That is, these curves are a subset of the rectifiable curves. Equivalently, a strictly convex curve is a curve that intersects any line in at most two points, or a simple curve in convex position, meaning that none of its points is a convex combination of any other subset of its points.Įvery convex curve that is the boundary of a closed convex set has a well-defined finite length. For this variation, a convex curve may have endpoints.Ī strictly convex curve is a convex curve that does not contain any line segments. Sometimes, a looser definition is used, in which a convex curve is a curve that forms a subset of the boundary of a convex set.

This definition is more restrictive than the definition in terms of tangent lines in particular, with this definition, a convex curve can have no endpoints. In other words, a convex curve is a curve that has a supporting line through each of its points.Ī convex curve may be defined as the boundary of a convex set in the Euclidean plane. A plane curve is called convex if it lies on one side of each of its tangent lines. We say that a curve C 'lies on one side of L' if it is entirely contained in one of the half-planes.

Definitions Definition by supporting lines Īny straight line L divides the Euclidean plane into two half-planes whose union is the entire plane and whose intersection is L.

Curved space graph