We present a framework for geometric warps and deformations. The framework provides a conceptual and mathematical foundation for analyzing known warps and for developing new warps, and serves as a common base for many warps and deformations. Our framework is composed of two components: a generic modular algorithm for warps and deformations; and a concise, geometrically meaningful formula that describes how warps are evaluated. Together, these two elements comprise a complete framework useful for analyzing, evaluating, designing, and implementing deformation algorithms. While the framework is independent of user-interfaces and geometric model representations and is formally capable of describing any warping algorithm, its design is geared toward the most prevalent class of user-controlled deformations: those computed using geometric operations. To demonstrate the expressive power of the framework, we cast several well-known warps in terms of the framework. To illustrate the framework's usefulness for analyzing and modifying existing warps, we present variations of these warps that provide additional functionality or improved behavior. To show the utility of the framework for developing new warps, we design a novel 3-D warping algorithm: a mesh warp ---useful as a modeling and animation tool---that allows users to deform a detailed surface by manipulating a low-resolution mesh of similar shape. Finally, to demonstrate the mathematical utility of the framework, we use the framework to develop guarantees of several mathematical properties such as commutativity and continuity for large classes of deformations.

We present a method for applying complex textures to hand-drawn characters in cel animation. The method correlates features in a simple, textured, 3-D model with features on a hand-drawn figure, and then distorts the model to conform to the hand-drawn artwork. The process uses two new algorithms: a silhouette detection scheme and a depth-preserving warp. The silhouette detection algorithm is simple and efficient, and it produces continuous, smooth, visible contours on a 3-D model. The warp distorts the model in only two dimensions to match the artwork from a given camera perspective, yet preserves 3-D effects such as self-occlusion and foreshortening. The entire process allows animators to combine complex textures with hand-drawn artwork, leveraging the strengths of 3-D computer graphics while retaining the expressiveness of traditional handdrawn cel animation.