Homotopic Morphing of Planar Curves

Abstract

This paper presents an algorithm for morphing between closed, planar piecewise‐C1 curves. The morph is guaranteed to be a regular homotopy, meaning that pinching will not occur in the intermediate curves. The algorithm is based on a novel convex characterization of the space of regular closed curves and a suitable symmetric length‐deviation energy. The intermediate curves constructed by the morphing algorithm are guaranteed to be regular due to the convexity and feasibility of the problem. We show that our method compares favorably with standard curve morphing techniques, and that these methods sometimes fail to produce a regular homotopy, and as a result produce an undesirable morph. We explore several applications and extensions of our approach, including morphing networks of curves with simple connectivity, morphing of curves with different turning numbers with minimal pinching, convex combination of several curves, and homotopic morphing of b‐spline curves via their control polygon.

Measuring similarity between curves on 2-manifolds via homotopy area

“…Here, a curve C : [0, 1] → M is regular if its first derivative C (t) is defined and not zero for all t ∈ [0, 1]. In our paper, similar to the setting of [23], we allow a curve C : [0, 1] → M to be piecewise-regular, meaning that C is continous and the above conditions on C holds for all t other than potentially a finite number of singular points…”

Blending of hyperbolic closed curves

“…Unfortunately, the method presented does not guaranty for intermediate curves to have no self-intersection. In both papers [SSHS14,DSL15], the authors approximate the source and target curve by their inscribed polygons, then they interpolate the discrete geodesic curvature and the edge lengths to reconstruct intermediate curves. Again, the intermediate polygon is not usually closed.…”

“…The most famous is the Poincaré disc, whose boundary represents infinity and in which the geodesics consist of all circular arcs contained within that disc that are orthogonal to the boundary of the disc, plus all diameters of the disc. Most existing closed curve blending methods were studied in the Euclidean plane [DSL15,SSHS14,SGWM93]. They consist in approximating the source and target closed curves by inscribed closed polygons.…”