Convex curve
The maximum number of grid points that can belong to a single curve is controlled by its length.
[2] For the next two millennia, there was little study of convexity:[2] its in-depth investigation began again only in the 19th century,[3] when Augustin-Louis Cauchy and others began using mathematical analysis instead of algebraic methods to put calculus on a more rigorous footing.
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.
[5] 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.
A plane curve is called convex if it has a supporting line through each of its points.
[12] Therefore, a smooth curve is convex if it lies on one side of each of its tangent lines.
Every connected subset of the boundary of a convex set has a support line through each of its points.
[20] They are the curves that can be formed as a connected subset of the boundary of a strictly convex set.
[24] Closed strictly convex curves can be defined as the simple closed curves that are locally equivalent (under an appropriate coordinate transformation) to the graphs of strictly convex functions.
[25][c] Smooth closed convex curves with an axis of symmetry, such as an ellipse or Moss's egg, may sometimes be called ovals.
In Euclidean geometry these are the smooth strictly convex closed curves, without any requirement of symmetry.
[20] Every bounded convex curve is a rectifiable curve, meaning that it has a well-defined finite arc length, and can be approximated in length by a sequence of inscribed polygonal chains.
For closed convex curves, the length may be given by a form of the Crofton formula as
[30] It is not possible for a strictly convex curve to pass through many points of the integer lattice.
, then according to a theorem of Vojtěch Jarník, the number of lattice points that it can pass through is at most
Because this estimate uses big O notation, it is accurate only in the limiting case of large lengths.
[31] A convex curve can have at most a countable set of singular points, where it has more than one supporting line.
This implies that the non-singular points form a dense set in the curve.
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.
More strongly, a smooth closed curve is convex if and only if it does not have three parallel tangent lines.
In the other direction, a non-convex smooth closed curve has at least one point with no support line.
[5] A smooth simple closed curve, with a regular parameterization, is convex if and only if its curvature has a consistent sign: always non-negative, or always non-positive.
For closed curves that are not convex, the total absolute curvature is always greater than
More generally, by Fenchel's theorem, the total absolute curvature of a closed smooth space curve is at least
[40][41] By the Alexandrov theorem, a non-smooth convex curve has a second derivative, and therefore a well-defined curvature, almost everywhere.
Although still unsolved in general, its solved cases include the convex curves.
A scaled and rotated copy of any rectangle or trapezoid can be inscribed in any given closed convex curve.
When the curve is smooth, a scaled and rotated copy of any cyclic quadrilateral can be inscribed in it.
However, the assumption of smoothness is necessary for this result, because some right kites cannot be inscribed in some obtuse isosceles triangles.
