Selfintersections

Self-intersections are points on a curve where the curve passes through the same location more than once. For a parametric curve r(t) in the plane or space, a self-intersection occurs when there exist two distinct parameters t1 ≠ t2 with r(t1) = r(t2). Such points indicate that the curve is not simple or embedded.

In the plane, a self-intersection means the curve meets itself at a location. A common example is the figure-eight or lemniscate, which crosses at a central point. In the context of plane algebraic curves, a self-intersection is a double point where two branches meet; if the tangents at the preimages are distinct, the intersection is called a node.

If the branches share the same tangent, the singularity may be a cusp or a higher-order contact rather than a true crossing. The key distinction is that a cusp arises from a single parameter value, not two distinct values.

In higher dimensions, self-intersections refer to double points of maps between manifolds; the set of pairs of distinct domain points mapping to the same image forms the double point locus and is studied in intersection theory and singularity theory.

In applications, detecting self-intersections matters in computer graphics, geometric modeling, and knot theory, where projections create crossings that are analyzed as over/under crossings; removing self-intersections can be necessary to obtain valid meshes or embeddings.