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Convex curve

A convex curve is a simple closed plane curve that bounds a convex domain, where a domain is convex if the line segment joining any two points within it lies entirely inside the domain.^[1] Equivalently, for a regular parametrized curve, it lies entirely on one side of each of its tangent lines.^[2] Convex curves exhibit non-negative curvature everywhere (or non-positive, depending on orientation), ensuring the curve does not "bend back" on itself.^[2] Classic examples include the circle, where curvature is constant, and the ellipse, which has varying but always positive curvature.^[2] A fundamental property is captured by the four-vertex theorem, which states that every simple closed convex plane curve of class C^3 possesses at least four vertices—points where the curvature attains a local maximum or minimum.^[3] This theorem, first proved for strictly convex curves by Syamadas Mukhopadhyaya in 1909, highlights the geometric rigidity of such curves.^[3] In applications, convex curves play a central role in differential geometry and variational problems. For instance, the isoperimetric inequality asserts that for a simple closed plane curve of length L enclosing area A, L^2 \geq 4\pi A, with equality if and only if the curve is a circle; this holds prominently for convex curves, maximizing enclosed area for given perimeter.^[4] Additionally, under flows like the curve shortening flow (the L^2-gradient flow of length), initial convex curves remain convex and evolve toward circularity before shrinking to a point.^[2] These properties extend to broader studies in convex geometry and optimization.

Preliminary concepts

Convex sets

In the Euclidean plane \mathbb{R}^2, a set C is defined as convex if, for any two points x, y \in C, the entire line segment connecting them, given by \{(1 - \theta)x + \theta y \mid \theta \in [0, 1]\}, is contained within C.^[5] This property ensures that convex sets have no "dents" or indentations, making them fundamental in geometry and optimization. The empty set and singletons are trivially convex under this definition.^[5] Common examples of convex sets in the plane include disks (the set of all points inside and on a circle), half-planes (all points on one side of a straight line, including the line), and convex polygons (such as triangles, squares, or regular n-gons where all interior angles are less than or equal to $180^\circ).^[6] Intersections of convex sets are always convex; for instance, the intersection of multiple half-planes forms a convex polygon.^[7] Convex hulls extend this idea: the convex hull \operatorname{conv}(S) of a set S \subseteq \mathbb{R}^2 is the smallest convex set containing S, explicitly given by

\operatorname{conv}(S) = \bigcap \{ K \mid K \text{ is convex and } S \subseteq K \}.

^[8] For finite point sets, the convex hull is the convex polygon with those points as vertices.^[9] Key properties of convex sets include closure under certain operations and separation results. Convex sets are closed under arbitrary intersections and convex combinations, meaning that if C_1, C_2, \dots are convex, so is \bigcap C_i.^[7] Closed convex sets, which are convex and topologically closed (containing all their limit points), play a central role in theorems like the supporting hyperplane theorem. A basic separation theorem states that if two nonempty convex sets C and D in \mathbb{R}^2 are disjoint, there exists a line (hyperplane) such that C lies entirely on one side and D on the other.^[10] For closed convex sets where at least one is compact, strict separation is possible, with the sets on opposite open half-planes.^[11]

Plane curves

A plane curve is a continuous mapping \gamma: I \to \mathbb{R}^2, where I \subseteq \mathbb{R} is an interval, tracing a path in the Euclidean plane.^[12] This parametrization allows the curve to be oriented by the direction of increasing parameter values along I. Alternatively, a plane curve can be regarded as the image set \gamma(I) \subset \mathbb{R}^2, which is unparametrized and focuses on the geometric locus rather than the traversal.^[13] Plane curves are classified by their topology and structure. A closed curve occurs when I = [a, b] is a closed interval and \gamma(a) = \gamma(b), forming a loop without endpoints.^[12] A simple closed curve is a closed curve that does not intersect itself, meaning \gamma(t_1) = \gamma(t_2) only when \{t_1, t_2\} = \{a, b\}.^[14] A Jordan curve is precisely such a simple closed plane curve, topologically equivalent to the unit circle.^[15] Basic properties of plane curves include continuity, which ensures the image is connected, and differentiability where applicable. Piecewise smooth curves consist of finitely many smooth segments, each parametrized by a C^1 mapping (continuously differentiable with continuous derivative), allowing tangent vectors \gamma'(t) to be defined almost everywhere.^[16] For closed curves, orientation is specified as clockwise or counterclockwise based on the parametrization direction; counterclockwise orientation typically places the enclosed region to the left of the traversal direction.^[17] A fundamental result is the Jordan curve theorem, which states that every Jordan curve separates the plane into a bounded interior region and an unbounded exterior region, with any continuous path connecting a point in the interior to one in the exterior intersecting the curve.^[18] The interior region bounded by such a curve may serve as the domain for convex sets in the plane.

