Selfintersecting

Self-intersecting is an adjective describing a curve, line, polygon, or other geometric object that crosses itself at one or more points. The crossing points are called self-intersection points and indicate that the object is not simple or injective in its presentation.

In mathematics, a self-intersecting plane curve is typically an immersion of a parameter set into the plane that is not injective, meaning distinct parameter values map to the same location. Familiar examples include the figure-eight (lemniscate) and various curves that cross themselves. Self-intersections create nodes or double points, and they complicate topological notions such as the interior: the Jordan curve theorem applies only to simple closed curves, so interior regions of self-intersecting curves are not uniquely defined. In analytic settings, the behavior around intersection points is described by intersection multiplicity, and tools like the winding number or planar graph decomposition help analyze the resulting regions.

In polygons, a polygon is self-intersecting if its boundary edges cross at nonconsecutive points. Classic examples include the bow-tie (hourglass) polygon and star polygons like the pentagram. Self-intersection affects area calculation and rendering: although a signed-area formula can be used, there is no single universally agreed interior. Graphic rendering and geometric modeling often employ fill rules (such as even-odd or nonzero winding) to determine which regions are considered inside.

Other contexts include typography, computer graphics, and CAD, where self-intersections in outlines can cause rendering, manufacturing, or interpretation challenges.