pass through is at most Because this estimate uses big O notation, it is accurate only in the limiting case of large lengths. Neither the leading constant nor the exponent in the error term can be improved. Supporting lines and support function A convex curve can have at most a countable set of singular points, where it has more than one supporting line. All of the remaining points must be non-singular, and the unique supporting line at these points is necessarily a tangent line. This implies that the non-singular points form a dense set in the curve. It is also possible to construct convex curves for which the singular points are dense. A closed strictly convex closed curve has a continuous support function, mapping each direction of supporting lines to their signed distance from the origin. It is an example of a hedgehog, a type of curve determined as the envelope of a system of lines with a continuous support function. The hedgehogs also include non-convex curves, such as the astroid, and even self-crossing curves, but the smooth strictly convex curves are the only hedgehogs that have no singular points. It is impossible for a convex curve to have three parallel tangent lines. More strongly, a smooth closed curve is convex if and only if it does not have three parallel tangent lines. In one direction, the middle of any three parallel tangent lines would separate the points of tangency of the other two lines, so it could not be a line of support. There could be no other line of support through its point of tangency, so a curve tangent to these three lines could not be convex. In the other direction, a non-convex smooth closed curve has at least one point with no support line. The tangent line through that point, and the two tangent supporting lines parallel to it, form a set of three parallel tangent lines. Curvature According to the four-vertex theorem, every smooth closed curve has at least four vertices, points that are local minima or local maxima of The original proof of the theorem, by Syamadas Mukhopadhyaya in 1909, considered only convex it was later extended to all smooth closed Curvature can be used to characterize the smooth closed curves that are The curvature depends in a trivial way on the parameterization of the curve: if a regularly parameterization of a curve is reversed,