Definitions

Via boundaries of convex sets

A convex curve can be defined as the topological boundary \partial K of a bounded convex set K \subset \mathbb{R}^2 with nonempty interior.^[19] Such a set K, often termed a convex body, is compact and closed, ensuring that its boundary forms a compact curve. This definition captures curves that enclose a convex region without self-intersections or indentations. The boundary \partial K inherits key topological properties from K: it is a simple closed curve, meaning it is continuous, non-self-intersecting, and returns to its starting point, and it is Jordan in the sense that it divides the plane into two distinct regions.^[20] Specifically, \partial K is compact due to the boundedness of K, and its simplicity follows from the convexity of K, which prevents the boundary from crossing itself.^[19] The interior \operatorname{int}(K) is itself convex, and \partial K serves as the separator between this bounded interior and the unbounded exterior of the plane.^[20] Every convex curve bounds a convex domain, and conversely, any simple closed curve that bounds a convex interior region qualifies as the boundary of a convex body.^[20] For any point x \in \partial K, there exists a supporting half-plane H such that x \in \partial H and K \subset H. This property underscores the global containment of K within half-planes tangent to its boundary, reinforcing the curve's role in defining the convex set.

Via supporting lines

A convex curve \gamma in the plane can be defined through the concept of supporting lines: at every point \gamma(t) on the curve, there exists a line L passing through \gamma(t) such that the entire curve lies on one side of L. This half-plane condition ensures that the curve does not "bend back" across the line, capturing the intuitive notion of convexity locally at each point.^[21] For a parametrized curve \gamma: I \to \mathbb{R}^2, this is formalized by the existence of a unit normal vector n(t) at \gamma(t) satisfying the inequality \langle \gamma(s) - \gamma(t), n(t) \rangle \leq 0 for all s \in I, where \langle \cdot, \cdot \rangle denotes the standard dot product; equality holds at s = t, and the supporting line L is given by \{ x \in \mathbb{R}^2 \mid \langle x - \gamma(t), n(t) \rangle = 0 \}. For smooth convex curves, the tangent line at each point \gamma(t) naturally serves as the supporting line, with the unit tangent vector T(t) perpendicular to n(t). At nonsmooth points, such as corners where the curve is only piecewise smooth, one-sided tangent lines or a range of supporting lines may apply, but the half-plane containment still holds. At smooth points, the direction of the supporting line (or equivalently, the normal vector n(t)) is unique, ensuring a well-defined local geometry.^[22] A key property of closed convex curves is that their supporting lines turn monotonically as one traverses the curve. Parametrizing by arc length s, the tangent angle \theta(s) is a strictly monotonic function, increasing (or decreasing, depending on orientation) by exactly $2\pi over the full length, reflecting the total turning angle of $360^\circ. This monotonicity distinguishes convex curves from non-convex ones, where the tangent angle may oscillate or reverse direction.^[23]

Via line intersections

A closed curve in the plane is defined to be convex if every straight line intersects it in at most two points.^[24] This global intersection condition captures the idea that the curve does not "bend back" on itself in a way that allows excessive crossings with straight lines. Supporting lines, which are tangent to the curve at intersection points, play a role in verifying this property locally.^[25] For simple closed curves, this intersection condition implies that the bounded region enclosed by the curve is convex.^[20] Specifically, if the interior were non-convex, there would exist two interior points whose connecting line segment exits and re-enters the region, resulting in the line intersecting the curve in more than two points, contradicting the assumption.^[20] This equivalence holds under the assumption that the curve is Jordan-simple, meaning it is continuous and non-self-intersecting except at the closure. The definition extends to non-closed curves, known as convex arcs, where every straight line intersects the arc in at most one connected segment.^[20] Transversality conditions, such as requiring intersections to be transverse (non-tangent) except possibly at endpoints, ensure that the intersections are well-defined and avoid degeneracies like infinite multiplicity at flat portions.^[26] A curve fails to be convex if there exists a line intersecting it in three or more points, which typically occurs in the presence of reflex angles or inward bends.^[27] For instance, curves with such features allow lines to enter, exit, and re-enter the structure multiple times. Ellipses exemplify convex curves under this definition, as any line intersects an ellipse—a quadratic curve—in at most two points by the fundamental theorem of algebra applied to the intersection equation.^[28] In contrast, cardioids, which are non-convex limaçons with a cusp and inward dimple, can be intersected by certain lines in up to four points due to their quartic algebraic degree.^[29]

