selftangent

A selftangent is a plane curve that intersects itself in such a way that the tangent line at the point of self-intersection is also tangent to the curve at another point. This means that at the point where the curve crosses itself, the direction of the curve is the same as the direction of the curve at another distinct point on the curve.

Selftangents are a type of singular point on a curve. Unlike a simple self-intersection, where two distinct branches of the curve pass through the same point, a selftangent implies a more specific geometric relationship between the curve and its tangent at that point.

A common example of a curve with a selftangent is the epitrochoid, a roulette traced by a point attached to a circle rolling around the outside of a fixed circle. Depending on the radii of the circles and the position of the point, the epitrochoid can exhibit self-tangent points. Another class of curves that can feature selftangents are certain algebraic curves.

The presence of a selftangent can complicate the analysis of a curve's properties, such as its curvature and its topological behavior. Detecting and characterizing selftangents is a topic of interest in differential geometry and algebraic geometry. The mathematical condition for a selftangent involves the vanishing of certain derivatives of the curve's parameterization.