| When a disk in 2D is stretched arbitrarily with possible self-overlaps, the boundary of the disk forms a complex self-intersecting closed loop curve, known as a self-overlapping curve. Examples of such curves can be found in animation applications, molecular structures of certain allotropes of carbon, like graphite, and many other physical models. Given a self-overlapping curve, a novel method to compute the deformation pattern (immersion) of the disk is proposed in this work, which is useful in applications like animation and compression.;The process of forming a disk immersion implies a natural morphing sequence. The second part of this work computes a smooth morphing between two self-overlapping curves. This is achieved by computing a compatible triangulation of the curve interiors using the earlier mentioned method of computing disk immersions, and then morphing each individual triangle from the source to target. The method automatically computes smooth morphs given the source and target shapes, wherein each intermediate morphed shape is a self-overlapping curve.;While morphing compatible triangulations of two self-overlapping curves produces satisfactory morphs in most cases, an even better morph is obtained by morphing the skeletons of both the curves. Since the triangulation of a disk, and hence the interior of a self-overlapping curve, is a tree, the medial axis or skeleton of a self-overlapping curve is a geometric tree. Morphing geometric trees requires the establishment of a good match between their node sets. In this context, two different geometric tree matching algorithms are proposed. The first one is generic in nature and computes matches between two arbitrary geometric trees by modeling the problem as minimum weight bipartite graph matching problem. Such trees may be found as representatives of actual physical systems like river bed patterns, respiratory trees of mammals etc. These trees may undergo structural changes with time or under the influence of external agents and a match between different instances of such trees helps to infer the change.;A second method for matching geometric trees, based on dynamic programming, is also proposed which is more suited to morphing applications, which prohibits cross-over of tree branches while morphing. The matches obtained from this second method are used to design a novel geometric tree morphing algorithm which is based on motion planning concepts. The morph is symmetric in nature and minimizes the distance travelled by the tree branches while morphing. |
