$pκ$-Curves: Interpolatory curves with curvature approximating a parabola

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.