ConicCurv: A curvature estimation algorithm for planar polygons

TL;DR

The curvature values estimated by ConicCurv are invariant to Euclidean changes of coordinates and reproduce the exact curvature values if the data are sampled from a conic.

Abstract

ConicCurv is a new derivative-free algorithm to estimate the curvature of a plane curve from a sample of data points. It is based on a known tangent estimator method grounded on classic results of Projective Geometry and Bézier rational conic curves. The curvature values estimated by ConicCurv are invariant to Euclidean changes of coordinates and reproduce the exact curvature values if the data are sampled from a conic. We show that ConicCurv< has convergence order $3$ and, if the sample points are uniformly arc-length distributed, the convergence order is $4$. The performance of ConicCurv is compared with some of the most frequently used algorithms to estimate curvatures and its performance is illustrated in the calculation of the elastic energy of subdivision curves and the location of L-curves corners.

Figures (15)

• Figure 1: Pascal's theorem
• Figure 2: Tangent estimation in $P_i$. This tangent is the limit of the secant $L_{P_i,P_{i+1}}$ in Fig. \ref{['fig:pascal']}.
• Figure 3: Nonconvex polygon divided into two sub-polygon inserting the inflection point $P$. The assigned tangent is drawn with discontinuous lines.
• Figure 4: Interpolating conics.
• Figure 5: Test curve: polynomial.

Theorems & Definitions (10)

• Theorem 2.1: Pascal's theorem
• Theorem 2.2
• Remark 2.1
• Remark 2.2
• Lemma 3.1
• proof
• Definition 3.1
• Theorem 4.1
• proof
• Remark 5.1