$p\kappa$-Curves: Interpolatory curves with curvature approximating a parabola

TL;DR

This paper presents $p\kappa$-curves, a new class of interpolatory curves with curvature profiles resembling parabolas, optimized for aesthetic appeal and efficient interactive modeling.

Contribution

Introduction of $p\kappa$-curves, combining curvature approximation to parabolas with an efficient algorithm for interactive curve design.

Findings

Curves exhibit curvature distributions closely matching parabolas.

The method enables interactive modeling without global optimization.

Visual comparisons show improved aesthetic quality.

Abstract

This paper introduces a novel class of fair and interpolatory curves called $p\kappa$-curves. These curves are comprised of smoothly stitched B\'ezier curve segments, where the curvature distribution of each segment is made to closely resemble a parabola, resulting in an aesthetically pleasing shape. Moreover, each segment passes through an interpolated point at a parameter where the parabola has an extremum, encouraging the alignment of interpolated points with curvature extrema. To achieve these properties, we tailor an energy function that guides the optimization process to obtain the desired curve characteristics. Additionally, we develop an efficient algorithm and an initialization method, enabling interactive modeling of the $p\kappa$-curves without the need for global optimization. We provide various examples and comparisons with existing state-of-the-art methods to demonstrate the curve modeling capabilities and visually pleasing appearance of $p\kappa$-curves.