3.9 3D Shape Interpolation
3.10 Chapter Summary
Chapter Summary 169
is said to be associated with the vertex on the other boundary at the same relative location.
Once the meshes have been associated, each mesh is recursively divided. One mesh is chosen for division, and a path of edges is found across it. Again, there are various ways to do this and the procedure is not dependent on the method chosen for its success. However, the results will have a slightly different quality depending on the method used. One good approach is to choose two vertices across the boundary from each other and try to find an existing path of edges between them.
An iterative procedure can be easily implemented that tries all pairs halfway around and then tries all pairs one less than halfway around, then two less, and so on. There will be no path only if the “mesh” is a single triangle—in which case the other mesh must be tried. There will be some path that exists on one of the two meshes unless both meshes are a single triangle, which is the terminating criterion for the recursion (and would have been previously tested for).
Once a path has been found across one mesh, then a path across the mesh it is associated with must be established between corresponding vertices. This may require creating new vertices and new edges (and, therefore, new faces) and is the trickiest part of the implementation because minimizing the number of new edges will help reduce the complexity of the resulting topologies. When these paths (one on each mesh) have been created, the meshes can be divided along these paths, cre- ating two pairs of new meshes. The boundary association, finding a path of edges, and mesh splitting are recursively applied to each new mesh until all of the meshes have been reduced to single triangles. At this point the new vertices and new edges have been added to one or both objects so that both objects have the same topol- ogy. Vertex-to-vertex interpolation of vertices can take place at this point in order to carry out the object interpolation.
170 3: Interpolation and Basic Techniques
pixel color, and shape parameters. The control of the interpolation process can be key framed, scripted, or analytically determined. But in any case, interpolation forms the foundation upon which most computer animation takes place, includ- ing those animations that use advanced algorithms.
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A
nimators are often more concerned with the general quality of the motion than with precisely controlling the position and orientation of each object in each frame. Such is the case with physical simulations; when dealing with a large num- ber of objects; when animating objects whose motion is constrained in one way or another; or when dealing with objects in the background whose precise motion is not of great importance to the animation. This chapter is concerned with the algo- rithms that employ some kind of structured model approach to producing motion. The structure of the model automatically enforces certain qualities or constraints on the motion to be generated. The use of a model eliminates the need for the animator to be constantly concerned with specifying details of the motion.Instead, those details are filled in by the model. Of course, by using these models, the animator typically loses some fine control over the motion of the objects. The model can take various forms, such as enforcing relative placement of geometric elements, enforcing nonpenetration constraints, calculating reaction to gravity and other forces, enforcing volume preservation, or following rules of behavior.
Motion is produced by the combination of the constraints and rules of the model with additional control information from the user.
This chapter discusses both kinematic and dynamic models. Kinematic control refers to the movement of objects irrespective of the forces involved in producing