Strict convexity

A convex curve is strictly convex if it is the boundary of a bounded strictly convex set K, meaning that for any two distinct points x, y \in \partial K, the open line segment connecting them lies entirely in the interior of K.^[11] Mathematically, this condition is expressed as (1-t)x + ty \in \operatorname{int}(K) for all t \in (0,1).^[11] Equivalently, a strictly convex curve contains no line segment in its entirety, ensuring that no three points on the curve are collinear.^[30] This property implies that every supporting line to a strictly convex curve intersects it at exactly one point, providing a unique supporting line at each point of the curve (with possible exceptions at endpoints for curve arcs).^[31] In contrast, convex curves more generally—sometimes termed weakly convex in this context—may include flat segments along their length, whereas strictly convex curves prohibit such features and, where differentiable, require positive curvature.^[30] Closed strictly convex curves are simple closed curves homeomorphic to the circle S^1 and possess unique supporting lines at every point.^[32] Regarding line intersections, the strict convexity condition ensures that any line meets the curve at most at two distinct points.^[30]

Properties

Support function

The support function of a convex set K \subset \mathbb{R}^2, or more specifically its boundary convex curve, provides a fundamental analytic representation in convex geometry. Defined as h_K(\theta) = \sup_{x \in K} \langle x, u(\theta) \rangle, where u(\theta) = (\cos \theta, \sin \theta) is the unit vector in direction \theta, it measures the signed distance from the origin to the supporting hyperplane perpendicular to u(\theta). This function encodes the position of all supporting lines and fully characterizes the convex set, with the boundary curve recovered parametrically for smooth cases as \gamma(\theta) = h(\theta) u(\theta) + h'(\theta) u(\theta)^\perp, where u(\theta)^\perp = (-\sin \theta, \cos \theta). Equivalently, the position of the boundary point in direction \theta is given by

x(\theta) = \left( h(\theta) \cos \theta - h'(\theta) \sin \theta, \, h(\theta) \sin \theta + h'(\theta) \cos \theta \right).

The support function h exhibits key properties that reflect the geometry of the convex curve. As a function on [0, 2\pi), h is convex, arising as the pointwise supremum of linear functions \langle x, u(\theta) \rangle over x \in K. For closed curves, h is periodic with period $2\pi, ensuring the parameterization traces the boundary exactly once. These attributes make h a versatile tool for analysis, distinguishing it from qualitative descriptions of supporting lines by providing a quantitative, differentiable framework when the curve is smooth. Applications of the support function include measuring directional widths and relating to global measures like perimeter. The width of the convex set in direction \theta is h(\theta) + h(\theta + \pi), capturing the distance between parallel supporting hyperplanes. The perimeter L connects to the support function via the arc length element, yielding L = \int_0^{2\pi} (h(\theta) + h''(\theta)) \, d\theta; due to periodicity, integration by parts simplifies this to L = \int_0^{2\pi} h(\theta) \, d\theta.

Curvature characteristics

For a smooth convex plane curve parametrized by arc length \gamma(s), the curvature \kappa(s) = \|\gamma''(s)\| \geq 0 at every point, with the second derivative \gamma''(s) pointing toward the interior of the convex region bounded by the curve. This non-negativity ensures that the curve bends consistently in one direction without reversing concavity. For a closed smooth convex curve, the Fenchel theorem states that the total curvature \int \kappa \, ds = 2\pi. Equality holds precisely when the curve is planar and convex, distinguishing it from non-convex closed curves where the total curvature exceeds $2\pi. The tangent angle \phi(s), defined as the angle between the tangent vector \gamma'(s) and a fixed direction, is non-decreasing along the curve, reflecting the monotonic turning of the tangent. The total turning angle over the closed curve is exactly $2\pi, with the turning at any point bounded by \pi. In the strict convexity case, the curvature satisfies \kappa > 0 almost everywhere, ensuring that supporting lines touch the curve at exactly one point. Convex curves exhibit no inflection points, as the non-negative curvature prevents sign changes that would indicate a reversal in bending direction. Consequently, the osculating circles at each point lie entirely on the convex side of the curve.

