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The intersection of two convex polygons is a convex polygon. A convex polygon may be triangulated in linear time through a fan triangulation, consisting in adding diagonals from one vertex to all other vertices. Helly's theorem: For every collection of at least three convex polygons: if all intersections of all but one polygon are nonempty ...
Convex regular icosahedron. Let P and Q be combinatorially equivalent 3-dimensional convex polytopes; that is, they are convex polytopes with isomorphic face lattices.Suppose further that each pair of corresponding faces from P and Q are congruent to each other, i.e. equal up to a rigid motion.
For a simple polygon (non-self-intersecting), regardless of whether it is convex or non-convex, this angle is called an internal angle (or interior angle) if a point within the angle is in the interior of the polygon. A polygon has exactly one internal angle per vertex. If every internal angle of a simple polygon is less than a straight angle ...
Carathéodory's theorem (convex hull) - If a point x of R d lies in the convex hull of a set P, there is a subset of P with d+1 or fewer points such that x lies in its convex hull. Choquet theory - an area of functional analysis and convex analysis concerned with measures with support on the extreme points of a convex set C .
Analogously to the two ears theorem, every non-convex simple polygon has at least one mouth. Polygons with the minimum number of principal vertices of both types, two ears and a mouth, are called anthropomorphic polygons. [7] Repeatedly finding and removing a mouth from a non-convex polygon will eventually turn it into the convex hull of the ...
Even though the word "polygon" is used to describe this region, in general it can be any convex shape with curved edges. The support polygon is invariant under translations and rotations about the gravity vector (that is, if the contact points and friction cones were translated and rotated about the gravity vector, the support polygon is simply translated and rotated).
For equilateral zonogons, a -sided one can be tiled by () rhombi.) In this tiling, there is a parallelogram for each pair of slopes of sides in the 2 n {\displaystyle 2n} -sided zonogon. At least three of the zonogon's vertices must be vertices of only one of the parallelograms in any such tiling. [ 5 ]
In the case of genus one, a fundamental convex polygon is sought for the action by translation of Λ = Z a ⊕ Z b on R 2 = C where a and b are linearly independent over R. (After performing a real linear transformation on R 2, it can be assumed if necessary that Λ = Z 2 = Z + Z i; for a genus one Riemann surface it can be taken to have the form Λ = Z 2 = Z + Z ω, with Im ω > 0.)