Crossangle

Crossangle is a term used in geometry and related fields to describe the angle formed at the intersection of two lines, curves, or trajectories. The crossing angle is the measure of the smallest angle between the two directions at the point where they cross. When two lines intersect, the four angles created consist of two equal vertical-angle pairs, and the adjacent angles are supplementary (sum to 180 degrees).

In two-dimensional space, if the lines have direction vectors u and v, the crossing angle θ is given by cos θ = (u·v)/(|u||v|). If lines are given by slopes m1 and m2, θ can also be computed as θ = arctan(|(m2−m1)/(1+m1m2)|). In three dimensions, the same dot-product relation applies to the angle between direction vectors, and the concept extends to the angle between curves at a common point via their tangent directions.

Crossangle appears in practice in computer graphics, computer-aided design (CAD), and robotics to quantify how sharply two features cross, to align components, or to control trajectories. It can also describe structural join angles, such as cross-bracing, where the angle between members affects strength and fit.

Terminology: crossangle is not universally standardized; many disciplines refer to the "angle of intersection," the "crossing angle," or the "interior angle at the intersection." The term is often context-dependent and may be used interchangeably with the acute or smallest angle formed by the crossing lines.