Arc length and enclosed area

The arc length L of a convex curve \gamma: [a, b] \to \mathbb{R}^2 parametrized by \gamma(t) = (x(t), y(t)) is given by the integral

L = \int_a^b \|\gamma'(t)\| \, dt = \int_a^b \sqrt{\left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2} \, dt,

where the parametrization traces the curve once. For a closed convex curve enclosing a region with inradius r (the radius of the largest inscribed circle), the perimeter satisfies L \geq 2\pi r, with equality if and only if the curve is a circle. The area A enclosed by a simple closed convex curve can be computed using Green's theorem in the form

A = \frac{1}{2} \oint (x \, dy - y \, dx),

where the line integral is taken over the boundary curve in the counterclockwise direction. For convex curves of fixed perimeter L, the isoperimetric inequality states that A \leq \frac{L^2}{4\pi}, with equality holding if and only if the curve is a circle; thus, among all such curves, the circle encloses the maximum area. Equivalently, the isoperimetric quotient satisfies

\frac{4\pi A}{L^2} \leq 1,

again with equality for the circle. When expressed via the support function h(\theta), where \theta \in [0, 2\pi) parametrizes the outward unit normal direction, the arc length of a closed convex curve is

L = \int_0^{2\pi} (h(\theta) + h''(\theta)) \, d\theta,

assuming h is sufficiently smooth (twice differentiable). The enclosed area in terms of the support function is

A = \frac{1}{2} \int_0^{2\pi} (h(\theta)^2 - h'(\theta)^2) \, d\theta.

Inscribed polygons

A polygon is inscribed in a convex curve if all its vertices lie on the curve and all its edges lie within the convex domain bounded by the curve. For a closed convex curve C, the inscribed convex polygon with n vertices that maximizes the enclosed area converges to C as n \to \infty, in the sense that the symmetric difference between the polygon and the domain bounded by C tends to zero. This convergence holds uniformly when the vertices are chosen appropriately, such as by spacing them evenly along the affine arc length parametrization of C, where the affine arc length is defined as s(t) = \int_0^t \kappa^{1/3}(\tau) \, d\tau with \kappa denoting the curvature. The error in this approximation is bounded in terms of the curvature of C; for a C^4 curve with positive curvature, the symmetric difference distance \delta_S(C, P_i^n) satisfies \delta_S(C, P_i^n) \sim \frac{1}{12} \lambda^3 n^{-2} - \frac{1}{2} \frac{\lambda^4}{5!} \int_0^\lambda k(s) \, ds \, n^{-4} + o(n^{-4}), where \lambda = \int_0^l \kappa^{1/3}(\tau) \, d\tau is the total affine perimeter and k(s) is the affine curvature. Affine-regular polygons, which are affine images of regular polygons, exhibit special symmetries when inscribed in convex curves; every plane convex set admits an inscribed affine-regular n-gon for each n \geq 3. One construction of an inscribed n-gon uses the support function h(\theta) to place vertices at angles \theta_k = \frac{2\pi k}{n} for k = 0, 1, \dots, n-1, with position x(\theta_k) = h(\theta_k) (\cos \theta_k, \sin \theta_k) + h'(\theta_k) (-\sin \theta_k, \cos \theta_k). This corresponds to equally spaced outward normal directions but does not generally yield the affine-regular polygon. The side lengths between consecutive vertices are then the Euclidean distances between these points, which depend on differences in h and its derivative h' over the angular increments.

Examples

Elementary cases

A fundamental example of a strictly convex closed curve is the circle, parametrized by \gamma(\theta) = (r \cos \theta, r \sin \theta) for $0 \leq \theta < 2\pi, where r > 0 is the radius. This curve bounds a disk, and any line segment joining two points on the circle lies entirely within the disk, satisfying the convexity condition. The circle exhibits constant curvature \kappa = 1/r.^[33] The ellipse provides another canonical instance of a strictly convex closed curve. It is parametrized by \gamma(\theta) = (a \cos \theta, b \sin \theta) for $0 \leq \theta < 2\pi, where a \geq b > 0. The support function of the ellipse, which describes the signed distance from the origin to the tangent line in direction \theta, is given by h(\theta) = \sqrt{a^2 \cos^2 \theta + b^2 \sin^2 \theta}.^[34] In contrast, the astroid, parametrized by x = a \cos^3 t, y = a \sin^3 t for $0 \leq t < 2\pi with a > 0, serves as a non-example of a convex curve due to its four cusps, which cause certain line segments between points on the curve to lie outside the region it bounds.^[35]

