Fact-checked by Grok 2 weeks ago

Convex polygon

A convex polygon is a simple closed plane figure formed by a finite number of straight line segments connected end-to-end, where all interior angles measure less than 180 degrees and the line segment joining any two points within the polygon lies entirely inside or on the boundary of the polygon.^[1]^[2] This distinguishes convex polygons from concave ones, which have at least one interior angle greater than 180 degrees and may allow line segments between interior points to extend outside the figure.^[1]^[3] Key properties of convex polygons include the fact that they form the boundary of a convex set, meaning the entire polygonal region is convex and any line through the interior intersects the boundary exactly twice unless tangent to an edge or vertex.^[2] The sum of their interior angles is given by the formula (n - 2) \times 180^\circ, where n is the number of sides, and the number of diagonals is \frac{n(n-3)}{2}, all of which lie entirely within the polygon.^[1] Convex polygons are always simple, with no self-intersections, and their vertices and sides satisfy the condition that the figure separates the plane into an interior and exterior region without indentations or "caved-in" sides.^[3]^[2] In mathematics and computational geometry, convex polygons serve as fundamental building blocks for studying convexity, optimization problems, and algorithms such as convex hull computation, due to their well-behaved geometric properties that ensure efficient intersection tests and triangulation.^[3]

Definition

Formal definition

A convex polygon is a simple polygon in which no line segment between two points on the boundary or in the interior ever exits the polygon.^[3] This ensures that the entire region bounded by the polygon, including both its boundary and interior, remains intact under such connections. Formally, a polygon P with vertices v_1, v_2, \dots, v_n (where n \geq 3) is convex if P is the boundary of a convex set C \subseteq \mathbb{R}^2, where a set C is convex if, for any two points p, q \in C, the line segment joining p and q lies entirely within C.^[4] Equivalently, the filled region (the convex hull) of P can be expressed as \operatorname{Conv}(\{v_1, \dots, v_n\}), the smallest convex set containing the vertices, assuming they are not collinear.^[4] This definition explicitly incorporates the boundary and interior in the convexity condition, distinguishing convex polygons from non-convex ones that may have indentations or self-intersections. The concept of convexity traces back to Euclidean geometry in ancient times, where properties of straight lines and figures implicitly relied on such ideas, but it was formalized in modern mathematical terms as part of convex set theory by Hermann Minkowski in the late 19th century (1897–1903).^[5]

Equivalent characterizations

A convex polygon admits several equivalent characterizations that provide alternative ways to verify or define its convexity, building upon the formal definition as a convex set. These characterizations are particularly useful for simple polygons in the Euclidean plane, where the boundary does not intersect itself.^[6] One characterization relies on the interior angles: a simple polygon is convex if and only if every interior angle is less than or equal to 180 degrees (or \pi radians).^[7] This condition ensures that the polygon does not "dent inward" at any vertex, maintaining the overall outward-pointing structure. Another equivalent condition involves the direction of turns along the boundary: when traversing the polygon's edges in order (either clockwise or counterclockwise), the turning angle at each vertex must be consistently in the same direction, such as all left turns for a counterclockwise traversal or all right turns otherwise.^[8] This can be computationally tested by checking that the perpendicular dot product of consecutive edge vectors has the same sign throughout, confirming uniform orientation without reversals.^[8] A convex polygon can also be characterized as the intersection of half-planes, each defined by one of its edges extended to a line, with the half-plane containing the polygon on the appropriate side of each line.^[6] Since the intersection of convex sets (half-planes) is convex, this representation directly yields a bounded convex region whose boundary consists of portions of the original edges. Finally, the vertices of a convex polygon lie in convex position, meaning that the convex hull of the vertices has all of them on its boundary, or equivalently, no vertex lies in the interior of the triangle formed by any three others.^[9] This property distinguishes convex polygons from non-convex ones, where at least one vertex would be inside the hull of the rest.

