Curvature along a path

Estimates the curvature at every point from the neighbouring points, using the standard formula \(\kappa = |x' \times x''| / |x'|^3\), which does not care how the points are spaced. On a circle of radius \(r\) the values come out as \(1/r\), and on a straight line as 0.

Usage

path_curvature(path, closed = FALSE)

Arguments

path

Nx3 numeric matrix of path points (or a length-3 vector).

closed

Whether the path loops back to its first point (default FALSE).

Value

A numeric vector with one curvature value per input point. The endpoints of an open path report the value of their only neighbour; fewer than 3 points yield a zero-length vector, since curvature is undefined there.

Details

This is a diagnostic tool rather than a rendering input: sweeping a tube of radius \(r\) along a curve whose curvature reaches \(\kappa\) folds the tube inside out, so a path can be swept safely up to a radius of \(1 / \max(\kappa)\), which is worth checking for tight turns in measured data (max(path_curvature(path))).

See also

spline_path, generate_tube

Examples

# Largest tube radius that does not fold this path onto itself:
path <- spline_path(matrix(c(0, 0, 0, 1, 1, 0, 2, 0, 0, 3, 1, 0),
                           ncol = 3, byrow = TRUE))
1 / max(path_curvature(path))
#> [1] 0.2171869