Polygons

- For rendering reasons, polygons are often used. Several deformation techniques are associated with polygons:

• Keyframing techniques: [33][41] Libraries of expressions are generated and the control is achieved by interpolation between two expressions.

• Paramatric deformations: [72][103][105] Deformations are grouped under parameters meaningful to the animator. Methods for control are varied, from numbers to data gloves (Section and Section ) Impressive results are obtained by puppeteers.

• Muscle-Based deformations: [107][144][124] Parametric deformations may be organized according to the arrangement of muscles in the face. The polygonal representation may be deformed by a muscle model, often taking the form of superposition of deformation basis functions.

• Interactive deformations: [75] Around 100 control points are selected on the face to control facial expressions. The propagation of the deformation generated by the displacement of these control points is calculated by projecting the face onto a 2D parametric space using a cylindrical projection. Then the 2D space generated by the set of control points is triangulated (Delaunay triangulation) and each intermediate point is recalculated: its displacement is the weighted sum of the displacement of the 3 closest control points.

• Free Form deformations: [65] Each deformation is associated with a deformation box (parallelepiped). The points of the face included in the deformation box are expressed in the coordinate system of the box. The position of the points are updated when the coordinates of the box are displaced. [129]

• Simplex Meshes: [32] Simplex meshes are connected meshes that are topologically dual of triangulation. Elastic behavior is modeled by local stabilizing functionals, controlling the mean curvature through the simplex angle extracted at each vertex. Skull, skin and muscles are represented with simplex meshes, characterized with a constant vertex to vertex connectivity. Furthermore, cutting surface parts may be easily achieved due to the local nature of simplex meshes: muscles attached to both skin and skull adjust the shape to the reconstructed skull.