Basic Properties

Interior angle properties

In a convex polygon with n sides, the sum of the interior angles is (n-2)\pi radians. This formula arises from triangulating the polygon into n-2 triangles, where each triangle contributes \pi radians to the total angle sum.^[10] Each interior angle of a convex polygon must be strictly less than \pi radians to maintain convexity, as an angle of \pi radians or greater would create a reflex angle indicative of concavity. This condition on interior angles ensures no "dents" form in the boundary, distinguishing convex polygons from concave ones.^[11]^[12] The exterior angles of a convex polygon, defined as the supplements to the interior angles at each vertex, sum to exactly $2\pi radians, corresponding to a full rotation around the boundary. Each exterior angle lies between $0 and \pi radians and has the same sign (typically positive for counterclockwise orientation), reflecting the consistent turning direction without reversal.^[13] For a regular convex polygon, where all sides and angles are equal, each interior angle measures \frac{(n-2)\pi}{n} radians. This uniform distribution exemplifies the angle properties while satisfying the overall sum constraint.^[10]

Line segment and intersection properties

A fundamental property of a convex polygon is that it is a convex set, meaning that for any two points in the polygon—whether in the interior or on the boundary—the entire line segment connecting them lies within the polygon. This containment ensures that the polygon has no "dents" or indentations that would force parts of such segments outside its boundaries. Formally, if P is a convex polygon and x, y \in P, then the set \{ \theta x + (1 - \theta) y \mid \theta \in [0, 1] \} \subseteq P.^[14] This line segment property directly implies that all diagonals of a convex polygon lie entirely inside the polygon. Since the vertices of the polygon are points on its boundary, any diagonal connecting non-adjacent vertices is a line segment fully contained within P, with no portion extending externally. Unlike concave polygons, where some diagonals may exit and re-enter the boundary, convex polygons exhibit no such external diagonals, reinforcing their global coherence.^[11] The boundary of a convex polygon intersects any straight line in at most two points. This follows from the convexity: if a line intersected the boundary at three or more points, the segments between those points would violate the line segment property by implying non-convex portions. For instance, in computational geometry contexts, this limits intersections during sweep line algorithms to at most two per convex component.^[15] Convex polygons are a special case of star-shaped polygons, where the kernel—the set of points from which the entire polygon is visible—coincides with the entire interior of the polygon. Every point in the interior can "see" all other points via line segments that remain inside, distinguishing convex polygons from more general star-shaped ones with smaller kernels.^[16] The boundary of a convex polygon forms a simple closed curve, and by the Jordan curve theorem, it divides the plane into exactly two regions: a bounded interior (the polygon itself) and an unbounded exterior, with no overlaps or ambiguities in defining the interior. This well-defined separation underscores the topological simplicity of convex polygons compared to self-intersecting or non-simple curves.^[17]

Geometric Measures

Perimeter and boundary

The perimeter of a convex polygon, denoted as P, is defined as the sum of the lengths of its sides, given by the formula P = \sum_{i=1}^{n} \|v_{i+1} - v_i\|, where v_1, v_2, \dots, v_n are the vertices of the polygon and v_{n+1} = v_1.^[18] This measurement directly quantifies the total boundary length, forming a closed loop that encloses the polygon's interior without any overlaps or indentations due to its convex nature. The boundary of a convex polygon consists of a closed convex chain of edges, where the sequence of vertices forms a counterclockwise (or clockwise) traversal in which each consecutive pair of edges turns monotonically in the same direction, ensuring consistent left (or right) turns relative to the interior.^[19] This structure guarantees that the boundary is simple and Jordan-closed, with no self-intersections, which simplifies perimeter computation to a direct summation of edge lengths without requiring unfolding or intersection checks that might be necessary for non-convex polygons.^[20] Among all simple n-gons with a fixed area and fixed number of sides n, the regular convex n-gon uniquely achieves the minimal perimeter, as established by the polygonal isoperimetric inequality.^[21] Furthermore, from any point exterior to the convex polygon, exactly two tangent lines can be drawn to its boundary, each touching at a single vertex or edge and separating the point from the interior.^[22]

Area formulas

The area of a convex polygon can be computed using several methods that exploit its boundary vertices and the absence of indentations, ensuring straightforward calculations without handling self-intersections. One efficient approach is the shoelace formula (also known as Gauss's shoelace formula), which applies to any simple polygon but is particularly reliable for convex polygons due to their monotonically ordered vertices in counterclockwise or clockwise traversal, avoiding sign inconsistencies from concavities.^[23] For a convex polygon with n vertices (x_1, y_1), (x_2, y_2), \dots, (x_n, y_n) listed in counterclockwise order, the area A is given by

A = \frac{1}{2} \left| \sum_{i=1}^{n} (x_i y_{i+1} - x_{i+1} y_i) \right|,

where (x_{n+1}, y_{n+1}) = (x_1, y_1).^[23] This formula yields a positive value for counterclockwise ordering and negative for clockwise, with the absolute value ensuring the magnitude; the convexity guarantees that the boundary does not cross itself, simplifying implementation in computational geometry.^[23] Another fundamental method decomposes the convex polygon into non-overlapping triangles, leveraging the property that any convex polygon can be triangulated by drawing diagonals from a single vertex to all non-adjacent vertices, forming a "fan" of n-2 triangles.^[24] The total area is then the sum of the areas of these triangles, each computed using the standard formula for the area of a triangle given three vertices, such as \frac{1}{2} |\det(\mathbf{v}_i - \mathbf{v}_0, \mathbf{v}_{i+1} - \mathbf{v}_0)| from a fixed vertex \mathbf{v}_0.^[24] This decomposition is efficient for convex polygons because all such diagonals lie entirely within the interior, requiring no checks for intersection or exterior placement.^[24] For regular convex n-gons, where all sides have equal length s and interior angles are uniform, a closed-form expression simplifies the area calculation. The area A is

A = \frac{n s^2}{4 \tan(\pi/n)}.

^[25] This formula derives from dividing the polygon into n identical isosceles triangles from the center, each with two sides equal to the circumradius and apex angle $2\pi/n.^[25] A specialized formula exists for cyclic convex quadrilaterals, which can be inscribed in a circle and thus form a subset of convex quadrilaterals. Brahmagupta's formula gives the area A in terms of the side lengths a, b, c, d and semiperimeter s = (a + b + c + d)/2 as

A = \sqrt{(s - a)(s - b)(s - c)(s - d)}.

^[26] This extends Heron's formula for triangles and applies specifically when the quadrilateral is cyclic, maximizing the area for given side lengths among all convex quadrilaterals.^[26]

Strict Convexity

Definition of strict convexity

A strictly convex polygon is a convex polygon in which all interior angles are strictly less than 180 degrees, distinguishing it from more general convex polygons that may include flat angles of exactly 180 degrees.^[27] This condition is equivalent to requiring that no three consecutive vertices are collinear, as collinearity of such vertices would result in an interior angle of precisely 180 degrees at the middle vertex. An equivalent characterization of strict convexity for a polygon is that the triangle formed by every set of three consecutive vertices has positive area. If three consecutive vertices were collinear, the area of this triangle would be zero, violating the strict inequality on interior angles. Strictly convex polygons are often described as having no flat sides in the sense that their boundaries turn strictly at each vertex, without any straight extensions across multiple vertices. All regular polygons with n \geq 3 sides satisfy this definition, as their uniform interior angles are always less than 180 degrees. Similarly, standard rectangles are strictly convex, though degenerate cases (such as collapsed to a line segment) would not be.

Implications for strictly convex polygons

Strictly convex polygons, characterized by no three vertices lying on the same straight line, possess unique supporting lines for each edge, as the line containing any edge passes through exactly two vertices and has all other vertices strictly on one side. This property eliminates degenerate cases in computational geometry algorithms, such as convex hull computations and intersection tests, where collinear points can cause numerical instability or require special handling. For instance, in algorithms for determining the convex position of point sets, assuming strict convexity ensures efficient O(n log n) time complexity without additional checks for collinearity. The absence of collinear vertices in strictly convex polygons facilitates efficient triangulation without the need for additional Steiner points, as any such polygon can be divided into n-2 triangles of positive area by fanning from a single vertex to all non-adjacent vertices. This results in a straightforward O(n construction, avoiding zero-area degenerate triangles that might arise in non-strict cases with flat vertices, and supports applications in mesh generation and finite element methods where non-degeneracy is crucial. Strictly convex polygons provide superior approximations to smooth strictly convex curves compared to general convex polygons, as their lack of flat segments allows closer uniform convergence in Hausdorff distance when inscribing vertices along the curve. For C^3-smooth strictly convex arcs with non-degenerate curvature, random lattice strictly convex polygons achieve optimal approximation rates, minimizing errors in perimeter and area estimates for applications like numerical integration over curved boundaries.^[28] In a strictly convex polygon, the boundary encloses a convex set where line segments connecting any two boundary points not on the same edge lie entirely in the interior except at the endpoints, enhancing properties like unique nearest-point projections on the boundary. This distinguishes it from non-strict convex polygons, where collinear vertices can place such segments partially on the boundary.^[29] The prohibition of flat parts in strictly convex polygons ensures no zero-area regions along the boundary, which maximizes the enclosed area for a fixed perimeter among polygons with the same number of sides in certain embeddings, approaching the isoperimetric optimum of the circle as the number of sides increases. This property is leveraged in optimization problems, such as tangential polygon constructions around convex bodies, where strict convexity guarantees the global maximum area without degenerate configurations.^[30]

References

[1]
[PDF] Page 1 of 5 Math 1312 Section 2.5 Convex Polygons Definition
Definition: A polygon is closed plane figure whose sides are line segments that intersect only at the endpoints. A convex polygon has two properties: a) Every ...
[2]
Definition of Polygons - Department of Mathematics at UTSA
Dec 11, 2021 · Convex: any line drawn through the polygon (and not tangent to an edge or corner) meets its boundary exactly twice. As a consequence, all its ...
[3]
Good Definitions as Biconditionals; Polygons - Andrews University
A polygon is convex if and only if its corresponding polygonal region is convex. We may use the word concave to describe a nonconvex polygon (but not a ...
[4]
[PDF] Introduction to Polygons - Cornell Mathematics
A (convex) polygon is a subset P of R2 of the form P = Conv({p1,...,pn}) for some points p1,...,pn not lying on a line.1. 4. Vertices and Edges. Based on the ...
[5]
1 Introduction - arXiv
May 28, 2024 · The notion of a convex set was introduced by Hermann Minkowski at the close of the 19th century. In 1897–1903, he published four papers ...
[6]
Polygon -- from Wolfram MathWorld
dimensions is called a polytope. PolygonInternalAngles. The sum I of interior angles in the top left diagram of a dissected polygon is. I=sum_(i=1)^n(alpha_i+ ...
[7]
Convex Polygon Definition - Math Open Reference
A convex polygon is defined as a polygon with all its interior angles less than 180°. This means that all the vertices of the polygon will point outwards.
[8]
Convex Polygon – Definition, Formula, Properties, Types, Examples
A convex polygon is a polygon with all interior angles less than 180 ∘ and vertices are pointed outwards.
[9]
Exterior Angle -- from Wolfram MathWorld
... interior angle bisector) is not given any special name. The sum of the angles gamma_i in a convex polygon is equal to 2pi radians ( 360 degrees ), since ...
[10]
[PDF] Lecture 3: September 4 3.1 Convex Sets
3.1 Convex Sets. Definition 3.1 A set C is convex if the line segment between any two points in C lies in C, i.e. ∀x1,x2 ∈ C, ∀θ ∈ [0, 1] θx1 + (1 − θ)x2 ∈ C. ...
[11]
[PDF] CMSC 754: Lecture 6 Halfplane Intersection and Point-Line Duality
By convexity, the sweep line intersects the boundary of each convex polygon Ki in at most two points, one for the upper chain and one for the lower chain. Hence ...
[12]
[PDF] an optimal visibility algorithm for a simple polygon with star-shaped ...
A convex polygon is the special case of a star-shaped polygon in which the kernel is the entire polygon. Figure 1. A star-shaped polygon and its kernel.Missing: interior | Show results with:interior
[13]
[PDF] 1 The Jordan Polygon Theorem - CGVR
The theorem states that any simple closed curve partitions the plane into two connected subsets, exactly one of which is bounded.
[14]
Perimeter Area - Department of Mathematics at UTSA
Dec 15, 2021 · A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length.
[15]
Polygon simplification by minimizing convex corners - ScienceDirect
Oct 29, 2019 · A convex chain of P is a path C i j = ( v i , v i + 1 , … , v j − 1 , v j ) of strictly convex vertices, where the indices are considered modulo ...<|separator|>
[16]
[PDF] 30 POLYGONS - CSUN
Jul 25, 2017 · Star-shaped polygon: The entire polygon is visible from some point inside the polygon. Orthogonal polygon: A polygon with sides parallel to ...
[17]
[PDF] on the stability of the polygonal isoperimetric inequality - CMU Math
the convex regular polygon uniquely minimizes the perimeter among all polygons subject to an area constraint. In other words, if L∗ denotes the perimeter ...
[18]
Convex Polygon - an overview | ScienceDirect Topics
A convex polygon is a polygon in which every pair of points within the polygon is visible to each other. It is a polygon that has no reflex vertices. AI ...<|control11|><|separator|>
[19]
Polygon Area -- from Wolfram MathWorld
The area of a convex polygon is defined to be positive if the points are arranged in a counterclockwise order and negative if they are in clockwise order.
[20]
[PDF] Polygon Triangulation - GMU CS Department
Every convex polygon may be triangulated as a “fan,” with all diagonals incident to a common vertex. The area of a polygon with vertices v. 0. , v. 1.
[21]
Regular Polygon -- from Wolfram MathWorld
Regular Polygon ; R · = 1/2acsc(pi/n) ; = rsec(pi/n) ; A, = 1/4na^2cot(pi/n) ; = nr^2tan(pi/n) ; = 1/2nR^2sin((2pi)/n).
[22]
Brahmagupta's Formula -- from Wolfram MathWorld
Brahmagupta's formula K=sqrt((s-a)(s-b)(s-c)(s-d)) (3) is a special case giving the area of a cyclic quadrilateral (i.e., a quadrilateral inscribed in a ...
[23]
[PDF] Strictly Convex Drawings of Planar Graphs - arXiv
Jun 21, 2006 · ... interior angles are less than 180◦. Theorem 1. (i) A three-connected planar graph with n vertices in which every face has at most k edges ...<|control11|><|separator|>
[24]
[PDF] The Parameterized Complexity of Guarding Almost Convex Polygons
... convex polygon, all interior angles are less than or equal to. 666. 180 degrees, while in a strictly convex polygon all interior angles are less than 180 ...<|separator|>
[25]
[PDF] Convex Polygons in Cartesian Products - DROPS
Abstract. We study several problems concerning convex polygons whose vertices lie in a Cartesian product of two sets of n real numbers (for short, grid).<|separator|>
[26]
[PDF] . OPTIMAL TRIANGULATION OF POLYGONS
The constrained Delaunay triangulation does the same for triangulating polygons without Steiner points (see. [32], [33]), and an algorithm using only ...
[27]
[PDF] Approximation of Convex Curves by Random Lattice Polygons
In the present work, we solve the approximation problem for a subclass of 0 consisting of C3-smooth strictly convex arcs γ ∈ 0 with non-degenerate curvature. 3.
[28]
[PDF] Lecture 4: Convexity 4.1 Basic Definitions
Definition 4.12 A convex set is strictly convex if for any two points in the set in general position, the line segment less the endpoints is contained in int C.
[29]
[PDF] Extremal Problems for Convex Polygons - GERAD
Increasing the number of sides gives in the limit the famous result that among all plane figures with fixed perimeter the circle encloses the largest area. 2.2.Missing: flats | Show results with:flats<|control11|><|separator|>