {23/6, 7/3, 3, 1}, {11/4, 11/2, 1/2, 1}, {7/2, 4, 2, 1}, {3, 6, 0, 1}} sqrender[netdd2, 4, 1, 1, 0, 0, 2, 0, 0, -0.2,6.1,-0.2,6.1,-0.2,6.1,0] trender[netdd2, 4, reftrig, 2, 0, 0,-0.2,6.1,-0.2,6.1,-0.2,6.1,0] sqrender[netddd, 4, 1, 1, 0, 0, 2, 0, 0, -0.2,6.1,-0.2,6.1,-0.2,6.1,0] trender[netddd, 4, reftrig, 2, 0, 0,0,6,0,6,0,3.1,0] (* trying consistent rows *) netf = {{0, 0, 0, 0}, {0, 0, 0, 0}, {4, 0, 2, 1}, {6, 0, 0, 1}, {0, 0, 0, 0}, {4, 0, 2, 1}, {5, 2, 2, 1}, {4, 0, 2, 1}, {4, 4, 2, 1}, {3, 6, 0, 1}}; netf2 = {{0, 0, 0, 0}, {0, 0, 0, 0}, {4, 0, 5, 1}, {6, 0, 0, 1}, {0, 0, 0, 0}, {4, 0, 5, 1}, {5, 2, 2, 1}, {4, 0, 5, 1}, {4, 4, 2, 1}, {3, 6, 0, 1}}; netf6 = {{6, 0, 0, 1}, {5, 2, 2, 1}, {4, 4, 2, 1}, {3, 6, 0, 1}, {4, 0, 5, 1}, {4, 0, 5, 1}, {4, 0, 5, 1}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}} netf4 = testblow2[netf2, 3, 2, 0] netf3 = {{6, 0, 0, 1}, {11/2, 0, 1/2, 1}, {5, 0, 1, 1}, {9/2, 0, 3/2, 1}, {4, 0, 2, 1}, {21/4, 3/2, 3/2, 1}, {5, 1, 3/2, 1}, {19/4, 1/2, 3/2, 1}, {9/2, 0, 3/2, 1}, {9/2, 3, 2, 1}, {9/2, 2, 2, 1}, {9/2, 1, 2, 1}, {15/4, 9/2, 3/2, 1}, {4, 3, 2, 1}, {3, 6, 0, 1}} netf5 = {{4, 0, 5, 1}, {4, 0, 5, 1}, {4, 0, 5, 1}, {4, 0, 5, 1}, {4, 0, 5, 1}, {15/4, 3/2, 15/4, 1}, {4, 1, 17/4, 1}, {17/4, 1/2, 17/4, 1}, {9/2, 0, 15/4, 1}, {7/2, 3, 5/2, 1}, {4, 2, 7/2, 1}, {9/2, 1, 7/2, 1}, {13/4, 9/2, 5/4, 1}, {4, 3, 11/4, 1}, {3, 6, 0, 1}} sqrender[netf3, 4, 1, 1, 0, 0, 2, 0, 0, 0,6,0,6,0,3.1,0] sqrender[netf6, 3, 1, 1, 0, 0, 2, 0, 0, 0,6,0,6,0,5,0] sqrender[netf5, 4, 1, 1, 0, 0, 2, 0, 0, 0,6,0,6,0,5,0] trender[netf5, 4, reftrig, 2, 0, 0,0,6,0,6.1,0,5,0] netfz = {{4, 0, 5, 1}, {4, 0, 5, 1}, {4, 0, 5, 1}, {6, 0, 0, 1}, {4, 0, 5, 1}, {4, 0, 5, 1}, {5, 2, 2, 1}, {4, 0, 5, 1}, {4, 4, 2, 1}, {3, 6, 0, 1}}; trender[netfz, 3, reftrig, 2, 0, 0,0,6,0,6.1,0,5,0] (* testing fzerocorner *) netg = {{0, 0, 0, 1}, {2, 0, 2, 1}, {4, 0, 2, 1}, {6, 0, 0, 1}, {1, 2, 2, 1}, {1, 2, 2, 1}, {1, 2, 2, 1}, {0, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}}; pair = {-1, 1, {1, 2, 2, 1}} pair = {-1} tutu = fzerocorner[netg, pair, 3, 0] (* more tests on the torus *) zut = testblow[tornet, 4, 2, 0] zut3 = {{{0, 1, 0, 4}, {0, 1, -1/9, 3}, {0, 98/97, -25/97, 97/45}, {0, 23/22, -5/11, 22/15}, {0, 8/7, -5/7, 14/15}, {0, 7/5, -1, 5/9}, {0, 2, -1, 1/3}, {-1, 1, -1, 4/3}, {-14/19, 21/19, -15/19, 19/15}, {-56/103, 122/103, -67/103, 103/90}, {-12/29, 36/29, -17/29, 29/30}, {-4/11, 14/11, -7/11, 11/15}, {-1/2, 5/4, -1, 4/9}, {-4/3, 1, -4/3, 4/5}, {-17/10, 11/10, -7/5, 2/3}, {-142/79, 96/79, -103/79, 79/135}, {-37/25, 33/25, -1, 5/9}, {-10/13, 18/13, -7/13, 26/45}, {-2, 1, -1, 4/5}, {-19/10, 7/10, -1, 2/3}, {-104/43, 22/43, -55/43, 43/90}, {-38/7, 4/7, -19/7, 7/30}, {-4, -1/3, -4/3, 4/5}, {-30/11, 1/11, -1, 11/15}, {-36/37, -2/37, -13/37, 37/45}, {-3, -3, -1, 4/3}, {-12, -3, -5, 1/3}, {0, -3, 0, 4}}, {{0, 3, 0, 4}, {0, 3, -1/9, 3}, {0, 290/97, -25/97, 97/45}, {0, 65/22, -5/11, 22/15}, {0, 20/7, -5/7, 14/15}, {0, 13/5, -1, 5/9}, {0, 2, -1, 1/3}, {3/2, 3, 0, 8/3}, {42/31, 3, -3/31, 31/15}, {164/139, 416/139, -31/139, 139/90}, {32/33, 98/33, -13/33, 11/10}, {8/11, 32/11, -7/11, 11/15}, {1/2, 11/4, -1, 4/9}, {24/7, 15/7, 0, 28/15}, {69/22, 51/22, -1/11, 22/15}, {418/151, 378/151, -31/151, 151/135}, {85/37, 99/37, -13/37, 37/45}, {22/13, 36/13, -7/13, 26/45}, {9/2, 0, 0, 8/5}, {9/2, 1/2, -1/9, 6/5}, {344/79, 88/79, -19/79, 79/90}, {74/19, 34/19, -7/19, 19/30}, {24/7, -15/7, 0, 28/15}, {78/19, -33/19, -3/19, 19/15}, {180/37, -38/37, -13/37, 37/45}, {3/2, -3, 0, 8/3}, {12/5, -3, -1/5, 5/3}, {0, -3, 0, 4}}} zut3 = {{{0, -2, -1, 1/3}, {-1, -2, -1, 2/9}, {-16/7, -10/7, -1, 7/45}, {-3, 0, -1, 2/15}, {-16/7, 10/7, -1, 7/45}, {-1, 2, -1, 2/9}, {0, 2, -1, 1/3}, {0, -13/5, -1, 5/9}, {-3/2, -35/16, -1, 16/45}, {-31/11, -1, -1, 11/45}, {-27/10, 7/10, -1, 2/9}, {-18/13, 19/13, -1, 13/45}, {-1/2, 5/4, -1, 4/9}, {0, -20/7, -5/7, 14/15}, {-32/13, -30/13, -17/13, 13/30}, {-59/26, -15/26, -10/13, 52/135}, {-80/23, 18/23, -35/23, 23/90}, {-10/13, 18/13, -7/13, 26/45}, {0, -65/22, -5/11, 22/15}, {-4, -100/41, -77/41, 41/90}, {-41/26, -7/26, -7/13, 26/45}, {-38/7, 4/7, -19/7, 7/30}, {0, -290/97, -25/97, 97/45}, {-126/19, -50/19, -55/19, 19/45}, {-36/37, -2/37, -13/37, 37/45}, {0, -3, -1/9, 3}, {-12, -3, -5, 1/3}, {0, -3, 0, 4}}, {{0, -2, -1, 1/3}, {1, -2, -1, 2/9}, {16/7, -10/7, -1, 7/45}, {3, 0, -1, 2/15}, {16/7, 10/7, -1, 7/45}, {1, 2, -1, 2/9}, {0, 2, -1, 1/3}, {0, -13/5, -1, 5/9}, {3/2, -41/16, -1, 16/45}, {37/11, -17/11, -1, 11/45}, {39/10, 7/10, -1, 2/9}, {30/13, 31/13, -1, 13/45}, {1/2, 11/4, -1, 4/9}, {0, -20/7, -5/7, 14/15}, {32/17, -48/17, -13/17, 17/30}, {107/26, -18/13, -10/13, 52/135}, {4, 48/35, -23/35, 7/18}, {22/13, 36/13, -7/13, 26/45}, {0, -65/22, -5/11, 22/15}, {164/77, -226/77, -41/77, 77/90}, {119/26, -31/26, -7/13, 26/45}, {74/19, 34/19, -7/19, 19/30}, {0, -290/97, -25/97, 97/45}, {126/55, -164/55, -19/55, 11/9}, {180/37, -38/37, -13/37, 37/45}, {0, -3, -1/9, 3}, {12/5, -3, -1/5, 5/3}, {0, -3, 0, 4}}} sqrender[zut3[[1]], 6, 1, 1, 0, 0, 2, 0, 0, -3.1,3.1,-3.1,3.1,-2.1,2.1,0] sqrender[zut3[[2]], 6, 1, 1, 0, 0, 2, 0, 0, -3.1,3.1,-3.1,3.1,-2.1,2.1,0] bzut = testblow3[tornet, 4, 2, 0] bzutt = {{{0, -2, -1, 1/3}, {-2, -2, -1, 1/6}, {-4, 0, -1, 1/9}, {-2, 2, -1, 1/6}, {0, 2, -1, 1/3}, {0, -3, -1, 1}, {-3, -2, -1, 1/2}, {-4, 1, -1, 1/3}, {-1, 2, -1, 1/2}, {0, 1, -1, 1}, {0, -3, 0, 4}, {-6, 0, -2, 0}, {0, 1, 0, 4/3}, {-2, 0, -2, 0}, {0, 1, 0, 4}}, {{0, -2, -1, 1/3}, {2, -2, -1, 1/6}, {4, 0, -1, 1/9}, {2, 2, -1, 1/6}, {0, 2, -1, 1/3}, {0, -3, -1, 1}, {3, -3, -1, 1/2}, {6, 0, -1, 1/3}, {3, 3, -1, 1/2}, {0, 3, -1, 1}, {0, -3, 0, 4}, {3, -3, 0, 2}, {6, 0, 0, 4/3}, {3, 3, 0, 2}, {0, 3, 0, 4}}} bzut1 = {{0, -2, -1, 1/3}, {-2, -2, -1, 1/6}, {-4, 0, -1, 1/9}, {-2, 2, -1, 1/6}, {0, 2, -1, 1/3}, {0, -3, -1, 1}, {-3, -2, -1, 1/2}, {-4, 1, -1, 1/3}, {-1, 2, -1, 1/2}, {0, 1, -1, 1}, {0, -3, 0, 4}, {-6, 0, -2, 0}, {0, 1, 0, 4/3}, {-2, 0, -2, 0}, {0, 1, 0, 4}} bzut2 = {{0, -2, -1, 1/3}, {2, -2, -1, 1/6}, {4, 0, -1, 1/9}, {2, 2, -1, 1/6}, {0, 2, -1, 1/3}, {0, -3, -1, 1}, {3, -3, -1, 1/2}, {6, 0, -1, 1/3}, {3, 3, -1, 1/2}, {0, 3, -1, 1}, {0, -3, 0, 4}, {3, -3, 0, 2}, {6, 0, 0, 4/3}, {3, 3, 0, 2}, {0, 3, 0, 4}} recrender[bzut1,2,4,2,0,1, -3.1,3.1,-3.1,3.1,-2.1,2.1,0] recrender[bzut1,2,4,3,0,0, -3.1,3.1,-3.1,3.1,-2.1,2.1,0] recrender[bzut1,2,4,1,0,0, -3.1,3.1,-3.1,3.1,-2.1,2.1,2] recrender[bzut2,2,4,3,0,0, -3.1,3.1,-3.1,3.1,-2.1,2.1,0] sqrender[tornet,4,1,1,-1,-1,3,0,0,-0.01,3.01,-3.01,3.01,-0.01,1.01,0] (* nice *) sqrender[tornet,4,1,1,-1,-1,3,1,0,-0.1,3.1,-3.1,3.1,-0.1,1.1,0] (* lite *) render1[tornet,4,3,0,0,0,2.01,0,2.01,0,1.01,0] render[tornet,4,3,0,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0] (* good) render[tornet,4,3,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0] (* lite *) render[tornet,4,1,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0] (* lite *) render3[tornet,4,3,0,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0] render3[tornet,4,2,0,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0] render3[tornet,4,1,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,-1] (* lite *) render3[tornet,4,2,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,-1] (* lite *) render3[tornet,4,2,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,-2] (* lite *) render3[tornet,4,3,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,1] (* lite *) render3[tornet,4,4,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,-2] (* lite *) render3[tornet,4,3,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,-1] (* lite *) render3[tornet,4,3,0,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0] render3[tornet,4,4,0,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,-1] render3[tornet,4,1,1,0,-3.5,3.5,-3.5,3.5,-2.5,2.5,1] (* lite *) render4[tornet,4,1,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,-1] (* lite *) render4[tornet,4,2,0,0,-3.5,3.5,-3.5,3.5,-2.5,2.5,0] render5[tornet,4,3,0,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0] (* lite *) render5[tornet,4,2,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,1] (* lite *) render5[tornet,4,2,0,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0] render6[tornet,4,2,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,-1] (* lite *) render6[tornet,4,3,0,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0] render34[tornet,4,3,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0] (************) render3[tornet,4,3,0,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0] (* good *) render4[tornet,4,3,0,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0] (* good *) render34[tornet,4,3,0,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0] (* good *) render56[tornet,4,3,0,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0] (* good *) render3to6[tornet,4,3,0,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0] (* good *) (***********) render34[tornet,4,4,0,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0] render34[tornet,4,2,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0] (* lite *) render34[tornet,4,2,0,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0] render56[tornet,4,2,0,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0] render56[tornet,4,2,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,1] (* lite *) render3to6[tornet,4,2,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0] (* lite *) srender3[tornet,4,3,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0,0.5,0.5] (* lite *) srender3[tornet,4,3,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0,0.4,0.4] (* lite *) srender3[tornet,4,3,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0,0.05,0.01] (* lite *) srender3[tornet,4,3,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,1,0.01,0.01] (* lite *) srender3[tornet,4,3,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0,0,0] (* lite *) srender3[tornet,4,1,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0,0,0] (* lite *) srender3[tornet,4,3,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0,0.02,0.02] (* lite *) srender3[tornet,4,3,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0,0.005,0.005] (* lite *) srender3[tornet,4,3,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0,0.0001,0.0001] (* better *) srender3[tornet,4,3,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0,0.00001,0.00001] (* better *) srender3[tornet,4,4,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0,0.00001,0.00001] (* better *) srender3[tornet,4,3,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0,0.000008,0.000008] (* better *) srender3[tornet,4,3,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0,0.000008,0.000008] (* nice *) srender3[tornet,4,1,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0,0.000008,0.000008] srender3[tornet,4,4,0,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0,0.000008,0.000008] (* nice *) srender3[tornet,4,3,0,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0,0.000008,0.000008] (* nice *) srender34[tornet,4,3,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0,0.000008,0.000008] srender34[tornet,4,3,0,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0,0.000008,0.000008] srender34[tornet,4,3,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0,0.001,0.001] srender34[tornet,4,3,0,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0,0.0001,0.0001] srender34[tornet,4,3,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0,0.01,0.01] srender34[tornet,4,3,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0,0.05,0.05] srender34[tornet,4,3,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0,0,0] srender56[tornet,4,3,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0,0,0] srender56[tornet,4,3,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0,0.1,0.1] srender3to6[tornet,4,3,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0,0.000001,0.000001] srender3to6[tornet,4,3,1,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0,0.0001,0.0001] srender3to6[tornet,4,3,0,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0,0,0] srender3to6[tornet,4,3,0,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0,0.1,0.1] lrender[tornet,4,5,2,-3.1,3.1,-3.1,3.1,-2.1,2.1,0] lfrender[tornet,4,5,1,-3.1,3.1,-3.1,3.1,-2.1,2.1,0] (* interesting *) lfrender[tornet,4,5,4,-3.1,3.1,-3.1,3.1,-2.1,2.1,0] runblow[tornet, 4, 6, 1, 1, 0, -3.1,3.1,-3.1,3.1,-2.1,2.1,-2] runblow[tornet, 4, 6, 2, 0, 0, -3.1,3.1,-3.1,3.1,-2.1,2.1,-2] runblow[tornet, 4, 6, 2, 1, 0, -3.1,3.1,-3.1,3.1,-2.1,2.1,-2] runblow3[tornet, 4, 6, 2, 0, 0, -3.1,3.1,-3.1,3.1,-2.1,2.1,0] reftrig = {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}; trender[fixnet,6,reftrig,2,0,0,-3.1,3.1,-3.1,3.1,-2.1,2.1,0] aa = 2; bb = 1; cc = 1; tor2x[u_, v_] = (aa*(u^2 + 1) - 2*bb*u)*(u^2 - v^2)/((u^2 + v^2)*(u^2 + 1)); tor2y[u_, v_] = 2*u*v*(aa*(u^2 + 1) - 2*bb*u)/((u^2 + v^2)*(u^2 + 1)); tor2z[u_, v_] = cc*(u^2 - 1)/(u^2 + 1); tor2 := ParametricPlot3D[{tor2x[u, v], tor2y[u, v], tor2z[u, v]}, {u, -1, 1}, {v, -1, 1}, DisplayFunction -> Identity]; viewtor2[lx_, mx_, ly_, my_, lz_, mz_] := Show[tor2, PlotRange -> {{lx, mx}, {ly, my}, {lz, mz}}, Axes -> True, AxesLabel -> {"x", "y", "z"}, DisplayFunction -> $DisplayFunction]; tor3x[u_, v_] = (aa*(u^2 + v^2) - 2*bb*u*v)*(v^2 -1)/((u^2 + v^2)*(v^2 + 1)); tor3y[u_, v_] = 2*v*(aa*(u^2 + v^2) - 2*bb*u*v)/((u^2 + v^2)*(v^2 + 1)); tor3z[u_, v_] = cc*(v^2 - u^2)/(u^2 + v^2); tor3 := ParametricPlot3D[{tor3x[u, v], tor3y[u, v], tor3z[u, v]}, {u, -1, 1}, {v, -1, 1}, DisplayFunction -> Identity]; viewtor3[lx_, mx_, ly_, my_, lz_, mz_] := Show[tor3, PlotRange -> {{lx, mx}, {ly, my}, {lz, mz}}, Axes -> True, AxesLabel -> {"x", "y", "z"}, DisplayFunction -> $DisplayFunction]; btor2x[u_, v_] = (aa*(u^2*(1 - v)^2 + 1) - 2*bb*u(1 - v))*(1 - 2*v)/ (((1 - v)^2 + v^2)*(u^2*(1 - v)^2 + 1)); btor2y[u_, v_] = 2*(1 - v)*v*(aa*(u^2*(1 - v)^2 + 1) - 2*bb*u(1 - v))/ (((1 - v)^2 + v^2)*(u^2*(1 - v)^2 + 1)); btor2z[u_, v_] = cc*(u^2*(1 - v)^2 - 1)/(u^2*(1 - v)^2 + 1); btor2 := ParametricPlot3D[{btor2x[u, v], btor2y[u, v], btor2z[u, v]}, {u, -2, 2}, {v, 0, 1}, DisplayFunction -> Identity]; btor2 := ParametricPlot3D[{btor2x[u, v], btor2y[u, v], btor2z[u, v]}, {u, -2, 2}, {v, -2, 2}, DisplayFunction -> Identity]; viewbtor2[lx_, mx_, ly_, my_, lz_, mz_] := Show[btor2, PlotRange -> {{lx, mx}, {ly, my}, {lz, mz}}, Axes -> True, AxesLabel -> {"x", "y", "z"}, DisplayFunction -> $DisplayFunction]; viewtor2[-3.1,3.1,-3.1,3.1,-2.1,2.1] viewtor3[-3.1,3.1,-3.1,3.1,-2.1,2.1] viewbtor2[-3.1,3.1,-3.1,3.1,-2.1,2.1] (* Elliptic Torus , degree 8 (in terms of tan theta/4 tan fi/4 *) poly1 = (aa * (1 + v^2)^2 - 4 * bb * v * (1 - v^2)) * (u^4 - 6 * u^2 + 1); poly2 = 4 * u * (1 - u^2) * (aa * (1 + v^2)^2 - 4 * bb * v * (1 - v^2)); poly3 = cc * (v^4 - 6 * v^2 + 1) * (1 + u^2)^2; poly4 = (1 + u^2)^2 * (1 + v^2)^2; pp1 = Expand[poly1]; p1 = InputForm[pp1]; p1 = aa - 6*aa*u^2 + aa*u^4 - 4*bb*v + 24*bb*u^2*v - 4*bb*u^4*v + 2*aa*v^2 - 12*aa*u^2*v^2 + 2*aa*u^4*v^2 + 4*bb*v^3 - 24*bb*u^2*v^3 + 4*bb*u^4*v^3 + aa*v^4 - 6*aa*u^2*v^4 + aa*u^4*v^4 pp2 = Expand[poly2]; p2 = InputForm[pp2]; p2 = 4*aa*u - 4*aa*u^3 - 16*bb*u*v + 16*bb*u^3*v + 8*aa*u*v^2 - 8*aa*u^3*v^2 + 16*bb*u*v^3 - 16*bb*u^3*v^3 + 4*aa*u*v^4 - 4*aa*u^3*v^4 pp3 = Expand[poly3]; p3 = InputForm[pp3]; p3 = cc + 2*cc*u^2 + cc*u^4 - 6*cc*v^2 - 12*cc*u^2*v^2 - 6*cc*u^4*v^2 + cc*v^4 + 2*cc*u^2*v^4 + cc*u^4*v^4 pp4 = Expand[poly4]; p4 = InputForm[pp4]; p4 = 1 + 2*u^2 + u^4 + 2*v^2 + 4*u^2*v^2 + 2*u^4*v^2 + v^4 + 2*u^2*v^4 + u^4*v^4 poly = {{{aa, 0, 0}, {-6*aa, 2, 0}, {aa, 4, 0}, {-4*bb, 0, 1}, {24*bb, 2, 1}, {-4*bb, 4, 1}, {2*aa, 0, 2}, {-12*aa, 2, 2}, {2*aa, 4, 2}, {4*bb, 0, 3}, {-24*bb, 2, 3}, {4*bb, 4, 3}, {aa, 0, 4}, {-6*aa, 2, 4}, {aa, 4, 4}}, {{4*aa, 1, 0}, {-4*aa, 3, 0}, {-16*bb, 1, 1}, {16*bb, 3, 1}, {8*aa, 1, 2}, {-8*aa, 3, 2}, {16*bb, 1, 3}, {-16*bb, 3, 3}, {4*aa, 1, 4}, {-4*aa, 3, 4}}, {{cc, 0, 0}, {2*cc, 2, 0}, {cc, 4, 0}, {-6*cc, 0, 2}, {-12*cc, 2, 2}, {-6*cc, 4, 2}, {cc, 0, 4}, {2*cc, 2, 4}, {cc, 4, 4}}, {{1, 0, 0}, {2, 2, 0}, {1, 4, 0}, {2, 0, 2}, {4, 2, 2}, {2, 4, 2}, {1, 0, 4}, {2, 2, 4}, {1, 4, 4}}}; tornet2 = contnet[poly, 8] (* This is the master control net for an elliptic torus depending *) (* on aa, bb, and cc. The numbers bb and cc specify the axes of the *) (* ellipse revolving around the z axis, and aa is the radius of the *) (* circle on which the center of the revolving ellipse is located *) (* Note: for aa <= bb or aa <= cc, the torus is self intersecting *) nettorus = {{aa, 0, cc, 1}, {aa - bb/2, 0, cc, 1}, {(14*((15*aa)/14 - bb))/15, 0, (11*cc)/15, 15/14}, {(14*((17*aa)/14 - (10*bb)/7))/17, 0, (5*cc)/17, 17/14}, {(70*((101*aa)/70 - (12*bb)/7))/101, 0, (-19*cc)/101, 101/70}, {(14*((25*aa)/14 - (25*bb)/14))/25, 0, (-3*cc)/5, 25/14}, {(7*((16*aa)/7 - (11*bb)/7))/16, 