A Construction of Interpolating Space Curves with Any Degree of Geometric Continuity

TL;DR

This paper presents a versatile method for constructing smooth, interpolating space curves with any degree of geometric continuity, ensuring desirable properties like $G^2$ smoothness, locality, and no self-intersection, adaptable to various geometric constraints.

Contribution

It introduces a novel construction framework for interpolating space curves with arbitrary geometric continuity, incorporating local, blending, redistributing, and gluing functions.

Findings

Produces $G^2$ smooth interpolating curves

Ensures curves are free of cusps and self-intersections

Demonstrates adaptability to convexity and sharp corners

Abstract

This paper outlines a methodology for constructing a geometrically smooth interpolatory curve in $\mathbb{R}^d$ applicable to oriented and flattenable points with $d\ge 2$. The construction involves four essential components: local functions, blending functions, redistributing functions, and gluing functions. The resulting curve possesses favorable attributes, including $G^2$ geometric smoothness, locality, the absence of cusps, and no self-intersection. Moreover, the algorithm is adaptable to various scenarios, such as preserving convexity, interpolating sharp corners, and ensuring sphere preservation. The paper substantiates the efficacy of the proposed method through the presentation of numerous numerical examples, offering a practical demonstration of its capabilities.