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

TL;DR

The paper addresses constructing interpolating space curves in $\mathbb{R}^d$ with high geometric smoothness while preserving locality. It introduces a four-component framework—local curves, redistribution, blending functions, and gluing—and uses $r$-blending functions to achieve arbitrary $G^r$ continuity, under suitable positivity/contractness conditions. A main theorem guarantees $G^r$ regularity and interpolation when the local QR curves, redistribution, and blending satisfy the prescribed properties, with potential $G^{r+1}$ gains for higher-smoothness local data. Numerical experiments in 3D illustrate sphere/corner preservation and curvature continuity, demonstrating the method’s practical capability to produce smooth interpolants for complex point clouds.

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.

Figures (24)

• Figure 1: A $C^2$ smooth interpolating curve(right) based on a Lissajous knot (left). The curvature of the curve is shown by the color on the curve. The curve change shows that the curvatures are all continuous.
• Figure 2: A $C^2$ smooth interpolating curve(bottom) based on a plane data (top).
• Figure 3: The differentiability of the parametric functions of $\Gamma(t)=(t^2,t^3), t\in [-1, 1]$ does not imply the graph is smooth.
• Figure 4: The graph of a curve is not necessary to be $G^r$ without the regularity of the parametric functions.
• Figure 5: A linear combination of two regular curves may not be regular

Theorems & Definitions (34)

• Definition 2.1
• Example 2.1
• Definition 2.2
• Proposition 2.2
• Remark 2.1
• Example 2.3
• Example 2.4
• Definition 2.3
• Remark 2.2
• Definition 3.1