0, (-7*cc)/8, 16/7}, {(3*aa - bb)/3, 0, -cc, 3}, {aa, 0, -cc, 4}, {aa, aa/2, cc, 1}, {aa - bb/2, aa/2 - (2*bb)/7, cc, 1}, {(14*((15*aa)/14 - bb))/15, (14*((23*aa)/42 - (4*bb)/7))/15, (11*cc)/15, 15/14}, {(14*((17*aa)/14 - (10*bb)/7))/17, (14*((9*aa)/14 - (4*bb)/5))/17, (5*cc)/17, 17/14}, {(70*((101*aa)/70 - (12*bb)/7))/101, (70*((4*aa)/5 - (32*bb)/35))/101, (-19*cc)/101, 101/70}, {(14*((25*aa)/14 - (25*bb)/14))/25, (14*((22*aa)/21 - (6*bb)/7))/25, (-3*cc)/5, 25/14}, {(7*((16*aa)/7 - (11*bb)/7))/16, (7*((10*aa)/7 - (4*bb)/7))/16, (-7*cc)/8, 16/7}, {(3*aa - bb)/3, (2*aa)/3, -cc, 3}, {(11*aa)/15, (14*aa)/15, cc, 15/14}, {(14*((11*aa)/14 - (5*bb)/14))/15, (14*(aa - (4*bb)/7))/15, cc, 15/14}, {(105*((29*aa)/35 - (5*bb)/7))/121, (105*((23*aa)/21 - (8*bb)/7))/121, (87*cc)/121, 121/105}, {(35*((32*aa)/35 - (73*bb)/70))/46, (35*((9*aa)/7 - (8*bb)/5))/46, (6*cc)/23, 46/35}, {(210*((73*aa)/70 - (46*bb)/35))/331, (210*((8*aa)/5 - (64*bb)/35))/331, (-77*cc)/331, 331/210}, {(42*((17*aa)/14 - (3*bb)/2))/83, (42*((44*aa)/21 - (12*bb)/7))/83, (-53*cc)/83, 83/42}, {(7*((10*aa)/7 - (11*bb)/7))/18, (7*((20*aa)/7 - (8*bb)/7))/18, (-8*cc)/9, 18/7}, {(5*aa)/17, (20*aa)/17, cc, 17/14}, {(14*((5*aa)/14 - bb/14))/17, (14*((10*aa)/7 - (4*bb)/5))/17, cc, 17/14}, {(35*((12*aa)/35 - bb/7))/46, (35*((109*aa)/70 - (8*bb)/5))/46, (16*cc)/23, 46/35}, {(35*((11*aa)/35 - (19*bb)/70))/53, (35*((127*aa)/70 - (79*bb)/35))/53, (11*cc)/53, 53/35}, {(70*((17*aa)/70 - (18*bb)/35))/129, (70*((78*aa)/35 - (92*bb)/35))/129, (-13*cc)/43, 129/70}, {(14*(aa/14 - (13*bb)/14))/33, (14*((20*aa)/7 - (18*bb)/7))/33, (-23*cc)/33, 33/14}, {(-19*aa)/101, (120*aa)/101, cc, 101/70}, {(70*((-19*aa)/70 + (12*bb)/35))/101, (70*((12*aa)/7 - (32*bb)/35))/101, cc, 101/70}, {(210*((-11*aa)/30 + (24*bb)/35))/331, (210*((194*aa)/105 - (64*bb)/35))/331, (219*cc)/331, 331/210}, {(70*((-39*aa)/70 + (6*bb)/7))/129, (70*((74*aa)/35 - (92*bb)/35))/129, (17*cc)/129, 129/70}, {(10*((-9*aa)/10 + (24*bb)/35))/23, (10*((88*aa)/35 - (16*bb)/5))/23, (-9*cc)/23, 23/10}, {(-3*aa)/5, aa, cc, 25/14}, {(14*((-15*aa)/14 + (6*bb)/7))/25, (14*((25*aa)/14 - (6*bb)/7))/25, cc, 25/14}, {(42*((-53*aa)/42 + (12*bb)/7))/83, (42*((79*aa)/42 - (12*bb)/7))/83, (51*cc)/83, 83/42}, {(14*((-23*aa)/14 + (16*bb)/7))/33, (14*((29*aa)/14 - (18*bb)/7))/33, cc/33, 33/14}, {(-7*aa)/8, (11*aa)/16, cc, 16/7}, {(7*(-2*aa + (10*bb)/7))/16, (7*((11*aa)/7 - (4*bb)/7))/16, cc, 16/7}, {(7*((-16*aa)/7 + (20*bb)/7))/18, (7*((11*aa)/7 - (8*bb)/7))/18, (5*cc)/9, 18/7}, {-aa, aa/3, cc, 3}, {(-3*aa + 2*bb)/3, aa/3, cc, 3}, {-aa, 0, cc, 4}}; (* Control net for a torus with aa = 2, bb = 1, cc = 1 *) aa = 2; bb = 1; cc = 1; tornet = {{2, 0, 1, 1}, {3/2, 0, 1, 1}, {16/15, 0, 11/15, 15/14}, {14/17, 0, 5/17, 17/14}, {82/101, 0, -19/101, 101/70}, {1, 0, -3/5, 25/14}, {21/16, 0, -7/8, 16/7}, {5/3, 0, -1, 3}, {2, 0, -1, 4}, {2, 1, 1, 1}, {3/2, 5/7, 1, 1}, {16/15, 22/45, 11/15, 15/14}, {14/17, 2/5, 5/17, 17/14}, {82/101, 48/101, -19/101, 101/70}, {1, 52/75, -3/5, 25/14}, {21/16, 1, -7/8, 16/7}, {5/3, 4/3, -1, 3}, {22/15, 28/15, 1, 15/14}, {17/15, 4/3, 1, 15/14}, {9/11, 10/11, 87/121, 121/105}, {55/92, 17/23, 6/23, 46/35}, {162/331, 288/331, -77/331, 331/210}, {39/83, 104/83, -53/83, 83/42}, {1/2, 16/9, -8/9, 18/7}, {10/17, 40/17, 1, 17/14}, {9/17, 144/85, 1, 17/14}, {19/46, 53/46, 16/23, 46/35}, {25/106, 48/53, 11/53, 53/35}, {-2/129, 128/129, -13/43, 129/70}, {-1/3, 4/3, -23/33, 33/14}, {-38/101, 240/101, 1, 101/70}, {-14/101, 176/101, 1, 101/70}, {-10/331, 392/331, 219/331, 331/210}, {-6/43, 112/129, 17/129, 129/70}, {-78/161, 128/161, -9/23, 23/10}, {-6/5, 2, 1, 25/14}, {-18/25, 38/25, 1, 25/14}, {-34/83, 86/83, 51/83, 83/42}, {-14/33, 2/3, 1/33, 33/14}, {-7/4, 11/8, 1, 16/7}, {-9/8, 9/8, 1, 16/7}, {-2/3, 7/9, 5/9, 18/7}, {-2, 2/3, 1, 3}, {-4/3, 2/3, 1, 3}, {-2, 0, 1, 4}}; sqrender[tornet,8,1,1,-1,-1,2,0,0,-3.2,3.2,-3.2,3.2,-1.1,1.1,0] (* great *) sqrender[tornet,8,1,1,-1,-1,2,1,0,-3.2,3.2,-3.2,3.2,-1.1,1.1,0] (* lite *) sqfractal[tornet,8,1,1,-1,-1,3,0,0,-3.2,3.2,-3.2,3.2,-1.1,1.1] (* very cute *) reftrig1 = {{-1, 1, 1}, {-1, -1, 3}, {1, 1, -1}}; reftrig2 = {{1, -1, 1}, {1, 1, -1}, {-1, -1, 3}}; trender[tornet,8,reftrig1,2,1,0,-3.2,3.2,-3.2,3.2,-1.1,1.1,0] (* lite *) trender[tornet,8,reftrig1,1,1,0,-4,4,-4,4,-3,3,0] (* lite *) trender[tornet,8,reftrig1,2,1,1,-4,4,-4,4,-3,3,0] (* lite *) trender[tornet,8,reftrig1,2,1,0,-4,4,-4,4,-3,3,0] (* Control net for a torus with aa = 1/2, bb = 1, cc = 1 *) (* this torus intersects itself *) aa = 1/2; bb = 1; cc = 1 tornet2= {{1/2, 0, 1, 1}, {0, 0, 1, 1}, {-13/30, 0, 11/15, 15/14}, {-23/34, 0, 5/17, 17/14}, {-139/202, 0, -19/101, 101/70}, {-1/2, 0, -3/5, 25/14}, {-3/16, 0, -7/8, 16/7}, {1/6, 0, -1, 3}, {1/2, 0, -1, 4}, {1/2, 1/4, 1, 1}, {0, -1/28, 1, 1}, {-13/30, -5/18, 11/15, 15/14}, {-23/34, -67/170, 5/17, 17/14}, {-139/202, -36/101, -19/101, 101/70}, {-1/2, -14/75, -3/5, 25/14}, {-3/16, 1/16, -7/8, 16/7}, {1/6, 1/3, -1, 3}, {11/30, 7/15, 1, 15/14}, {1/30, -1/15, 1, 15/14}, {-63/242, -125/242, 87/121, 121/105}, {-41/92, -67/92, 6/23, 46/35}, {-333/662, -216/331, -77/331, 331/210}, {-75/166, -28/83, -53/83, 83/42}, {-1/3, 1/9, -8/9, 18/7}, {5/34, 10/17, 1, 17/14}, {3/34, -6/85, 1, 17/14}, {1/46, -5/8, 16/23, 46/35}, {-4/53, -189/212, 11/53, 53/35}, {-55/258, -106/129, -13/43, 129/70}, {-25/66, -16/33, -23/33, 33/14}, {-19/202, 60/101, 1, 101/70}, {29/202, -4/101, 1, 101/70}, {211/662, -190/331, 219/331, 331/210}, {27/86, -110/129, 17/129, 129/70}, {33/322, -136/161, -9/23, 23/10}, {-3/10, 1/2, 1, 25/14}, {9/50, 1/50, 1, 25/14}, {91/166, -65/166, 51/83, 83/42}, {41/66, -43/66, 1/33, 33/14}, {-7/16, 11/32, 1, 16/7}, {3/16, 3/32, 1, 16/7}, {2/3, -5/36, 5/9, 18/7}, {-1/2, 1/6, 1, 3}, {1/6, 1/6, 1, 3}, {-1/2, 0, 1, 4}}; sqrender[tornet2,8,1,1,-1,-1,2,0,0,-1.7,1.7,-1.7,1.7,-1.1,1.1,0] (* great *) (* Control net for an elliptic torus with aa = 2, bb = 1/2, cc = 1 *) aa = 2; bb = 1/2; cc = 1; tornet3 = {{2, 0, 1, 1}, {7/4, 0, 1, 1}, {23/15, 0, 11/15, 15/14}, {24/17, 0, 5/17, 17/14}, {142/101, 0, -19/101, 101/70}, {3/2, 0, -3/5, 25/14}, {53/32, 0, -7/8, 16/7}, {11/6, 0, -1, 3}, {2, 0, -1, 4}, {2, 1, 1, 1}, {7/4, 6/7, 1, 1}, {23/15, 34/45, 11/15, 15/14}, {24/17, 62/85, 5/17, 17/14}, {142/101, 80/101, -19/101, 101/70}, {3/2, 14/15, -3/5, 25/14}, {53/32, 9/8, -7/8, 16/7}, {11/6, 4/3, -1, 3}, {22/15, 28/15, 1, 15/14}, {13/10, 8/5, 1, 15/14}, {273/242, 170/121, 87/121, 121/105}, {183/184, 31/23, 6/23, 46/35}, {300/331, 480/331, -77/331, 331/210}, {141/166, 140/83, -53/83, 83/42}, {29/36, 2, -8/9, 18/7}, {10/17, 40/17, 1, 17/14}, {19/34, 172/85, 1, 17/14}, {43/92, 81/46, 16/23, 46/35}, {69/212, 175/106, 11/53, 53/35}, {16/129, 220/129, -13/43, 129/70}, {-3/22, 62/33, -23/33, 33/14}, {-38/101, 240/101, 1, 101/70}, {-26/101, 208/101, 1, 101/70}, {-82/331, 584/331, 219/331, 331/210}, {-16/43, 68/43, 17/129, 129/70}, {-102/161, 240/161, -9/23, 23/10}, {-6/5, 2, 1, 25/14}, {-24/25, 44/25, 1, 25/14}, {-70/83, 122/83, 51/83, 83/42}, {-10/11, 40/33, 1/33, 33/14}, {-7/4, 11/8, 1, 16/7}, {-23/16, 5/4, 1, 16/7}, {-11/9, 1, 5/9, 18/7}, {-2, 2/3, 1, 3}, {-5/3, 2/3, 1, 3}, {-2, 0, 1, 4}} render1[tornet3,8,2,0,0,-2.6,2.6,-2.6,2.6,-1.1,1.1,0] sqrender[tornet3,8,1,1,-1,-1,2,0,0,-2.6,2.6,-2.6,2.6,-1.1,1.1,0] (* great *) sqrender[tornet3,8,1,1,-1,-1,3,0,0,-2.6,2.6,-2.6,2.6,-1.1,1.1,0] (* great *) trender[tornet3,8,reftrig1,2,0,0,-3.2,3.2,-3.2,3.2,-1.1,1.1,0] (* Moebius strip , degree 7 *) poly = {{{2, 0, 0}, {-10, 2, 0}, {-10, 4, 0}, {2, 6, 0}, {2, 1, 1}, {-12, 3, 1}, {2, 5, 1}}, {{8, 1, 0}, {-8, 5, 0}, {8, 2, 1}, {-8, 4, 1}}, {{1, 0, 1}, {1, 2, 1}, {-1, 4, 1}, {-1, 6, 1}}, {{1, 0, 0}, {3, 2, 0}, {3, 4, 0}, {1, 6, 0}}}; net = contnet[poly, 7]; mobnet = InputForm[net]; mobnet = {{2, 0, 0, 1}, {2, 0, 1/7, 1}, {2, 0, 2/7, 1}, {2, 0, 3/7, 1}, {2, 0, 4/7, 1}, {2, 0, 5/7, 1}, {2, 0, 6/7, 1}, {2, 0, 1, 1}, {2, 8/7, 0, 1}, {43/21, 8/7, 1/7, 1}, {44/21, 8/7, 2/7, 1}, {15/7, 8/7, 3/7, 1}, {46/21, 8/7, 4/7, 1}, {47/21, 8/7, 5/7, 1}, {16/7, 8/7, 6/7, 1}, {4/3, 2, 0, 8/7}, {17/12, 31/15, 2/15, 8/7}, {3/2, 32/15, 4/15, 8/7}, {19/12, 11/5, 2/5, 8/7}, {5/3, 34/15, 8/15, 8/7}, {7/4, 7/3, 2/3, 8/7}, {2/5, 12/5, 0, 10/7}, {11/25, 64/25, 3/25, 10/7}, {12/25, 68/25, 6/25, 10/7}, {13/25, 72/25, 9/25, 10/7}, {14/25, 76/25, 12/25, 10/7}, {-10/17, 40/17, 0, 68/35}, {-2/3, 130/51, 5/51, 68/35}, {-38/51, 140/51, 10/51, 68/35}, {-14/17, 50/17, 5/17, 68/35}, {-22/15, 28/15, 