Abstract. The art of animation relies uopn modeling objects that change over time. A sequence of static images is displayed to produce an illusion of motion, which is frequently trusted to be topologically meaningful. A careful analysis exposes that formal topological guarantees are often lacking. This lack of formal justification can lead to subtle, but significant, flaws regarding topological integrity. A modified approach is proposed that integrates topological rigor with a continuous model of time. Examples will be given for splines widely used in many applications, with particular emphasis upon scientific visualization for molecular modeling. Moreover, the approach of choosing a family of functions and studying their topological properties over time should be broadly applicable to other domains. Prototype animations are available for viewing over the web.

An example is presented of a cubic Bézier curve that is the unknot (a knot with no crossings), but whose control polygon is knotted. It is also shown that there is no upper bound on the number of crossings in the control polygon for an unknotted Bézier curve. These examples complement known upper bounds on the number of subdivisions sufficient for a control polygon to be ambient isotopic to its Bézier curve. There can be substantial topological differences between a curve and its control polygon, as depicted in Figure 1, which has control polygon P0, P1,... P5, P0. (a)

For a rich class of composite cubic Bézier curves, an a priori bound exists on the number of subdivisions to achieve ambient isotopy between the curve and its control polygon. The authors of that theorem did not present any examples when the original control polygon is not ambient isotopic to the curve. An example is given here of a composite cubic Bézier curve that is the unknot (a knot with no crossings), but whose control polygon is knotted. It is also shown that there is no upper bound on the number of crossings in the control polygon for an unknotted composite Bézier curve. There can be substantial topological differences between a curve and its control polygon, as depicted in Figure 1 and explained, below. A knot will be considered to be

Abstract: Ambient isotopic approximations are fundamental for correct representation of the embedding of geometric objects in R 3, with a detailed geometric construction given here. Using that geometry, an algorithm is presented for efficient update of these isotopic approximations for dynamic visualization with a molecular simulation. 1 Approximation and Topology for Visualization Figure 1(a) depicts a knot 5 and Figure 1(b) shows a visually similar protein model 6. prompting two criteria for efficient algorithms for visualization: