selfintersects

Self-intersections occur when a geometric object intersects itself at a point that lies on multiple parts of the object. For curves or parametric representations, this means there exist parameters t1 ≠ t2 with f(t1) = f(t2). For polygons, a self-intersecting polygon has nonconsecutive edges that cross.

Common examples include the lemniscate of Bernoulli, a curve with a figure-eight shape that crosses at the origin, and the pentagram, a star polygon whose edges cross. In contrast, a simple closed curve or a non-self-intersecting polygon has no such crossings. Self-intersections can be classified by the nature of the contact: a transverse crossing (a node) where branches cross with distinct tangents, or a tangential contact (a cusp or tacnode) where the contact is by a shared tangent.

In topology and geometry, the concept relates to embeddings and immersions. An embedding is a map that is injective and has no self-intersections, while an immersion may have self-intersections but remains locally a smooth map. Self-intersections often indicate interesting or problematic topology, such as non-simplicity in a polygon or non-injectivity in a parameterization.

Detection and handling are common in computational geometry and computer graphics. Self-intersection tests reduce to line segment or curve-curve intersection checks, with algorithms like sweep-line methods. Self-intersections can complicate area calculations, rendering, and mesh or curve operations; remedies include splitting the object at intersection points, performing boolean operations, or reparameterizing to remove overlaps and produce non-self-intersecting representations.