Linesvary

Linesvary is a metric used in computational geometry to quantify how much a given set of lines deviates from a reference configuration. It is applicable to planar line data from drawings, image detections, and geometric datasets where assessing alignment and spread aids analysis and quality control.

A typical definition represents each line in normal form by angle theta_i in [0, pi) and offset p_i. The angular component uses circular statistics: compute the mean direction mu_theta and the resultant length R = sqrt((sum cos theta_i)^2 + (sum sin theta_i)^2)/n, with var_theta = 1 - R. The positional component uses var_p, the usual variance of the offsets p_i around their mean m_p. A normalized linesvary is linesvary = w_theta * var_theta_norm + w_p * var_p_norm, where var_theta_norm and var_p_norm scale var_theta and var_p to [0,1], and w_theta + w_p = 1.

Variants may use unnormalized components or alternative line representations, and can be made invariant to translation or scale by pre-aligning lines or using relative offsets. The metric can be extended to three dimensions or to sets of line segments instead of infinite lines.

Applications include evaluating line-detection and feature-extraction algorithms, quality control for technical drawings, and guiding procedural generation of line patterns in graphics. It is related to, but distinct from, line-fitting error, Hough-space clustering, and other dispersion measures.