0, 20/7}, {-5/3, 2, 1/15, 20/7}, {-28/15, 32/15, 2/15, 20/7}, {-2, 1, 0, 32/7}, {-9/4, 1, 0, 32/7}, {-2, 0, 0, 8}}; sqrender[mobnet,7,1,1,-1,-1,2,0,0,-3.2,3.2,-3.2,3.2,-2.2,2.2,0] sqrender[mobnet,7,1,1,-1,-1,2,0,0,-3.1,3,-3,3,-2.2,2.2,0] (* gorgeous *) sqrender[mobnet,7,1,1,-1,-1,2,2,0,-3.1,3,-3,3,-2.2,2.2,0] (* in/outside colors *) sqrender[mobnet,7,1,1,-1,-1,2,4,0,-3.1,3,-3,3,-2.2,2.2,0] (* 4 colors *) sqfractal[mobnet,7,1,1,-1,-1,2,0,0,-3.1,3,-3,3,-2.2,2.2] sqfractal[mobnet,7,1,1,-1,-1,3,1,0,-3.1,3,-3,3,-2.2,2.2] (* lite! *) sqfractal[mobnet,7,1,1,-1,-1,3,0,0,-3.1,3,-3.1,3.1,-2.2,2.2] (* cute *) trender[mobnet,7,reftrig1,2,0,0,-3.2,3.2,-3.2,3.2,-2.2,2.2,0] trender[mobnet,7,reftrig1,2,4,0,-3.2,3.2,-3.2,3.2,-2.2,2.2,0] trender[mobnet,7,reftrig1,2,2,0,-3.2,3.2,-3.2,3.2,-2.2,2.2,0] trender[mobnet,7,reftrig2,2,2,0,-3.2,3.2,-3.2,3.2,-2.2,2.2,0] (* Moebius strip, rectangular, degree (6, 1) *) poly = {{{2, 0, 0}, {-10, 2, 0}, {-10, 4, 0}, {2, 6, 0}, {2, 1, 1}, {-12, 3, 1}, {2, 5, 1}}, {{8, 1, 0}, {-8, 5, 0}, {8, 2, 1}, {-8, 4, 1}}, {{1, 0, 1}, {1, 2, 1}, {-1, 4, 1}, {-1, 6, 1}}, {{1, 0, 0}, {3, 2, 0}, {3, 4, 0}, {1, 6, 0}}}; rmobnet2 = gbicontnet[poly, 6,1, -1, 1, -1, 1]; recmobnet2 = InputForm[rmobnet2]; recmobnet2 = {{-3, 0, 0, 8}, {-1, 0, 0, 8}, {0, -16, -8/3, 0}, {0, -16/3, 8/3, 0}, {17/3, 0, 0, 8/5}, {1, 0, 0, 8/5}, {0, 16/5, -8/5, 0}, {0, -16/5, 8/5, 0}, {1, 0, 0, 8/5}, {17/3, 0, 0, 8/5}, {0, 16/3, -8/3, 0}, {0, 16, 8/3, 0}, {-1, 0, 0, 8}, {-3, 0, 0, 8}} polmob = bipolarizesurf[poly, 6, 1] rmobnet = bicontnet[poly, 6,1]; recmobnet = InputForm[rmobnet]; recmobnet = {{2, 0, 0, 1}, {2, 0, 1, 1}, {2, 4/3, 0, 1}, {7/3, 4/3, 1, 1}, {10/9, 20/9, 0, 6/5}, {5/3, 8/3, 8/9, 6/5}, {0, 5/2, 0, 8/5}, {1/4, 7/2, 3/4, 8/5}, {-10/9, 20/9, 0, 12/5}, {-14/9, 10/3, 5/9, 12/5}, {-2, 4/3, 0, 4}, {-3, 2, 1/3, 4}, {-2, 0, 0, 8}, {-3, 0, 0, 8}} recmob4 = newrecnet[recmobnet, 6, 1, -1, 1, -1, 1, 2] recmob5 = InputForm[recmob4] recmobnet2 = {{-3, 0, 0, 8}, {-1, 0, 0, 8}, {0, -16, -8/3, 0}, {0, -16/3, 8/3, 0}, {17/3, 0, 0, 8/5}, {1, 0, 0, 8/5}, {0, 16/5, -8/5, 0}, {0, -16/5, 8/5, 0}, {1, 0, 0, 8/5}, {17/3, 0, 0, 8/5}, {0, 16/3, -8/3, 0}, {0, 16, 8/3, 0}, {-1, 0, 0, 8}, {-3, 0, 0, 8}} recrender[recmobnet2,6,1,3,2,0,-3,3,-3,3,-2.2,2.2,0] (* 2 colors *) recrender[recmobnet2,6,1,1,0,0,-3,3,-3,3,-2.2,2.2,0] (* cute *) recrender[recmobnet2,6,1,2,0,0,-3,3,-3,3,-2.2,2.2,0] (* cute *) bbmob2 = convrectr[recmobnet, 6, 1, 0] trmob = InputForm[bbmob] trmob = {{2, 0, 0, 1}, {2, 0, 1/7, 1}, {2, 0, 2/7, 1}, {2, 0, 3/7, 1}, {2, 0, 4/7, 1}, {2, 0, 5/7, 1}, {2, 0, 6/7, 1}, {2, 0, 1, 1}, {2, 8/7, 0, 1}, {43/21, 8/7, 1/7, 1}, {44/21, 8/7, 2/7, 1}, {15/7, 8/7, 3/7, 1}, {46/21, 8/7, 4/7, 1}, {47/21, 8/7, 5/7, 1}, {16/7, 8/7, 6/7, 1}, {4/3, 2, 0, 8/7}, {17/12, 31/15, 2/15, 8/7}, {3/2, 32/15, 4/15, 8/7}, {19/12, 11/5, 2/5, 8/7}, {5/3, 34/15, 8/15, 8/7}, {7/4, 7/3, 2/3, 8/7}, {2/5, 12/5, 0, 10/7}, {11/25, 64/25, 3/25, 10/7}, {12/25, 68/25, 6/25, 10/7}, {13/25, 72/25, 9/25, 10/7}, {14/25, 76/25, 12/25, 10/7}, {-10/17, 40/17, 0, 68/35}, {-2/3, 130/51, 5/51, 68/35}, {-38/51, 140/51, 10/51, 68/35}, {-14/17, 50/17, 5/17, 68/35}, {-22/15, 28/15, 0, 20/7}, {-5/3, 2, 1/15, 20/7}, {-28/15, 32/15, 2/15, 20/7}, {-2, 1, 0, 32/7}, {-9/4, 1, 0, 32/7}, {-2, 0, 0, 8}} (* again the mobius strip, triangular net *) mobnet = {{2, 0, 0, 1}, {2, 0, 1/7, 1}, {2, 0, 2/7, 1}, {2, 0, 3/7, 1}, {2, 0, 4/7, 1}, {2, 0, 5/7, 1}, {2, 0, 6/7, 1}, {2, 0, 1, 1}, {2, 8/7, 0, 1}, {43/21, 8/7, 1/7, 1}, {44/21, 8/7, 2/7, 1}, {15/7, 8/7, 3/7, 1}, {46/21, 8/7, 4/7, 1}, {47/21, 8/7, 5/7, 1}, {16/7, 8/7, 6/7, 1}, {4/3, 2, 0, 8/7}, {17/12, 31/15, 2/15, 8/7}, {3/2, 32/15, 4/15, 8/7}, {19/12, 11/5, 2/5, 8/7}, {5/3, 34/15, 8/15, 8/7}, {7/4, 7/3, 2/3, 8/7}, {2/5, 12/5, 0, 10/7}, {11/25, 64/25, 3/25, 10/7}, {12/25, 68/25, 6/25, 10/7}, {13/25, 72/25, 9/25, 10/7}, {14/25, 76/25, 12/25, 10/7}, {-10/17, 40/17, 0, 68/35}, {-2/3, 130/51, 5/51, 68/35}, {-38/51, 140/51, 10/51, 68/35}, {-14/17, 50/17, 5/17, 68/35}, {-22/15, 28/15, 0, 20/7}, {-5/3, 2, 1/15, 20/7}, {-28/15, 32/15, 2/15, 20/7}, {-2, 1, 0, 32/7}, {-9/4, 1, 0, 32/7}, {-2, 0, 0, 8}}; (* testing gbicontnet and newrecnet *) mobnet4 = gbicontnet[poly, 6,1, -1, 0, -1, 0]; mobnet5 = InputForm[mobnet4]; mobnet5 = {{-3, 0, 0, 8}, {-2, 0, 0, 8}, {-3, -2, -1/3, 4}, {-2, -4/3, 0, 4}, {-14/9, -10/3, -5/9, 12/5}, {-10/9, -20/9, 0, 12/5}, {1/4, -7/2, -3/4, 8/5}, {0, -5/2, 0, 8/5}, {5/3, -8/3, -8/9, 6/5}, {10/9, -20/9, 0, 6/5}, {7/3, -4/3, -1, 1}, {2, -4/3, 0, 1}, {2, 0, -1, 1}, {2, 0, 0, 1}} recmob6 = newrecnet[recmobnet, 6, 1, -1, 0, -1, 0, 0] recmob7 = InputForm[recmob6] recmob7 = {{-3, 0, 0, 8}, {-2, 0, 0, 8}, {-3, -2, -1/3, 4}, {-2, -4/3, 0, 4}, {-14/9, -10/3, -5/9, 12/5}, {-10/9, -20/9, 0, 12/5}, {1/4, -7/2, -3/4, 8/5}, {0, -5/2, 0, 8/5}, {5/3, -8/3, -8/9, 6/5}, {10/9, -20/9, 0, 6/5}, {7/3, -4/3, -1, 1}, {2, -4/3, 0, 1}, {2, 0, -1, 1}, {2, 0, 0, 1}} (* Wrong Funny Klein bottle , degree 8 *) poly1 = aa*(u^4-6*u^2+1)*(1+v^2)^2+rr*(v^4-6*v^2+1)*(1-u^4); poly2 = 4*bb*u*(1-u^2)*(1+v^2)^2+2*rr*u*(v^4-6*v^2+1)*(1+u^2); poly3 = 4*rr*v(1-v^2)*((1+u^2)^2); poly4 = (1+u^2)^2*(1+v^2)^2; dudo2 = 4*bb*u*(1-u^2)*(1+v^2)^2+2*rr*u*(v^4-6*v^2+1)*(1+u^2); p1 = Expand[poly1]; p1 = InputForm[p1]; p1 = aa + rr - 6*aa*u^2 + aa*u^4 - rr*u^4 + 2*aa*v^2 - 6*rr*v^2 - 12*aa*u^2*v^2 + 2*aa*u^4*v^2 + 6*rr*u^4*v^2 + aa*v^4 + rr*v^4 - 6*aa*u^2*v^4 + aa*u^4*v^4 - rr*u^4*v^4 p2 = Expand[poly2]; p2 = InputForm[p2]; p2 = 4*bb*u + 2*rr*u - 4*bb*u^3 + 2*rr*u^3 + 8*bb*u*v^2 - 12*rr*u*v^2 - 8*bb*u^3*v^2 - 12*rr*u^3*v^2 + 4*bb*u*v^4 + 2*rr*u*v^4 - 4*bb*u^3*v^4 + 2*rr*u^3*v^4 p3 = Expand[poly3]; p3 = InputForm[p3]; p3 = 4*rr*v + 8*rr*u^2*v + 4*rr*u^4*v - 4*rr*v^3 - 8*rr*u^2*v^3 - 4*rr*u^4*v^3; p4 = Expand[poly4]; p4 = InputForm[p4]; p4 = 1 + 2*u^2 + u^4 + 2*v^2 + 4*u^2*v^2 + 2*u^4*v^2 + v^4 + 2*u^2*v^4 + u^4*v^4; poly = {{{aa + rr, 0, 0}, {-6*aa, 2, 0}, {aa - rr, 4, 0}, {2*aa - 6*rr, 0, 2}, {12*aa, 2, 2}, {2*aa + 6*rr, 4, 2}, {aa + rr, 0, 4}, {-6*aa, 2, 4}, {aa - rr, 4, 4}}, {{4*bb + 2*rr, 1, 0}, {-4*bb + 2*rr, 3, 0}, {8*bb - 12*rr, 1, 2}, {-8*bb - 12*rr, 3, 2}, {4*bb + 2*rr, 1, 4}, {-4*bb + 2*rr, 3, 4}}, {{4*rr, 0, 1}, {8*rr, 2, 1}, {4*rr, 4, 1}, {-4*rr, 0, 3}, {-8*rr, 2, 3}, {-4*rr, 4, 3}}, {{1, 0, 0}, {2, 2, 0}, {1, 4, 0}, {2, 0, 2}, {4, 2, 2}, {2, 4, 2}, {1, 0, 4}, {2, 2, 4}, {1, 4, 4}}}; (* In the above poly, in p1, coeff of u^2v^2 is 12*aa instead of -12*aa *) poly = {{{aa + rr, 0, 0}, {-6*aa, 2, 0}, {aa - rr, 4, 0}, {2*aa - 6*rr, 0, 2}, {-12*aa, 2, 2}, {2*aa + 6*rr, 4, 2}, {aa + rr, 0, 4}, {-6*aa, 2, 4}, {aa - rr, 4, 4}}, {{4*bb + 2*rr, 1, 0}, {-4*bb + 2*rr, 3, 0}, {8*bb - 12*rr, 1, 2}, {-8*bb - 12*rr, 3, 2}, {4*bb + 2*rr, 1, 4}, {-4*bb + 2*rr, 3, 4}}, {{4*rr, 0, 1}, {8*rr, 2, 1}, {4*rr, 4, 1}, {-4*rr, 0, 3}, {-8*rr, 2, 3}, {-4*rr, 4, 3}}, {{1, 0, 0}, {2, 2, 0}, {1, 4, 0}, {2, 0, 2}, {4, 2, 2}, {2, 4, 2}, {1, 0, 4}, {2, 2, 4}, {1, 4, 4}}}; net = contnet[poly, 8]; kleinet = InputForm[net]; (* Master net *) wkleinet = {{aa + rr, 0, 0, 1}, {aa + rr, 0, rr/2, 1}, {(14*(aa + (2*aa - 6*rr)/28 + rr))/15, 0, (14*rr)/15, 15/14}, {(14*(aa + (3*(2*aa - 6*rr))/28 + rr))/17, 0, (20*rr)/17, 17/14}, {(70*(aa + (3*(2*aa - 6*rr))/14 + rr + (aa + rr)/70))/101, 0, (120*rr)/101, 101/70}, {(14* (aa + (5*(2*aa - 6*rr))/14 + rr + (aa + rr)/14))/25, 0, rr, 25/14}, {(7*(aa + (15*(2*aa - 6*rr))/28 + rr + (3*(aa + rr))/14))/16, 0, (11*rr)/16, 16/7}, {(aa + (3*(2*aa - 6*rr))/4 + rr + (aa + rr)/2)/3, 0, rr/3, 3}, {(4*aa - 4*rr)/4, 0, 0, 4}, {aa + rr, (4*bb + 2*rr)/8, 0, 1}, {aa + rr, (4*bb + 2*rr)/8, rr/2, 1}, {(14*(aa + (2*aa - 6*rr)/28 + rr))/15, (14*((8*bb - 12*rr)/168 + (4*bb + 2*rr)/8))/15, (14*rr)/15, 15/14}, {(14*(aa + (3*(2*aa - 6*rr))/28 + rr))/17, (14*((8*bb - 12*rr)/56 + (4*bb + 2*rr)/8))/17, (20*rr)/17, 17/14}, {(70*(aa + (3*(2*aa - 6*rr))/14 + rr + (aa + rr)/70))/101, (70*((8*bb - 12*rr)/28 + (9*(4*bb + 2*rr))/70))/101, (120*rr)/101, 101/70}, {(14*(aa + (5*(2*aa - 6*rr))/14 + rr + (aa + rr)/14))/25, (14*((5*(8*bb - 12*rr))/84 + (4*bb + 2*rr)/7))/25, rr, 25/14}, {(7*(aa + (15*(2*aa - 6*rr))/28 + rr + (3*(aa + rr))/14))/16, (7*((5*(8*bb - 12*rr))/56 + (5*(4*bb + 2*rr))/28))/16, (11*rr)/16, 16/7}, {(aa + (3*(2*aa - 6*rr))/4 + rr + (aa + rr)/2)/3, ((8*bb - 12*rr)/8 + (4*bb + 2*rr)/4)/3, rr/3, 3}, {(14*((11*aa)/14 + rr))/15, (7*(4*bb + 2*rr))/30, 0, 15/14}, {(14*((11*aa)/14 + rr))/15, (7*(4*bb + 2*rr))/30, (23*rr)/45, 15/14}, {(105*((57*aa)/70 + (2*aa - 6*rr)/28 + rr))/121, (105*((8*bb - 12*rr)/84 + (4*bb + 2*rr)/4))/121, (115*rr)/121, 121/105}, {(35*((61*aa)/70 + (3*(2*aa - 6*rr))/28 + rr))/46, (35*((8*bb - 12*rr)/28 + (4*bb + 2*rr)/4))/46, (109*rr)/92, 46/35}, {(210*((33*aa)/35 + (3*(2*aa - 6*rr))/14 + rr + (aa + rr)/70))/331, (210*((8*bb - 12*rr)/14 + (9*(4*bb + 2*rr))/35))/331, (388*rr)/331, 331/210}, {(42*(aa + (5*(2*aa - 6*rr))/14 + rr + (aa + rr)/14))/83, (42*((5*(8*bb - 12*rr))/42 + (2*(4*bb + 2*rr))/7))/83, (79*rr)/83, 83/42}\ , {(7*(aa + (15*(2*aa - 6*rr))/28 + rr + (3*(aa + rr))/14))/18, (7*((5*(8*bb - 12*rr))/28 + (5*(4*bb + 2*rr))/14))/18, (11*rr)/18, 18/7}, {(14*((5*aa)/14 + rr))/17, (14* ((-4*bb + 2*rr)/56 + (3*(4*bb + 2*rr))/8))/17, 0, 17/14}, {(14*((5*aa)/14 + rr))/17, (14* ((-4*bb + 2*rr)/56 + (3*(4*bb + 2*rr))/8))/17, (9*rr)/17, 17/14}, {(35*((31*aa)/70 + (2*aa - 6*rr)/28 + rr))/46, (35*((-8*bb - 12*rr)/560 + (8*bb - 12*rr)/56 + (-4*bb + 2*rr)/56 + (3*(4*bb + 2*rr))/8))/46, (45*rr)/46, 46/35}, {(35*((43*aa)/70 + (3*(2*aa - 6*rr))/28 + rr))/53, (35*((3*(-8*bb - 12*rr))/560 + (3*(8*bb - 12*rr))/56 + (-4*bb + 2*rr)/56 + (3*(4*bb + 2*rr))/8))/53, (127*rr)/106, 53/35}, {(70*((29*aa)/35 + (3*(2*aa - 6*rr))/14 + rr + (aa + rr)/70))/129, (70*((3*(-8*bb - 12*rr))/280 + (3*(8*bb - 12*rr))/28 + (3*(-4*bb + 2*rr))/140 + (27*(4*bb + 2*rr))/70))/129, (148*rr)/129, 129/70}, {(14*(aa + (5*(2*aa - 6*rr))/14 + rr + (aa + rr)/14))/33, (14*((-8*bb - 12*rr)/56 + (5*(8*bb - 12*rr))/28 + (-4*bb + 2*rr)/28 + (3*(4*bb + 2*rr))/7))/33, (29*rr)/33, 33/14}, {(70*((-2*aa)/7 + (aa - rr)/70 + rr))/101, (70*((-4*bb + 2*rr)/14 + (4*bb + 2*rr)/2))/101, 0, 101/70}, {(70*((-2*aa)/7 + (aa - rr)/70 + rr))/101, (70*((-4*bb + 2*rr)/14 + (4*bb + 2*rr)/2))/101, (56*rr)/101, 101/70}, {(210*((-4*aa)/35 + (2*aa - 6*rr)/28 + (aa - rr)/70 + rr + (2*aa + 6*rr)/420))/331, (210*((-8*bb - 12*rr)/140 + (8*bb - 12*rr)/42 + (-4*bb + 2*rr)/14 + (4*bb + 2*rr)/2))/331, (336*rr)/331, 331/210}, {(70*((8*aa)/35 + (3*(2*aa - 6*rr))/28 + (aa - rr)/70 + rr + (2*aa + 6*rr)/140))/129, (70*((3*(-8*bb - 12*rr))/140 + (8*bb - 12*rr)/14 + (-4*bb + 2*rr)/14 + (4*bb + 2*rr)/2))/129, (52*rr)/43, 129/70}, {(10*((23*aa)/35 + (3*(2*aa - 6*rr))/14 + (aa - rr)/35 + rr + (aa + rr)/70 + (2*aa + 6*rr)/70))/23, (10*((3*(-8*bb - 12*rr))/70 + (8*bb - 12*rr)/7 + (3*(-4*bb + 2*rr))/35 + (18*(4*bb + 2*rr))/35))/23, (176*rr)/161, 23/10}, {(14*((-8*aa)/7 + (aa - rr)/14 + rr))/25, (14*((5*(-4*bb + 2*rr))/28 + (5*(4*bb + 2*rr))/8))/25, 0, 25/14}, {(14*((-8*aa)/7 + (aa - rr)/14 + rr))/25, (14*((5*(-4*bb + 2*rr))/28 + (5*(4*bb + 2*rr))/8))/25, (44*rr)/75, 25/14}\ , {(42*((-6*aa)/7 + (2*aa - 6*rr)/28 + (aa - rr)/14 + rr + (2*aa + 6*rr)/84))/83, (42*((-8*bb - 12*rr)/56 + (5*(8*bb - 12*rr))/168 + (5*(-4*bb + 2*rr))/28 + (5*(4*bb + 2*rr))/8))/83, (88*rr)/83, 83/42}\ , {(14*((-2*aa)/7 + (3*(2*aa - 6*rr))/28 + (aa - rr)/14 + rr + (2*aa + 6*rr)/28))/33, (14*((3*(-8*bb - 12*rr))/56 + (5*(8*bb - 12*rr))/56 + (5*(-4*bb + 2*rr))/28 + (5*(4*bb + 2*rr))/8))/33, (40*rr)/33, 33/14}\ , {(7*((-31*aa)/14 + (3*(aa - rr))/14 + rr))/16, (7*((5*(-4*bb + 2*rr))/14 + (3*(4*bb + 2*rr))/4))/16, 0, 16/7}, {(7*((-31*aa)/14 + (3*(aa - rr))/14 + rr))/16, (7*((5*(-4*bb + 2*rr))/14 + (3*(4*bb + 2*rr))/4))/16, (5*rr)/8, 16/7}, {(7*((-25*aa)/14 + (2*aa - 6*rr)/28 + (3*(aa - rr))/14 + rr + (2*aa + 6*rr)/28))/18, (7*((-8*bb - 12*rr)/28 + (8*bb - 12*rr)/28 + (5*(-4*bb + 2*rr))/14 + (3*(4*bb + 2*rr))/4))/18, (10*rr)/9, 18/7}, {((-7*aa)/2 + (aa - rr)/2 + rr)/3, ((5*(-4*bb + 2*rr))/8 + (7*(4*bb + 2*rr))/8)/3, 0, 3}, {((-7*aa)/2 + (aa - rr)/2 + rr)/3, ((5*(-4*bb + 2*rr))/8 + (7*(4*bb + 2*rr))/8)/3, (2*rr)/3, 3}, {-aa, rr, 0, 4}}; (* Case rr = 1; aa = 4; bb = 1 *) rr = 1; aa = 4; bb = 1; wklein = {{5, 0, 0, 1}, {5, 0, 1/2, 1}, {71/15, 0, 14/15, 15/14}, {73/17, 0, 20/17, 17/14}, {385/101, 0, 120/101, 101/70}, {17/5, 0, 1, 25/14}, {25/8, 0, 11/16, 16/7}, {3, 0, 1/3, 3}, {3, 0, 0, 4}, {5, 3/4, 0, 1}, {5, 3/4, 1/2, 1}, {71/15, 61/90, 14/15, 15/14}, {73/17, 19/34, 20/17, 17/14}, {385/101, 44/101, 120/101, 101/70}, {17/5, 26/75, 1, 25/14}, {25/8, 5/16, 11/16, 16/7}, {3, 1/3, 1/3, 3}, {58/15, 7/5, 0, 15/14}, {58/15, 7/5, 23/45, 15/14}, {909/242, 305/242, 115/121, 121/105}, {329/92, 95/92, 109/92, 46/35}, {1107/331, 264/331, 388/331, 331/210}, {255/83, 52/83, 79/83, 83/42}, {25/9, 5/9, 11/18, 18/7}, {2, 31/17, 0, 17/14}, {2, 31/17, 9/17, 17/14}, {199/92, 295/184, 45/46, 46/35}, {257/106, 5/4, 127/106, 53/35}, {337/129, 38/43, 148/129, 129/70}, {85/33, 20/33, 29/33, 33/14}, {-7/101, 200/101, 0, 101/70}, {-7/101, 200/101, 56/101, 101/70}, {145/331, 550/331, 336/331, 331/210}, {53/43, 50/43, 52/43, 129/70}, {309/161, 104/161, 176/161, 23/10}, {-47/25, 19/10, 0, 25/14}, {-47/25, 19/10, 44/75, 25/14}, {-1, 245/166, 88/83, 83/42}, {1/3, 5/6, 40/33, 33/14}, {-101/32, 53/32, 0, 16/7}, {-101/32, 53/32, 5/8, 16/7}, {-23/12, 41/36, 10/9, 18/7}, {-23/6, 4/3, 0, 3}, {-23/6, 4/3, 2/3, 3}, {-4, 1, 0, 4}}; sqrender[wklein,8,1,1,-1,-1,2,0,0,-6.1,6.1,-2.1,2.1,-2.1,2.1,0] (* bizarre *) trender[wklein,8,reftrig1,2,0,0,-6.1,6.1,-2.1,2.1,-2.1,2.1,0] sqrender[wklein,8,1,1,-1,0,2,0,0,-6.1,6.1,-2.1,2.1,-2.1,2.1,0] (* Also very good viewpoint: ViewPoint -> {-15, -10, 5} *) (* Show[%148,ViewPoint -> {-15, -5, 2}] amazing *) (* Funny Klein bottle , degree 8 *) poly1 = aa*(u^4-6*u^2+1)*(1+v^2)^2+rr*(v^4-6*v^2+1)*(1-u^4); poly2 = 4*bb*u*(1-u^2)*(1+v^2)^2+2*rr*u*(v^4-6*v^2+1)*(1+u^2); poly3 = 4*rr*v*(1-v^2)*((1+u^2)^2); poly4 = (1+u^2)^2*(1+v^2)^2; dudo2 = 4*bb*u*(1-u^2)*(1+v^2)^2+2*rr*u*(v^4-6*v^2+1)*(1+u^2); p1 = Expand[poly1]; p1 = InputForm[p1]; p1 = aa + rr - 6*aa*u^2 + aa*u^4 - rr*u^4 + 2*aa*v^2 - 6*rr*v^2 - 12*aa*u^2*v^2 + 2*aa*u^4*v^2 + 6*rr*u^4*v^2 + aa*v^4 + rr*v^4 - 6*aa*u^2*v^4 + aa*u^4*v^4 - rr*u^4*v^4 p2 = Expand[poly2]; p2 = InputForm[p2]; p2 = 4*bb*u + 2*rr*u - 4*bb*u^3 + 2*rr*u^3 + 8*bb*u*v^2 - 12*rr*u*v^2 - 8*bb*u^3*v^2 - 12*rr*u^3*v^2 + 4*bb*u*v^4 + 2*rr*u*v^4 - 4*bb*u^3*v^4 + 2*rr*u^3*v^4 p3 = Expand[poly3]; p3 = InputForm[p3]; p3 = 4*rr*v + 8*rr*u^2*v + 4*rr*u^4*v - 4*rr*v^3 - 8*rr*u^2*v^3 - 4*rr*u^4*v^3; p4 = Expand[poly4]; p4 = InputForm[p4]; p4 = 1 + 2*u^2 + u^4 + 2*v^2 + 4*u^2*v^2 + 2*u^4*v^2 + v^4 + 2*u^2*v^4 + u^4*v^4; poly = {{{aa + rr, 0, 0}, {-6*aa, 2, 0}, {aa - rr, 4, 0}, {2*aa - 6*rr, 0, 2}, {-12*aa, 2, 2}, {2*aa + 6*rr, 4, 2}, {aa + rr, 0, 4}, {-6*aa, 2, 4}, {aa - rr, 4, 4}}, {{4*bb + 2*rr, 1, 0}, {-4*bb + 2*rr, 3, 0}, {8*bb - 12*rr, 1, 2}, {-8*bb - 12*rr, 3, 2}, {4*bb + 2*rr, 1, 4}, {-4*bb + 2*rr, 3, 4}}, {{4*rr, 0, 1}, {8*rr, 2, 1}, {4*rr, 4, 1}, {-4*rr, 0, 3}, {-8*rr, 2, 3}, {-4*rr, 4, 3}}, {{1, 0, 0}, {2, 2, 0}, {1, 4, 0}, {2, 0, 2}, {4, 2, 2}, {2, 4, 2}, {1, 0, 4}, {2, 2, 4}, {1, 4, 4}}}; net = contnet[poly, 8]; kleinet = InputForm[net]; (* Master net *) kleinet = {{aa + rr, 0, 0, 1}, {aa + rr, 0, rr/2, 1}, {(14*(aa + (2*aa - 6*rr)/28 + rr))/15, 0, (14*rr)/15, 15/14}, {(14*(aa + (3*(2*aa - 6*rr))/28 + rr))/17, 0, (20*rr)/17, 17/14}, {(70*(aa + (3*(2*aa - 6*rr))/14 + rr + (aa + rr)/70))/101, 0, (120*rr)/101, 101/70}, {(14* (aa + (5*(2*aa - 6*rr))/14 + rr + (aa + rr)/14))/25, 0, rr, 25/14}, {(7*(aa + (15*(2*aa - 6*rr))/28 + rr + (3*(aa + rr))/14))/16, 0, (11*rr)/16, 16/7}, {(aa + (3*(2*aa - 6*rr))/4 + rr + (aa + rr)/2)/3, 0, rr/3, 3}, {(4*aa - 4*rr)/4, 0, 0, 4}, {aa + rr, (4*bb + 2*rr)/8, 0, 1}, {aa + rr, (4*bb + 2*rr)/8, rr/2, 1}, {(14*(aa + (2*aa - 6*rr)/28 + rr))/15, (14*((8*bb - 12*rr)/168 + (4*bb + 2*rr)/8))/15, (14*rr)/15, 15/14}, {(14*(aa + (3*(2*aa - 6*rr))/28 + rr))/17, (14*((8*bb - 12*rr)/56 + (4*bb + 2*rr)/8))/17, (20*rr)/17, 17/14}, {(70*(aa + (3*(2*aa - 6*rr))/14 + rr + (aa + rr)/70))/101, (70*((8*bb - 12*rr)/28 + (9*(4*bb + 2*rr))/70))/101, (120*rr)/101, 101/70}, {(14*(aa + (5*(2*aa - 6*rr))/14 + rr + (aa + rr)/14))/25, (14*((5*(8*bb - 12*rr))/84 + (4*bb + 2*rr)/7))/25, rr, 25/14}, {(7*(aa + (15*(2*aa - 6*rr))/28 + rr + (3*(aa + rr))/14))/16, (7*((5*(8*bb - 12*rr))/56 + (5*(4*bb + 2*rr))/28))/16, (11*rr)/16, 16/7}, {(aa + (3*(2*aa - 6*rr))/4 + rr + (aa + rr)/2)/3, ((8*bb - 12*rr)/8 + (4*bb + 2*rr)/4)/3, rr/3, 3}, {(14*((11*aa)/14 + rr))/15, (7*(4*bb + 2*rr))/30, 0, 15/14}, {(14*((11*aa)/14 + rr))/15, (7*(4*bb + 2*rr))/30, (23*rr)/45, 15/14}, {(105*((53*aa)/70 + (2*aa - 6*rr)/28 + rr))/121, (105*((8*bb - 12*rr)/84 + (4*bb + 2*rr)/4))/121, (115*rr)/121, 121/105}, {(35*((7*aa)/10 + (3*(2*aa - 6*rr))/28 + rr))/46, (35*((8*bb - 12*rr)/28 + (4*bb + 