# Floating Tangents for Approximating Spatial Curves with $G^1$ Piecewise Helices

Computer-Aided Geometric Design, June 2013

#### Abstract

Curves are widely used in computer science to describe real-life objects such as slender deformable structures. Using only 3 parameters per element, piecewise helices offer an interesting and compact way of representing digital curves. In this paper, we present a robust and fast algorithm to approximate Bézier curves with $G^1$ piecewise helices. Our approximation algorithm takes a Bézier spline as input along with an integer N and returns a piecewise helix with N elements that closely approximates the input curve. The key idea of our method is to take N+1 evenly distributed points along the curve, together with their tangents, and interpolate these tangents with helices by slightly relaxing the points. Building on previous work, we generalize the proof for Ghosh's co-helicity condition, which serves us to guarantee the correctness of our algorithm in the general case. Finally, we demonstrate both the efficiency and robustness of our method by successfully applying it on various datasets of increasing complexity, ranging from synthetic curves created by an artist to automatic image-based reconstructions of real data such as hair, heart muscular fibers or magnetic field lines of a star.

Erratum: The original publication contained a typo in Appendix A (p. 46, above Remark 1, conditions for expressing $\varphi *$ have to be switched), which led to an incorrect formulation in the 2d case. This has been fixed in the revised version. Be sure to download the PDF version above.