Rapid blending of closed curves based on curvature flow

Summary

Shape blending (or shape morphing), which describes the continuous and gradual transition of solid models, has been actively investigated in computer graphics. In particular, the closed curve blending problem has received a great deal of attention because of its wide range of applications in areas including computer-assisted animation. The closed curve blending problem can be decomposed into two consecutive subproblems, namely the vertex correspondence problem and the vertex path problem. In this study, we focus only on the vertex path problem: computing the actual path by which each vertex moves to its counterpart specified by vertex correspondence.

A blending algorithm needs to be equipped with several properties. For animation-oriented applications, a sequence of blended curves should be aesthetic: the source shape should appear to undergo a natural transition to the target shape. It is also desirable to calculate blended curves as quickly as possible so as to let animators modulate the entire collection of blended curves interactively.

We propose a new closed blending technique based on curvature flow. Curvature flow is a type of geometric flow that is often defined as a gradient flow associated to a functional on a manifold wherein a geometric value (e.g., curvature) is defined. We find that curvature flow has great potential for applications in shape blending, but it is currently applied only in shape fairing. Further- more, designing curvature flow provides great flexibility upon blending and conveys special properties described below.

Thanks to the problem formulation in curvature space, our method performs stable and fast calculation, namely two orders of magnitude faster than state-of-the-art methods. Our method can be applied for a pair of curves wherein the turning numbers are both same (Fig. 1) and different (Fig. 2). Especially for curves with different turning numbers, incorporating curvature smoothing flow enables as smooth as possible blending, which could have not been realized so far.

Reference

• Masahiro Hirano, Yoshihiro Watanabe, Masatoshi Ishikawa: Rapid blending of closed curves based on curvature flow, Computer Aided Geometric Design (Special issue on GMP2017), Vol.52-53, pp.217-230, 2017. [doi: http://dx.doi.org/10.1016/j.cagd.2017.03.005]
• Masahiro Hirano, Yoshihiro Watanabe, Masatoshi Ishikawa: Rapid blending of closed curves based on curvature flow, International Conference on Geometric Modeling and Processing (GMP2017), (Xiamen, China, 2017.4.19)/Proceedings (2017)
• Masahiro Hirano, Yoshihiro Watanabe, Masatoshi Ishikawa: Closed curve blending based on curvature flow, The Symposium on Solid & Physical Modeling 2016 (SPM2016), (Berlin, Germany, 2016.6.21)/Abstracts (2016)