Approximating the Nonlocal Curvature of Planar Curves

Abstract

Here we establish several results on the nonlocal curvature of planar curves. First we show how to express the nonlocal curvature of a curve relative to a point in terms of the nonlocal curvatures of simpler components of that curve relative to the same point. To obtain these results, it is necessary to extend the definition of nonlocal curvature to points off of the curve. We also find a formula for the nonlocal curvature of a line segment relative to any point in the plane in terms of the incomplete beta function. These results are then used to prove an approximation theorem, which states that the nonlocal curvature of a planar curve with some Hölder regularity can be approximated by the nonlocal curvature of a linear interpolating spline associated with the curve.

Figures (4)

• Figure 1: Depiction of the sets ${{\cal A}}_e(z)$ and ${{\cal A}}_i(z)$ for a given curve ${{\cal C}}$ and point $z$. The grey region depicts ${{\cal A}}_i(z)$ while the white region depicts ${{\cal A}}_e(z)$, up to sets of zero area. The curve ${{\cal C}}$ is the solid line while the dashed line segments depict other parts of the boundary between ${{\cal A}}_e(z)$ and ${{\cal A}}_i(z)$.
• Figure 2: Depiction of the line segment ${{\cal L}}$ along with $z$ and ${\bf u}$ in the plane, and the region ${{\cal A}}_i(z)\cap{}{{\cal P}}_+$.
• Figure 3: Depiction of ${{\cal C}}_\rho$ along with $z$, where the leftmost dark gray region, the center light gray region, and the rightmost dark gray region denote the regions of the first, second, and third double integrals on the right-hand side of \ref{['threeint']}, respectively.
• Figure 4: Depiction of a curve ${{\cal C}}$ that is decomposed into six pieces. A point $z$ and unit vector ${\bf u}$ are also shown. The dashed lines illustrate how the decomposition of the curve is relate to the point $z$ and the unit vector ${\bf u}$.

Theorems & Definitions (15)

• proposition 1
• proof
• proposition 2
• proof
• proposition 3
• proof
• proposition 4
• proof
• lemma 1
• proof