Parametric representations

Convex curves, being the boundaries of convex sets in the plane, can be represented parametrically by functions \mathbf{r}(t) = (x(t), y(t)) where t varies over an interval, typically [0, 2\pi) for closed curves, ensuring the curve is simple and the enclosed region is convex. Such parametrizations often use trigonometric functions to maintain smoothness and convexity, with the parameter t corresponding to an angular or arc-length measure. These representations facilitate computations like curvature analysis and are widely used in geometry and computer-aided design.^[36] A fundamental example is the circle, a strictly convex curve of constant curvature. For a circle of radius a centered at the origin, the parametric equations are

x(t) = a \cos t, \quad y(t) = a \sin t, \quad t \in [0, 2\pi).

This parametrization traces the curve counterclockwise, with the tangent vector maintaining unit speed when scaled appropriately.^[33] The ellipse generalizes the circle while preserving convexity. For an ellipse centered at the origin with semimajor axis a along the x-axis and semiminor axis b \leq a along the y-axis, the parametric equations are

x(t) = a \cos t, \quad y(t) = b \sin t, \quad t \in [0, 2\pi).

This form arises from affine transformations of the unit circle and ensures the curve is closed and convex, with eccentricity e = \sqrt{1 - (b/a)^2}.^[37] Another representative convex curve is the cardioid, generated as the epicycloid of a circle rolling on another of equal radius. Its parametric equations, for a cardioid with "cusp radius" a, are

x(t) = a \cos t (1 - \cos t), \quad y(t) = a \sin t (1 - \cos t), \quad t \in [0, 2\pi).

The cardioid is convex, with a cusp at the origin where curvature becomes infinite, yet the enclosed region remains convex due to non-negative curvature.^[38] For non-smooth convex curves, the Reuleaux triangle exemplifies constant-width sets. Constructed from an equilateral triangle of side length 1, its boundary consists of three circular arcs. A parametric representation over one arc (from \beta = 0 to \pi/3) is

x(\beta) = 1 - \cos \beta - \sqrt{3} \sin \beta, \quad y(\beta) = 1 - \sin \beta - \sqrt{3} \cos \beta,

with cyclic permutations for the other arcs to complete the closed curve. This curve has constant width 1 and is strictly convex except at vertices.^[39] Superellipses provide a family of convex curves interpolating between squares and circles. For exponents n \geq 1, the parametric equations are

x(t) = a |\cos t|^{2/n} \operatorname{sgn}(\cos t), \quad y(t) = b |\sin t|^{2/n} \operatorname{sgn}(\sin t), \quad t \in [0, 2\pi),

yielding convex shapes; as n \to \infty, the curve approaches a rectangle (in the limit, with sharp corners).^[40]

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[43]
[PDF] Synopsis and Exercises for the Theory of Convex Sets
Apr 28, 2009 · (e) Give the support function of σ1 + σ2 as a function x and y. 9–6 In R3 let σi be the line segment joining −ei to ei, i = 1,2,3, where e1 ...
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[PDF] Archimedes' quadrature of the parabola and the method of exhaustion
ellipse circle parabola. 2. ARCHIMEDES' THEOREM. A segment of a convex curve (such as a parabola, ellipse or hy- perbola) is a region bounded by a straight ...
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Astroid -- from Wolfram MathWorld
### Summary of Astroid Convexity and Parametric Equation
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Parametric Equations -- from Wolfram MathWorld
... equation of a circle in Cartesian coordinates can be given by r^2=x^2+y^2, one set of parametric equations for the circle are given by x = rcost (1) y ...
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Cardioid -- from Wolfram MathWorld
/2. The cardioid has Cartesian equation (x^2+y^2+ax)^2=a^2(x^2+y^2), (3) and the parametric equations x = acost(1-cost) (4) y = asint(1-cost). (5) The cardioid ...
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Reuleaux Triangle -- from Wolfram MathWorld
A curve of constant width constructed by drawing arcs from each polygon vertex of an equilateral triangle between the other two vertices.
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Superellipse -- from Wolfram MathWorld
A superellipse is a curve with a Cartesian equation, described parametrically by x=acos^(2/r)t and y=bsin^(2/r)t.