plane. Intuitively, it is a set of points that could be traced out by a moving point. More specifically, smooth curves generally at least require that the function from the interval to the plane be continuously differentiable, and in some contexts are defined to require higher derivatives. The function parameterizing a smooth curve is often assumed to be regular, meaning that its derivative stays away from zero; intuitively, the moving point never slows to a halt or reverses direction. Each interior point of a smooth curve has a tangent line. If, in addition, the second derivative exists everywhere, then each of these points has a well-defined curvature. A plane curve is closed if the two endpoints of the interval are mapped to the same point in the plane, and it is simple if no other two points coincide. Less commonly, a simple plane curve may be said to be open if it is topologically equivalent to a line, neither having an endpoint nor forming any limiting point that does not belong to it, and dividing the plane into two unbounded regions. However, this terminology is ambiguous as other sources refer to a curve with two distinct endpoints as an open curve. Here, we use the topological-line meaning of an open curve. Supporting lines A supporting line is a line containing at least one point of the curve, for which the curve is contained in one of the two half-planes bounded by the line. A plane curve is called convex if it has a supporting line through each of its points. For example, the graph of a convex function has a supporting line below the graph through each of its points. More strongly, at the points where the function has a derivative, there is exactly one supporting line, the tangent line. Supporting lines and tangent lines are not the same thing, but for convex curves, every tangent line is a supporting line. At a point of a curve where a tangent line exists, there can only be one supporting line, the tangent line. Therefore, a smooth curve is convex if it lies on one side of each of its tangent lines. This may be used as an equivalent definition of convexity for smooth curves, or more generally for piecewise smooth curves. Boundaries of convex sets A convex curve may be alternatively defined as a connected subset of the boundary of a