hoppingshape

Hoppingshape is a term used in geometry and computer graphics to describe a parameterized family of shapes that evolves over time with occasional discrete changes in topology. In this concept, the shapes change smoothly most of the time, but at a finite set of parameter values, the evolution includes a topological event, or hop, that alters the structure of the shape.

Formal framework for hopping shapes involves a family S(t) of compact sets in Euclidean space, t in [0,1]. For all t not in a finite set H, the evolution is continuous with respect to the Hausdorff metric. At each h in H, S(h) is related to the nearby shapes by a local surgery that creates or annihilates components, merges components, or changes the genus of the boundary. The hops are typically localized so that the rest of the shape remains stable through the transition.

Examples include a pair of disks connected by a narrowing bridge that disappears at a certain parameter value, causing two components to merge into a single disk, or an annulus whose inner hole closes, turning into a disk. More complex hops can involve splitting a component into two or the sudden appearance of a new handle.

Properties and analysis of hopping shapes often focus on the finite hop set, the locality of each hop, and how the sequence of shapes behaves between hops. They are used to model abrupt yet controlled transitions in animations, assist in topology-aware geometric modeling, and provide simplified scenarios for studying feature birth and death in topological data analysis.

See also: morphing, topology change, persistent homology, shape optimization.