2*rr)/4))/46, (109*rr)/92, 46/35}, {(210*((3*aa)/5 + (3*(2*aa - 6*rr))/14 + rr + (aa + rr)/70))/331, (210*((8*bb - 12*rr)/14 + (9*(4*bb + 2*rr))/35))/331, (388*rr)/331, 331/210}, {(42*((3*aa)/7 + (5*(2*aa - 6*rr))/14 + rr + (aa + rr)/14))/ 83, (42*((5*(8*bb - 12*rr))/42 + (2*(4*bb + 2*rr))/7))/83, (79*rr)/83, 83/42}, {(7*(aa/7 + (15*(2*aa - 6*rr))/28 + rr + (3*(aa + rr))/14))/18, (7*((5*(8*bb - 12*rr))/28 + (5*(4*bb + 2*rr))/14))/18, (11*rr)/18, 18/7}, {(14*((5*aa)/14 + rr))/17, (14* ((-4*bb + 2*rr)/56 + (3*(4*bb + 2*rr))/8))/17, 0, 17/14}, {(14*((5*aa)/14 + rr))/17, (14* ((-4*bb + 2*rr)/56 + (3*(4*bb + 2*rr))/8))/17, (9*rr)/17, 17/14}, {(35*((19*aa)/70 + (2*aa - 6*rr)/28 + rr))/46, (35*((-8*bb - 12*rr)/560 + (8*bb - 12*rr)/56 + (-4*bb + 2*rr)/56 + (3*(4*bb + 2*rr))/8))/46, (45*rr)/46, 46/35}, {(35*(aa/10 + (3*(2*aa - 6*rr))/28 + rr))/53, (35*((3*(-8*bb - 12*rr))/560 + (3*(8*bb - 12*rr))/56 + (-4*bb + 2*rr)/56 + (3*(4*bb + 2*rr))/8))/53, (127*rr)/106, 53/35}, {(70*(-aa/5 + (3*(2*aa - 6*rr))/14 + rr + (aa + rr)/70))/129, (70*((3*(-8*bb - 12*rr))/280 + (3*(8*bb - 12*rr))/28 + (3*(-4*bb + 2*rr))/140 + (27*(4*bb + 2*rr))/70))/129, (148*rr)/129, 129/70}, {(14*((-5*aa)/7 + (5*(2*aa - 6*rr))/14 + rr + (aa + rr)/14))/ 33, (14*((-8*bb - 12*rr)/56 + (5*(8*bb - 12*rr))/28 + (-4*bb + 2*rr)/28 + (3*(4*bb + 2*rr))/7))/33, (29*rr)/33, 33/14}, {(70*((-2*aa)/7 + (aa - rr)/70 + rr))/101, (70*((-4*bb + 2*rr)/14 + (4*bb + 2*rr)/2))/101, 0, 101/70}, {(70*((-2*aa)/7 + (aa - rr)/70 + rr))/101, (70*((-4*bb + 2*rr)/14 + (4*bb + 2*rr)/2))/101, (56*rr)/101, 101/70}, {(210*((-16*aa)/35 + (2*aa - 6*rr)/28 + (aa - rr)/70 + rr + (2*aa + 6*rr)/420))/331, (210*((-8*bb - 12*rr)/140 + (8*bb - 12*rr)/42 + (-4*bb + 2*rr)/14 + (4*bb + 2*rr)/2))/331, (336*rr)/331, 331/210}, {(70*((-4*aa)/5 + (3*(2*aa - 6*rr))/28 + (aa - rr)/70 + rr + (2*aa + 6*rr)/140))/129, (70*((3*(-8*bb - 12*rr))/140 + (8*bb - 12*rr)/14 + (-4*bb + 2*rr)/14 + (4*bb + 2*rr)/2))/129, (52*rr)/43, 129/70}, {(10*((-7*aa)/5 + (3*(2*aa - 6*rr))/14 + (aa - rr)/35 + rr + (aa + rr)/70 + (2*aa + 6*rr)/70))/23, (10*((3*(-8*bb - 12*rr))/70 + (8*bb - 12*rr)/7 + (3*(-4*bb + 2*rr))/35 + (18*(4*bb + 2*rr))/35))/23, (176*rr)/161, 23/10}, {(14*((-8*aa)/7 + (aa - rr)/14 + rr))/25, (14*((5*(-4*bb + 2*rr))/28 + (5*(4*bb + 2*rr))/8))/25, 0, 25/14}, {(14*((-8*aa)/7 + (aa - rr)/14 + rr))/25, (14*((5*(-4*bb + 2*rr))/28 + (5*(4*bb + 2*rr))/8))/25, (44*rr)/75, 25/14}\ , {(42*((-10*aa)/7 + (2*aa - 6*rr)/28 + (aa - rr)/14 + rr + (2*aa + 6*rr)/84))/83, (42*((-8*bb - 12*rr)/56 + (5*(8*bb - 12*rr))/168 + (5*(-4*bb + 2*rr))/28 + (5*(4*bb + 2*rr))/8))/83, (88*rr)/83, 83/42}\ , {(14*(-2*aa + (3*(2*aa - 6*rr))/28 + (aa - rr)/14 + rr + (2*aa + 6*rr)/28))/33, (14*((3*(-8*bb - 12*rr))/56 + (5*(8*bb - 12*rr))/56 + (5*(-4*bb + 2*rr))/28 + (5*(4*bb + 2*rr))/8))/33, (40*rr)/33, 33/14}\ , {(7*((-31*aa)/14 + (3*(aa - rr))/14 + rr))/16, (7*((5*(-4*bb + 2*rr))/14 + (3*(4*bb + 2*rr))/4))/16, 0, 16/7}, {(7*((-31*aa)/14 + (3*(aa - rr))/14 + rr))/16, (7*((5*(-4*bb + 2*rr))/14 + (3*(4*bb + 2*rr))/4))/16, (5*rr)/8, 16/7}, {(7*((-37*aa)/14 + (2*aa - 6*rr)/28 + (3*(aa - rr))/14 + rr + (2*aa + 6*rr)/28))/18, (7*((-8*bb - 12*rr)/28 + (8*bb - 12*rr)/28 + (5*(-4*bb + 2*rr))/14 + (3*(4*bb + 2*rr))/4))/18, (10*rr)/9, 18/7}, {((-7*aa)/2 + (aa - rr)/2 + rr)/3, ((5*(-4*bb + 2*rr))/8 + (7*(4*bb + 2*rr))/8)/3, 0, 3}, {((-7*aa)/2 + (aa - rr)/2 + rr)/3, ((5*(-4*bb + 2*rr))/8 + (7*(4*bb + 2*rr))/8)/3, (2*rr)/3, 3}, {-aa, rr, 0, 4}}; (* Case rr = 1; aa = 4; bb = 1 *) rr = 1; aa = 4; bb = 1; klein = {{5, 0, 0, 1}, {5, 0, 1/2, 1}, {71/15, 0, 14/15, 15/14}, {73/17, 0, 20/17, 17/14}, {385/101, 0, 120/101, 101/70}, {17/5, 0, 1, 25/14}, {25/8, 0, 11/16, 16/7}, {3, 0, 1/3, 3}, {3, 0, 0, 4}, {5, 3/4, 0, 1}, {5, 3/4, 1/2, 1}, {71/15, 61/90, 14/15, 15/14}, {73/17, 19/34, 20/17, 17/14}, {385/101, 44/101, 120/101, 101/70}, {17/5, 26/75, 1, 25/14}, {25/8, 5/16, 11/16, 16/7}, {3, 1/3, 1/3, 3}, {58/15, 7/5, 0, 15/14}, {58/15, 7/5, 23/45, 15/14}, {861/242, 305/242, 115/121, 121/105}, {281/92, 95/92, 109/92, 46/35}, {819/331, 264/331, 388/331, 331/210}, {159/83, 52/83, 79/83, 83/42}, {13/9, 5/9, 11/18, 18/7}, {2, 31/17, 0, 17/14}, {2, 31/17, 9/17, 17/14}, {151/92, 295/184, 45/46, 46/35}, {113/106, 5/4, 127/106, 53/35}, {49/129, 38/43, 148/129, 129/70}, {-1/3, 20/33, 29/33, 33/14}, {-7/101, 200/101, 0, 101/70}, {-7/101, 200/101, 56/101, 101/70}, {-143/331, 550/331, 336/331, 331/210}, {-1, 50/43, 52/43, 129/70}, {-267/161, 104/161, 176/161, 23/10}, {-47/25, 19/10, 0, 25/14}, {-47/25, 19/10, 44/75, 25/14}, {-179/83, 245/166, 88/83, 83/42}, {-85/33, 5/6, 40/33, 33/14}, {-101/32, 53/32, 0, 16/7}, {-101/32, 53/32, 5/8, 16/7}, {-13/4, 41/36, 10/9, 18/7}, {-23/6, 4/3, 0, 3}, {-23/6, 4/3, 2/3, 3}, {-4, 1, 0, 4}}; sqrender[klein,8,1,1,-1,-1,2,0,0,-5.1,5.1,-2.1,2.1,-2.1,2.1,0] trender[klein,8,reftrig1,2,0,0,-5.1,5.1,-2.1,2.1,-2.1,2.1,0] (* Case rr = 2; aa = 6; bb = 3 *) rr = 2; aa = 6; bb = 3; klein = {{8, 0, 0, 1}, {8, 0, 1, 1}, {112/15, 0, 28/15, 15/14}, {112/17, 0, 40/17, 17/14}, {568/101, 0, 240/101, 101/70}, {24/5, 0, 2, 25/14}, {17/4, 0, 11/8, 16/7}, {4, 0, 2/3, 3}, {4, 0, 0, 4}, {8, 2, 0, 1}, {8, 2, 1, 1}, {112/15, 28/15, 28/15, 15/14}, {112/17, 28/17, 40/17, 17/14}, {568/101, 144/101, 240/101, 101/70}, {24/5, 32/25, 2, 25/14}, {17/4, 5/4, 11/8, 16/7}, {4, 4/3, 2/3, 3}, {94/15, 56/15, 0, 15/14}, {94/15, 56/15, 46/45, 15/14}, {687/121, 420/121, 230/121, 121/105}, {217/46, 70/23, 109/46, 46/35}, {1200/331, 864/331, 776/331, 331/210}, {216/83, 192/83, 158/83, 83/42}, {16/9, 20/9, 11/9, 18/7}, {58/17, 82/17, 0, 17/14}, {58/17, 82/17, 18/17, 17/14}, {127/46, 101/23, 45/23, 46/35}, {91/53, 196/53, 127/53, 53/35}, {64/129, 128/43, 296/129, 129/70}, {-8/11, 80/33, 58/33, 33/14}, {24/101, 520/101, 0, 101/70}, {24/101, 520/101, 112/101, 101/70}, {-132/331, 1488/331, 672/331, 331/210}, {-60/43, 448/129, 104/43, 129/70}, {-408/161, 384/161, 352/161, 23/10}, {-64/25, 24/5, 0, 25/14}, {-64/25, 24/5, 88/75, 25/14}, {-252/83, 324/83, 176/83, 83/42}, {-124/33, 28/11, 80/33, 33/14}, {-73/16, 4, 0, 16/7}, {-73/16, 4, 5/4, 16/7}, {-85/18, 26/9, 20/9, 18/7}, {-17/3, 3, 0, 3}, {-17/3, 3, 4/3, 3}, {-6, 2, 0, 4}}; (* Case rr = 2; aa = 6; bb = 3 *) sqrender[klein,8,1,1,-1,-1,2,0,0,-8.1,8.1,-5.1,5.1,-2.2,2.2,0] (* great *) (* Also very good viewpoint: ViewPoint -> {-15, -10, 5} *) (* Show[%835, ViewPoint -> {-12, -8, 2}] *) (* Show[%12, ViewPoint -> {-12, -5, 2}] *) sqrender[klein,8,1,0,-1,-1,2,0,0,-8.1,8.1,-5.1,5.1,-2.2,2.2,0] (* cute, half klein *) trender[klein,8,reftrig1,2,0,0,-8.1,8.1,-5.1,5.1,-2.2,2.2,0] (* cute *) render1[klein,8,2,0,0,-8.1,8.1,-5.1,5.1,-2.2,2.2,0] (* cute *) sqfractal[klein,8,1,1,-1,-1,2,0,0,-8.1,8.1,-5.1,5.1,-2.2,2.2] sqfractal[klein,8,1,1,-1,-1,3,0,0,-8.1,8.1,-5.1,5.1,-2.2,2.2] pta = {1, 1, -1}; ptb = {-1, -1, 3}; scurv3D[klein,8,5,pta,ptb,-8.1,8.1,-5.1,5.1,-2.2,2.2,0] clocurv3D[klein,8,5,pta,ptb,-8.1,8.1,-5.1,5.1,-2.2,2.2,0] (* very nice *) (* Experiencing with Klein bottle , degree 8 (from Do Carmo) *) p1 = ((aa + rr)*v^4 + 2*(aa - 3*rr)*v^2 + aa + rr)*(u^4 -6*u^2 + 1); p2 = ((aa + rr)*v^4 + 2*(aa - 3*rr)*v^2 + aa + rr)*4*u*(1 - u^2); p3 = 4*rr*v*(1 - v^2)*(1 - u^4); p4 = 8*rr*u*v*(1 - v^2)*(1 + u^2); p5 = (u^2 + 1)^2*(v^2 + 1)^2; p1 = Expand[p1]; p1 = InputForm[p1] p2 = Expand[p2]; p2 = InputForm[p2] p3 = Expand[p3]; p3 = InputForm[p3] p4 = Expand[p4]; p4 = InputForm[p4] p5 = Expand[p5]; p5 = InputForm[p5] p1 = aa + rr - 6*aa*u^2 - 6*rr*u^2 + aa*u^4 + rr*u^4 + 2*aa*v^2 - 6*rr*v^2 - 12*aa*u^2*v^2 + 36*rr*u^2*v^2 + 2*aa*u^4*v^2 - 6*rr*u^4*v^2 + aa*v^4 + rr*v^4 - 6*aa*u^2*v^4 - 6*rr*u^2*v^4 + aa*u^4*v^4 + rr*u^4*v^4 p2 = 4*aa*u + 4*rr*u - 4*aa*u^3 - 4*rr*u^3 + 8*aa*u*v^2 - 24*rr*u*v^2 - 8*aa*u^3*v^2 + 24*rr*u^3*v^2 + 4*aa*u*v^4 + 4*rr*u*v^4 - 4*aa*u^3*v^4 - 4*rr*u^3*v^4 p3 = 4*rr*v - 4*rr*u^4*v - 4*rr*v^3 + 4*rr*u^4*v^3; p4 = 8*rr*u*v + 8*rr*u^3*v - 8*rr*u*v^3 - 8*rr*u^3*v^3; p5 = u^4*v^4 + 2*u^4*v^2 + u^4 + 2*u^2*v^4 + 4*u^2*v^2 + 2*u^2 + v^4 + 2*v^2 + 1; (* projection along x axis *) polyx = {{{4*aa + 4*rr, 1, 0}, {-4*aa - 4*rr, 3, 0}, {8*aa - 24*rr, 1, 2}, {-8*aa + 24*rr, 3, 2}, {4*aa + 4*rr, 1, 4}, {-4*aa - 4*rr, 3, 4}}, {{4*rr, 0, 1}, {-4*rr, 4, 1}, {-4*rr, 0, 3}, {4*rr, 4, 3}}, {{8*rr, 1, 1}, {8*rr, 3, 1}, {-8*rr, 1, 3}, {-8*rr, 3, 3}}, {{1, 4, 4}, {2, 4, 2}, {1, 4, 0}, {2, 2, 4}, {4, 2, 2}, {2, 2, 0}, {1, 0, 4}, {2, 0, 2}, {1, 0, 0}}}; p1 = aa + rr - 6*aa*u^2 - 6*rr*u^2 + aa*u^4 + rr*u^4 + 2*aa*v^2 - 6*rr*v^2 - 12*aa*u^2*v^2 + 36*rr*u^2*v^2 + 2*aa*u^4*v^2 - 6*rr*u^4*v^2 + aa*v^4 + rr*v^4 - 6*aa*u^2*v^4 - 6*rr*u^2*v^4 + aa*u^4*v^4 + rr*u^4*v^4 (* projection along y axis *) polyy = {{{aa + rr, 0, 0}, {-6*aa - 6*rr, 2, 0}, {aa + rr, 4, 0}, {2*aa - 6*rr, 0, 2}, {-12*aa + 36*rr, 2, 2}, {2*aa - 6*rr, 4, 2}, {aa + rr, 0, 4}, {-6*aa - 6*rr, 2, 4}, {aa + rr, 4, 4}}, {{4*rr, 0, 1}, {-4*rr, 4, 1}, {-4*rr, 0, 3}, {4*rr, 4, 3}}, {{8*rr, 1, 1}, {8*rr, 3, 1}, {-8*rr, 1, 3}, {-8*rr, 3, 3}}, {{1, 4, 4}, {2, 4, 2}, {1, 4, 0}, {2, 2, 4}, {4, 2, 2}, {2, 2, 0}, {1, 0, 4}, {2, 0, 2}, {1, 0, 0}}}; (* projection along z axis *) polyz = {{{aa + rr, 0, 0}, {-6*aa - 6*rr, 2, 0}, {aa + rr, 4, 0}, {2*aa - 6*rr, 0, 2}, {-12*aa + 36*rr, 2, 2}, {2*aa - 6*rr, 4, 2}, {aa + rr, 0, 4}, {-6*aa - 6*rr, 2, 4}, {aa + rr, 4, 4}}, {{4*aa + 4*rr, 1, 0}, {-4*aa - 4*rr, 3, 0}, {8*aa - 24*rr, 1, 2}, {-8*aa + 24*rr, 3, 2}, {4*aa + 4*rr, 1, 4}, {-4*aa - 4*rr, 3, 4}}, {{8*rr, 1, 1}, {8*rr, 3, 1}, {-8*rr, 1, 3}, {-8*rr, 3, 3}}, {{1, 4, 4}, {2, 4, 2}, {1, 4, 0}, {2, 2, 4}, {4, 2, 2}, {2, 2, 0}, {1, 0, 4}, {2, 0, 2}, {1, 0, 0}}}; (* projection along t axis *) polyt = {{{aa + rr, 0, 0}, {-6*aa - 6*rr, 2, 0}, {aa + rr, 4, 0}, {2*aa - 6*rr, 0, 2}, {-12*aa + 36*rr, 2, 2}, {2*aa - 6*rr, 4, 2}, {aa + rr, 0, 4}, {-6*aa - 6*rr, 2, 4}, {aa + rr, 4, 4}}, {{4*aa + 4*rr, 1, 0}, {-4*aa - 4*rr, 3, 0}, {8*aa - 24*rr, 1, 2}, {-8*aa + 24*rr, 3, 2}, {4*aa + 4*rr, 1, 4}, {-4*aa - 4*rr, 3, 4}}, {{4*rr, 0, 1}, {-4*rr, 4, 1}, {-4*rr, 0, 3}, {4*rr, 4, 3}}, {{1, 4, 4}, {2, 4, 2}, {1, 4, 0}, {2, 2, 4}, {4, 2, 2}, {2, 2, 0}, {1, 0, 4}, {2, 0, 2}, {1, 0, 0}}}; netx = contnet[polyx, 8]; kleinetx = InputForm[netx]; kleinetx = {{0, 0, 0, 1}, {0, rr/2, 0, 1}, {0, (14*rr)/15, 0, 15/14}, {0, (20*rr)/17, 0, 17/14}, {0, (120*rr)/101, 0, 101/70}, {0, rr, 0, 25/14}, {0, (11*rr)/16, 0, 16/7}, {0, rr/3, 0, 3}, {0, 0, 0, 4}, {(4*aa + 4*rr)/8, 0, 0, 1}, {(4*aa + 4*rr)/8, rr/2, rr/7, 1}, {(14*((8*aa - 24*rr)/168 + (4*aa + 4*rr)/8))/15, (14*rr)/15, (4*rr)/15, 15/14}, {(14*((8*aa - 24*rr)/56 + (4*aa + 4*rr)/8))/17, (20*rr)/17, (28*rr)/85, 17/14}, {(70*((8*aa - 24*rr)/28 + (9*(4*aa + 4*rr))/70))/ 101, (120*rr)/101, (32*rr)/101, 101/70}, {(14*((5*(8*aa - 24*rr))/84 + (4*aa + 4*rr)/7))/25, rr, (6*rr)/25, 25/14}, {(7*((5*(8*aa - 24*rr))/56 + (5*(4*aa + 4*rr))/28))/16, (11*rr)/16, rr/8, 16/7}, {((8*aa - 24*rr)/8 + (4*aa + 4*rr)/4)/3, rr/3, 0, 3}, {(7*(4*aa + 4*rr))/30, 0, 0, 15/14}, {(7*(4*aa + 4*rr))/30, (7*rr)/15, (4*rr)/15, 15/14}, {(105*((8*aa - 24*rr)/84 + (4*aa + 4*rr)/4))/121, (105*rr)/121, (60*rr)/121, 121/105}, {(35*((8*aa - 24*rr)/28 + (4*aa + 4*rr)/4))/46, (25*rr)/23, (14*rr)/23, 46/35}, {(210*((8*aa - 24*rr)/14 + (9*(4*aa + 4*rr))/35))/331, (360*rr)/331, (192*rr)/331, 331/210}, {(42* ((5*(8*aa - 24*rr))/42 + (2*(4*aa + 4*rr))/7))/83, (75*rr)/83, (36*rr)/83, 83/42}, {(7*((5*(8*aa - 24*rr))/28 + (5*(4*aa + 4*rr))/14))/ 18, (11*rr)/18, (2*rr)/9, 18/7}, {(14*((-4*aa - 4*rr)/56 + (3*(4*aa + 4*rr))/8))/17, 0, 0, 17/14}, {(14*((-4*aa - 4*rr)/56 + (3*(4*aa + 4*rr))/8))/17, (7*rr)/17, (32*rr)/85, 17/14}, {(35*((8*aa - 24*rr)/56 + (-4*aa - 4*rr)/56 + (3*(4*aa + 4*rr))/8 + (-8*aa + 24*rr)/560))/46, (35*rr)/46, (16*rr)/23, 46/35}, {(35*((3*(8*aa - 24*rr))/56 + (-4*aa - 4*rr)/56 + (3*(4*aa + 4*rr))/8 + (3*(-8*aa + 24*rr))/560))/53, (50*rr)/53, (89*rr)/106, 53/35}, {(70*((3*(8*aa - 24*rr))/28 + (3*(-4*aa - 4*rr))/140 + (27*(4*aa + 4*rr))/70 + (3*(-8*aa + 24*rr))/280))/129, (40*rr)/43, (100*rr)/129, 129/70}, {(14*((5*(8*aa - 24*rr))/28 + (-4*aa - 4*rr)/28 + (3*(4*aa + 4*rr))/7 + (-8*aa + 24*rr)/56))/33, (25*rr)/33, (6*rr)/11, 33/14}, {(70*((-4*aa - 4*rr)/14 + (4*aa + 4*rr)/2))/101, 0, 0, 101/70}, {(70*((-4*aa - 4*rr)/14 + (4*aa + 4*rr)/2))/101, (34*rr)/101, (48*rr)/101, 101/70}, {(210* ((8*aa - 24*rr)/42 + (-4*aa - 4*rr)/14 + (4*aa + 4*rr)/2 + (-8*aa + 24*rr)/140))/331, (204*rr)/331, (288*rr)/331, 331/210}, {(70*((8*aa - 24*rr)/14 + (-4*aa - 4*rr)/14 + (4*aa + 4*rr)/2 + (3*(-8*aa + 24*rr))/140))/129, (98*rr)/129, (44*rr)/43, 129/70}, {(10*((8*aa - 24*rr)/7 + (3*(-4*aa - 4*rr))/35 + (18*(4*aa + 4*rr))/35 + (3*(-8*aa + 24*rr))/70))/23, (120*rr)/161, (144*rr)/161, 23/10}, {(14* ((5*(-4*aa - 4*rr))/28 + (5*(4*aa + 4*rr))/8))/25, 0, 0, 25/14}, {(14*((5*(-4*aa - 4*rr))/28 + (5*(4*aa + 4*rr))/8))/25, (6*rr)/25, (14*rr)/25, 25/14}, {(42*((5*(8*aa - 24*rr))/168 + (5*(-4*aa - 4*rr))/28 + (5*(4*aa + 4*rr))/8 + (-8*aa + 24*rr)/56))/ 83, (36*rr)/83, (84*rr)/83, 83/42}, {(14*((5*(8*aa - 24*rr))/56 + (5*(-4*aa - 4*rr))/28 + (5*(4*aa + 4*rr))/8 + (3*(-8*aa + 24*rr))/56))/33, (6*rr)/11, (38*rr)/33, 33/14}, {(7*((5*(-4*aa - 4*rr))/14 + (3*(4*aa + 4*rr))/4))/ 16, 0, 0, 16/7}, {(7*((5*(-4*aa - 4*rr))/14 + (3*(4*aa + 4*rr))/4))/16, rr/8, (5*rr)/8, 16/7}, {(7* ((8*aa - 24*rr)/28 + (5*(-4*aa - 4*rr))/14 + (3*(4*aa + 4*rr))/4 + (-8*aa + 24*rr)/28))/18, (2*rr)/9, (10*rr)/9, 18/7}, {((5*(-4*aa - 4*rr))/8 + (7*(4*aa + 4*rr))/8)/3, 0, 0, 3}, {((5*(-4*aa - 4*rr))/8 + (7*(4*aa + 4*rr))/8)/3, 0, (2*rr)/3, 3}, {0, 0, 0, 4}}; (* Case rr = 1, aa = 2 *) kleinx = {{0, 0, 0, 1}, {0, 1/2, 0, 1}, {0, 14/15, 0, 15/14}, {0, 20/17, 0, 17/14}, {0, 120/101, 0, 101/70}, {0, 1, 0, 25/14}, {0, 11/16, 0, 16/7}, {0, 1/3, 0, 3}, {0, 0, 0, 4}, {3/2, 0, 0, 1}, {3/2, 1/2, 1/7, 1}, {61/45, 14/15, 4/15, 15/14}, {19/17, 20/17, 28/85, 17/14}, {88/101, 120/101, 32/101, 101/70}, {52/75, 1, 6/25, 25/14}, {5/8, 11/16, 1/8, 16/7}, {2/3, 1/3, 0, 3}, {14/5, 0, 0, 15/14}, {14/5, 7/15, 4/15, 15/14}, {305/121, 105/121, 60/121, 121/105}, {95/46, 25/23, 14/23, 46/35}, {528/331, 360/331, 192/331, 331/210}, {104/83, 75/83, 36/83, 83/42}, {10/9, 11/18, 2/9, 18/7}, {60/17, 0, 0, 17/14}, {60/17, 7/17, 32/85, 17/14}, {291/92, 35/46, 16/23, 46/35}, {273/106, 50/53, 89/106, 53/35}, {84/43, 40/43, 100/129, 129/70}, {16/11, 25/33, 6/11, 33/14}, {360/101, 0, 0, 101/70}, {360/101, 34/101, 48/101, 101/70}, {1052/331, 204/331, 288/331, 331/210}, {332/129, 98/129, 44/43, 129/70}, {304/161, 120/161, 144/161, 23/10}, {3, 0, 0, 25/14}, {3, 6/25, 14/25, 25/14}, {221/83, 36/83, 84/83, 83/42}, {71/33, 6/11, 38/33, 33/14}, {33/16, 0, 0, 16/7}, {33/16, 1/8, 5/8, 16/7}, {11/6, 2/9, 10/9, 18/7}, {1, 0, 0, 3}, {1, 0, 2/3, 3}, {0, 0, 0, 4}}; rendersq[kleinx,8,-1,1,-1,1,2,0,0,-3.1,3.1,-1.1,1.1,-1.1,1.1,0] rendersq[kleinx,8,-1,1,-1,1,2,0,0,-2,2,-1.1,1.1,-1.1,1.1,0] rendersq[kleinx,8,-1,1,-1,1,2,0,0,-1,2,-1.1,1.1,-1.1,1.1,0] rendersq[kleinx,8,-1,1,-1,1,2,0,0,0,2,-1.1,1.1,-1.1,1.1,0] rendersq[kleinx,8,0,1,-1,1,2,2,0,-3.1,3.1,-1.1,1.1,-1.1,1.1,0] rendersq[kleinx,8,-1,0,-1,1,2,2,0,-3.1,3.1,-1.1,1.1,-1.1,1.1,0] rendersq[kleinx,8,-1,1,0,1,2,0,0,-3.1,3.1,-1.1,1.1,-1.1,1.1,0] rendersq[kleinx,8,-1,1,-1,0,2,0,0,-3.1,3.1,-1.1,1.1,-1.1,1.1,0] nety = contnet[polyy, 8]; kleinety = InputForm[nety]; kleinety = {{aa + rr, 0, 0, 1}, {aa + rr, rr/2, 0, 1}, {(14*(aa + (2*aa - 6*rr)/28 + rr))/15, (14*rr)/15, 0, 15/14}, {(14*(aa + (3*(2*aa - 6*rr))/28 + rr))/17, (20*rr)/17, 0, 17/14}, {(70*(aa + (3*(2*aa - 6*rr))/14 + rr + (aa + rr)/70))/101, (120*rr)/101, 0, 101/70}, {(14*(aa + (5*(2*aa - 6*rr))/14 + rr + (aa + rr)/14))/25, rr, 0, 25/14}, {(7*(aa + (15*(2*aa - 6*rr))/28 + rr + (3*(aa + rr))/14))/16, (11*rr)/16, 0, 16/7}, {(aa + (3*(2*aa - 6*rr))/4 + rr + (aa + rr)/2)/3, rr/3, 0, 3}, {(4*aa - 4*rr)/4, 0, 0, 4}, {aa + rr, 0, 0, 1}, {aa + rr, rr/2, rr/7, 1}, {(14*(aa + (2*aa - 6*rr)/28 + rr))/15, (14*rr)/15, (4*rr)/15, 15/14}, {(14*(aa + (3*(2*aa - 6*rr))/28 + rr))/17, (20*rr)/17, (28*rr)/85, 17/14}, {(70*(aa + (3*(2*aa - 6*rr))/14 + rr + (aa + rr)/70))/101, (120*rr)/101, (32*rr)/101, 101/70}, {(14* (aa + (5*(2*aa - 6*rr))/14 + rr + (aa + rr)/14))/25, rr, (6*rr)/25, 25/14}, {(7*(aa + (15*(2*aa - 6*rr))/28 + rr + (3*(aa + rr))/14))/16, (11*rr)/16, rr/8, 16/7}, {(aa + (3*(2*aa - 6*rr))/4 + rr + (aa + rr)/2)/ 3, rr/3, 0, 3}, {(14*(aa + (-6*aa - 6*rr)/28 + rr))/15, 0, 0, 15/14}, {(14*(aa + (-6*aa - 6*rr)/28 + rr))/15, (7*rr)/15, (4*rr)/15, 15/14}, {(105*(aa + (-6*aa - 6*rr)/28 + (2*aa - 6*rr)/28 + rr + (-12*aa + 36*rr)/420))/121, (105*rr)/121, (60*rr)/121, 121/105}, {(35*(aa + (-6*aa - 6*rr)/28 + (3*(2*aa - 6*rr))/28 + rr + (-12*aa + 36*rr)/140))/46, (25*rr)/23, (14*rr)/23, 46/35}, {(210*(aa + (4*(-6*aa - 6*rr))/105 + (3*(2*aa - 6*rr))/14 + rr + (aa + rr)/70 + (-12*aa + 36*rr)/70))/331, (360*rr)/331, (192*rr)/331, 331/210}, {(42* (aa + (-6*aa - 6*rr)/21 + (5*(2*aa - 6*rr))/14 + rr + (aa + rr)/14 + (-12*aa + 36*rr)/42))/83, (75*rr)/83, (36*rr)/83, 83/42}, {(7*(aa + (-6*aa - 6*rr)/14 + (15*(2*aa - 6*rr))/28 + rr + (3*(aa + rr))/14 + (-12*aa + 36*rr)/28))/18, (11*rr)/18, (2*rr)/9, 18/7}, {(14*(aa + (3*(-6*aa - 6*rr))/28 + rr))/17, 0, 0, 17/14}, {(14*(aa + (3*(-6*aa - 6*rr))/28 + rr))/17, (7*rr)/17, (32*rr)/85, 17/14}, {(35*(aa + (3*(-6*aa - 6*rr))/28 + (2*aa - 6*rr)/28 + rr + (-12*aa + 36*rr)/140))/46, (35*rr)/46, (16*rr)/23, 46/35}, {(35*(aa + (3*(-6*aa - 6*rr))/28 + (3*(2*aa - 6*rr))/28 + rr + (3*(-12*aa + 36*rr))/140))/53, (50*rr)/53, (89*rr)/106, 53/35}, {(70*(aa + (4*(-6*aa - 6*rr))/35 + (3*(2*aa - 6*rr))/14 + rr + (aa + rr)/70 + (3*(-12*aa + 36*rr))/70))/129, (40*rr)/43, (100*rr)/129, 129/70}, {(14* (aa + (-6*aa - 6*rr)/7 + (5*(2*aa - 6*rr))/14 + rr + (aa + rr)/14 + (-12*aa + 36*rr)/14))/33, (25*rr)/33, (6*rr)/11, 33/14}, {(70*(aa + (3*(-6*aa - 6*rr))/14 + rr + (aa + rr)/70))/101, 0, 0, 101/70}, {(70*(aa + (3*(-6*aa - 6*rr))/14 + rr + (aa + rr)/70))/101, (34*rr)/101, (48*rr)/101, 101/70}, {(210* (aa + (3*(-6*aa - 6*rr))/14 + (4*(2*aa - 6*rr))/105 + rr + (aa + rr)/70 + (-12*aa + 36*rr)/70))/331, (204*rr)/331, (288*rr)/331, 331/210}, {(70* (aa + (3*(-6*aa - 6*rr))/14 + (4*(2*aa - 6*rr))/35 + rr + (aa + rr)/70 + (3*(-12*aa + 36*rr))/70))/129, (98*rr)/129, (44*rr)/43, 129/70}, {(10*(aa + (8*(-6*aa - 6*rr))/35 + (8*(2*aa - 6*rr))/35 + rr + (3*(aa + rr))/70 + (3*(-12*aa + 36*rr))/35))/23, (120*rr)/161, (144*rr)/161, 23/10}, {(14*(aa + (5*(-6*aa - 6*rr))/14 + rr + (aa + rr)/14))/25, 0, 0, 25/14}, {(14*(aa + (5*(-6*aa - 6*rr))/14 + rr + (aa + rr)/14))/25, (6*rr)/25, (14*rr)/25, 25/14}, {(42*(aa + (5*(-6*aa - 6*rr))/14 + (2*aa - 6*rr)/21 + rr + (aa + rr)/14 + (-12*aa + 36*rr)/42))/83, (36*rr)/83, (84*rr)/83, 83/42}, {(14*(aa + (5*(-6*aa - 6*rr))/14 + (2*aa - 6*rr)/7 + rr + (aa + rr)/14 + (-12*aa + 36*rr)/14))/33, (6*rr)/11, (38*rr)/33, 33/14}, {(7*(aa + (15*(-6*aa - 6*rr))/28 + rr + (3*(aa + rr))/14))/16, 0, 0, 16/7}, {(7*(aa + (15*(-6*aa - 6*rr))/28 + rr + (3*(aa + rr))/14))/ 16, rr/8, (5*rr)/8, 16/7}, {(7*(aa + (15*(-6*aa - 6*rr))/28 + (2*aa - 6*rr)/14 + rr + (3*(aa + rr))/14 + (-12*aa + 36*rr)/28))/18, (2*rr)/9, (10*rr)/9, 18/7}, {(aa + (3*(-6*aa - 6*rr))/4 + rr + (aa + rr)/2)/3, 0, 0, 3}, {(aa + (3*(-6*aa - 6*rr))/4 + rr + (aa + rr)/2)/3, 0, (2*rr)/3, 3}, {(-4*aa - 4*rr)/4, 0, 0, 4}} (* Case rr = 1, aa = 2 *) kleiny = {{3, 0, 0, 1}, {3, 1/2, 0, 1}, {41/15, 14/15, 0, 15/14}, {39/17, 20/17, 0, 17/14}, {183/101, 120/101, 0, 101/70}, {7/5, 1, 0, 25/14}, {9/8, 11/16, 0, 16/7}, {1, 1/3, 0, 3}, {1, 0, 0, 4}, {3, 0, 0, 1}, {3, 1/2, 1/7, 1}, {41/15, 14/15, 4/15, 15/14}, {39/17, 20/17, 28/85, 17/14}, {183/101, 120/101, 32/101, 101/70}, {7/5, 1, 6/25, 25/14}, {9/8, 11/16, 1/8, 16/7}, {1, 1/3, 0, 3}, {11/5, 0, 0, 15/14}, {11/5, 7/15, 4/15, 15/14}, {243/121, 105/121, 60/121, 121/105}, {39/23, 25/23, 14/23, 46/35}, {441/331, 360/331, 192/331, 331/210}, {81/83, 75/83, 36/83, 83/42}, {2/3, 11/18, 2/9, 18/7}, {15/17, 0, 0, 17/14}, {15/17, 7/17, 32/85, 17/14}, {19/23, 35/46, 16/23, 46/35}, {39/53, 50/53, 89/106, 53/35}, {25/43, 40/43, 100/129, 129/70}, {1/3, 25/33, 6/11, 33/14}, {-57/101, 0, 0, 101/70}, {-57/101, 34/101, 48/101, 101/70}, {-151/331, 204/331, 288/331, 331/210}, {-37/129, 98/129, 44/43, 129/70}, {-29/161, 120/161, 144/161, 23/10}, {-9/5, 0, 0, 25/14}, {-9/5, 6/25, 14/25, 25/14}, {-127/83, 36/83, 84/83, 83/42}, {-37/33, 6/11, 38/33, 33/14}, {-21/8, 0, 0, 16/7}, {-21/8, 1/8, 5/8, 16/7}, {-20/9, 2/9, 10/9, 18/7}, {-3, 0, 0, 3}, {-3, 0, 2/3, 3}, {-3, 0, 0, 4}}; rendersq[kleiny,8,-1,1,-1,1,2,0,0,-3.1,3.1,-1.1,1.1,-1.1,1.1,0] (* very nice *) rendersq[kleiny,8,-1,1,-1,1,2,2,0,-3.1,3.1,-1.1,1.1,-1.1,1.1,0] rendersq[kleiny,8,-1,1,-1,1,2,0,0,-0.1,3.1,-1.1,1.1,-1.1,1.1,0] rendersq[kleiny,8,0,1,-1,1,2,0,0,-3.1,3.1,-1.1,1.1,-1.1,1.1,0] rendersq[kleiny,8,-1,0,-1,1,2,0,0,-3.1,3.1,-1.1,1.1,-1.1,1.1,0] rendersq[kleiny,8,-1,1,0,1,2,0,0,-3.1,3.1,-1.1,1.1,-1.1,1.1,0] rendersq[kleiny,8,-1,1,-1,0,2,0,0,-3.1,3.1,-1.1,1.1,-1.1,1.1,0] netz = contnet[polyz, 8]; kleinetz = InputForm[netz]; kleinetz = {{aa + rr, 0, 0, 1}, {aa + rr, 0, 0, 1}, {(14*(aa + (2*aa - 6*rr)/28 + rr))/15, 0, 0, 15/14}, {(14*(aa + (3*(2*aa - 6*rr))/28 + rr))/17, 0, 0, 17/14}, {(70*(aa + (3*(2*aa - 6*rr))/14 + rr + (aa + rr)/70))/101, 0, 0, 101/70}, {(14*(aa + (5*(2*aa - 6*rr))/14 + rr + (aa + rr)/14))/25, 0, 0, 25/14}, {(7*(aa + (15*(2*aa - 6*rr))/28 + rr + (3*(aa + rr))/14))/16, 0, 0, 16/7}, {(aa + (3*(2*aa - 6*rr))/4 + rr + (aa + rr)/2)/3, 0, 0, 3}, {(4*aa - 4*rr)/4, 0, 0, 4}, {aa + rr, (4*aa + 4*rr)/8, 0, 1}, {aa + rr, (4*aa + 4*rr)/8, rr/7, 1}, {(14*(aa + (2*aa - 6*rr)/28 + rr))/15, (14*((8*aa - 24*rr)/168 + (4*aa + 4*rr)/8))/15, (4*rr)/15, 15/14}, {(14*(aa + (3*(2*aa - 6*rr))/28 + rr))/17, (14*((8*aa - 24*rr)/56 + (4*aa + 4*rr)/8))/17, (28*rr)/85, 17/14}, {(70*(aa + (3*(2*aa - 6*rr))/14 + rr + (aa + rr)/70))/101, (70*((8*aa - 24*rr)/28 + (9*(4*aa + 4*rr))/70))/101, (32*rr)/101, 101/70}, {(14*(aa + (5*(2*aa - 6*rr))/14 + rr + (aa + rr)/14))/25, (14*((5*(8*aa - 24*rr))/84 + (4*aa + 4*rr)/7))/25, (6*rr)/25, 25/14}, {(7*(aa + (15*(2*aa - 6*rr))/28 + rr + (3*(aa + rr))/14))/16, (7*((5*(8*aa - 24*rr))/56 + (5*(4*aa + 4*rr))/28))/16, rr/8, 16/7}, {(aa + (3*(2*aa - 6*rr))/4 + rr + (aa + rr)/2)/3, ((8*aa - 24*rr)/8 + (4*aa + 4*rr)/4)/3, 0, 3}, {(14*(aa + (-6*aa - 6*rr)/28 + rr))/15, (7*(4*aa + 4*rr))/30, 0, 15/14}, {(14*(aa + (-6*aa - 6*rr)/28 + rr))/15, (7*(4*aa + 4*rr))/30, (4*rr)/15, 15/14}, {(105*(aa + (-6*aa - 6*rr)/28 + (2*aa - 6*rr)/28 + rr + (-12*aa + 36*rr)/420))/121, (105*((8*aa - 24*rr)/84 + (4*aa + 4*rr)/4))/121, (60*rr)/121, 121/105}, {(35*(aa + (-6*aa - 6*rr)/28 + (3*(2*aa - 6*rr))/28 + rr + (-12*aa + 36*rr)/140))/46, (35*((8*aa - 24*rr)/28 + (4*aa + 4*rr)/4))/46, (14*rr)/23, 46/35}, {(210*(aa + (4*(-6*aa - 6*rr))/105 + (3*(2*aa - 6*rr))/14 + rr + (aa + rr)/70 + (-12*aa + 36*rr)/70))/331, (210*((8*aa - 24*rr)/14 + (9*(4*aa + 4*rr))/35))/331, (192*rr)/331, 331/210}, {(42*(aa + (-6*aa - 6*rr)/21 + (5*(2*aa - 6*rr))/14 + rr + (aa + rr)/14 + (-12*aa + 36*rr)/42))/83, (42*((5*(8*aa - 24*rr))/42 + (2*(4*aa + 4*rr))/7))/83, (36*rr)/83, 83/42}, {(7*(aa + (-6*aa - 6*rr)/14 + (15*(2*aa - 6*rr))/28 + rr + (3*(aa + rr))/14 + (-12*aa + 36*rr)/28))/18, (7*((5*(8*aa - 24*rr))/28 + (5*(4*aa + 4*rr))/14))/18, (2*rr)/9, 18/7}, {(14*(aa + (3*(-6*aa - 6*rr))/28 + rr))/17, (14*((-4*aa - 4*rr)/56 + (3*(4*aa + 4*rr))/8))/17, 0, 17/14}, {(14*(aa + (3*(-6*aa - 6*rr))/28 + rr))/17, (14*((-4*aa - 4*rr)/56 + (3*(4*aa + 4*rr))/8))/17, (32*rr)/85, 17/14}, {(35*(aa + (3*(-6*aa - 6*rr))/28 + (2*aa - 6*rr)/28 + rr + (-12*aa + 36*rr)/140))/46, (35*((8*aa - 24*rr)/56 + (-4*aa - 4*rr)/56 + (3*(4*aa + 4*rr))/8 + (-8*aa + 24*rr)/560))/46, (16*rr)/23, 46/35}, {(35*(aa + (3*(-6*aa - 6*rr))/28 + (3*(2*aa - 6*rr))/28 + rr + (3*(-12*aa + 36*rr))/140))/53, (35*((3*(8*aa - 24*rr))/56 + (-4*aa - 4*rr)/56 + (3*(4*aa + 4*rr))/8 + (3*(-8*aa + 24*rr))/560))/53, (89*rr)/106, 53/35}, {(70*(aa + (4*(-6*aa - 6*rr))/35 + (3*(2*aa - 6*rr))/14 + rr + (aa + rr)/70 + (3*(-12*aa + 36*rr))/70))/129, (70*((3*(8*aa - 24*rr))/28 + (3*(-4*aa - 4*rr))/140 + (27*(4*aa + 4*rr))/70 + (3*(-8*aa + 24*rr))/280))/129, (100*rr)/129, 129/70}, {(14* (aa + (-6*aa - 6*rr)/7 + (5*(2*aa - 6*rr))/14 + rr + (aa + rr)/14 + (-12*aa + 36*rr)/14))/33, (14*((5*(8*aa - 24*rr))/28 + (-4*aa - 4*rr)/28 + (3*(4*aa + 4*rr))/7 + (-8*aa + 24*rr)/56))/33, (6*rr)/11, 33/14}, {(70*(aa + (3*(-6*aa - 6*rr))/14 + rr + (aa + rr)/70))/101, (70*((-4*aa - 4*rr)/14 + (4*aa + 4*rr)/2))/101, 0, 101/70}, {(70*(aa + (3*(-6*aa - 6*rr))/14 + rr + (aa + rr)/70))/101, (70*((-4*aa - 4*rr)/14 + (4*aa + 4*rr)/2))/101, (48*rr)/101, 101/70}, {(210*(aa + (3*(-6*aa - 6*rr))/14 + (4*(2*aa - 6*rr))/105 + rr + (aa + rr)/70 + (-12*aa + 36*rr)/70))/331, (210*((8*aa - 24*rr)/42 + (-4*aa - 4*rr)/14 + (4*aa + 4*rr)/2 + (-8*aa + 24*rr)/140))/331, (288*rr)/331, 331/210}, {(70*(aa + (3*(-6*aa - 6*rr))/14 + (4*(2*aa - 6*rr))/35 + rr + (aa + rr)/70 + (3*(-12*aa + 36*rr))/70))/129, (70*((8*aa - 24*rr)/14 + (-4*aa - 4*rr)/14 + (4*aa + 4*rr)/2 + (3*(-8*aa + 24*rr))/140))/129, (44*rr)/43, 129/70}, {(10*(aa + (8*(-6*aa - 6*rr))/35 + (8*(2*aa - 6*rr))/35 + rr + (3*(aa + rr))/70 + (3*(-12*aa + 36*rr))/35))/23, (10*((8*aa - 24*rr)/7 + (3*(-4*aa - 4*rr))/35 + (18*(4*aa + 4*rr))/35 + (3*(-8*aa + 24*rr))/70))/23, (144*rr)/161, 23/10}, {(14*(aa + (5*(-6*aa - 6*rr))/14 + rr + (aa + rr)/14))/25, (14*((5*(-4*aa - 4*rr))/28 + (5*(4*aa + 4*rr))/8))/25, 0,