The polygon formed by connecting the Bézier points with lines, starting with P < sub > 0 </ sub > and finishing with P < sub > n </ sub >, is called the Bézier polygon ( or control polygon ).

* The start ( end ) of the curve is tangent to the first ( last ) section of the Bézier polygon.

While at university, Gauss independently rediscovered several important theorems ; his breakthrough occurred in 1796 when he showed that any regular polygon with a number of sides which is a Fermat prime ( and, consequently, those polygons with any number of sides which is the product of distinct Fermat primes and a power of 2 ) can be constructed by compass and straightedge.

Each wall, the floor and the ceiling is a simple polygon, in this case, a rectangle.

:# The polygon EFGH is a face of the dual polyhedron.

For example, every polygon is topologically self-dual ( it has the same number of vertices as edges, and these are switched by duality ), but will not in general be geometrically self-dual ( up to rigid motion, for instance ) – regular polygons are geometrically self-dual ( all angles are congruent, as are all edges, so under duality these congruences swap ), but irregular polygons may not be geometrically self-dual.

The structure of C < sub > 60 </ sub > is a truncated icosahedron, which resembles an association football ball of the type made of twenty hexagons and twelve pentagons, with a carbon atom at the vertices of each polygon and a bond along each polygon edge.

Hexa – is a prefix from the Greek word for ' six ' ( hexa, ἕξ ), e. g. in hexagon, a polygon with six corners.

In geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i. e., not a Platonic solid, Archimedean solid, prism or antiprism.

There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex.

Although there is no obvious restriction that any given regular polygon cannot be a face of a Johnson solid, it turns out that the faces of Johnson solids always have 3, 4, 5, 6, 8, or 10 sides.

This square is a fundamental polygon of the Klein bottle.

The generalization of the Klein bottle to higher genus is given in the article on the fundamental polygon.

A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions ( such as a polychoron in four dimensions ).

For example, a polygon is a 2-polytope, a polyhedron is a 3-polytope, and a polychoron is a 4-polytope.

In geometry a polygon () is a flat shape consisting of straight lines that are joined to form a closed chain or

In geometry, polytope means the generalization to any dimension of the sequence: polygon in two dimensions, polyhedron in three dimensions, and polychoron in four dimensions.

The wheels can be any regular polygon except a triangle, but the catenary must have parameters corresponding to the shape and dimensions of the wheels.

As one wraps around the polygon, these triangles with positive and negative area will overlap, and the areas between the origin and the polygon will be cancelled out and sum to 0, while only the area inside the reference triangle remains.

While it is possible to construct analogies to the Penrose triangle with other regular polygons to create a Penrose polygon, the visual effect is not as striking, and as the sides increase, the object seems merely to be warped or twisted.

While point clouds can be directly rendered and inspected, usually point clouds themselves are generally not directly usable in most 3D applications, and therefore are usually converted to polygon or triangle mesh models, NURBS surface models, or CAD models through a process commonly referred to as surface reconstruction.

For a convex polygon ( such as a triangle ), a surface normal can be calculated as the vector cross product of two ( non-parallel ) edges of the polygon.

* Clear pseudocode for calculating a surface normal from either a triangle or polygon.

* The maximum allowed size of the approximation polygon ( for triangulations it can be maximum allowed length of triangle sides ).

Consider a polygon P and a triangle T, with one edge in common with P. Assume Pick's theorem is true for both P and T separately ; we want to show that it is also true to the polygon PT obtained by adding T to P. Since P and T share an edge, all the boundary points along the edge in common are merged to interior points, except for the two endpoints of the edge, which are merged to boundary points.

The last step uses the fact that if the theorem is true for the polygon PT and for the triangle T, then it's also true for P ; this can be seen by a calculation very much similar to the one shown above.

Curves of constant width can be generated by joining circular arcs centered on the vertices of a regular or irregular convex polygon with an odd number of sides ( triangle, pentagon, heptagon, etc.

A Reuleaux triangle is the simplest and best known Reuleaux polygon, a curve of constant width.

The Reuleaux triangle can be generalized to regular polygons with an odd number of sides, yielding a Reuleaux polygon.

The basic polygon is often ( but not necessarily ) a convex plane-filling polygon, such as a square or a triangle.

Since a reflection flips the triangle and changes the orientation, we can join the triangles in pairs and obtain an orientation-preserving tiling polygon.

Just as the other regular cupolas have an alternating sequence of squares and triangles surrounding a single polygon at the top ( triangle, square or pentagon ), each half of the gyrobifastigium consists of just alternating squares and triangles, connected at the top only by a ridge.

Like all cupolae, the base polygon has twice as many sides as the top ( in this case, the bottom polygon is a hexagon because the top is a triangle ).

The basic polygon is often ( but not necessarily ) a convex plane-filling polygon, such as a square or a triangle.

As the polygon generating program feeds triangles to the PowerVR ( driver ), it stores them in memory in a triangle strip or an indexed format.

The base of a pyramid can be trilateral, quadrilateral, or any polygon shape, meaning that a pyramid has at least three outer triangular surfaces ( at least four faces including the base ).

For incircles of non-triangle polygons, see Tangential quadrilateral or Tangential polygon.

A square is a special case of a rhombus ( equal sides, opposite equal angles ), a kite ( two pairs of adjacent equal sides ), a parallelogram ( opposite sides parallel ), a quadrilateral or tetragon ( four-sided polygon ), and a rectangle ( opposite sides equal, right-angles ) and therefore has all the properties of all these shapes, namely:

Tracing all the way around the polygon makes one full turn, so the sum of the exterior angles must be 360 °.

An ordinary polygon is unbounded because the sequence closes back in itself in a loop or circuit, while an apeirogon ( infinite polygon ) is unbounded because it goes on for ever so you can never reach any bounding end point.

The interior ( body ) of the polygon is another element, and ( for technical reasons ) so is the null polytope or nullitope.

VRML is a text file format where, e. g., vertices and edges for a 3D polygon can be specified along with the surface color, UV mapped textures, shininess, transparency, and so on.

This correction makes it so that in parts of the polygon that are closer to the viewer the difference from pixel to pixel between texture coordinates is smaller ( stretching the texture wider ), and in parts that are farther away this difference is larger ( compressing the texture ).

But what makes the game look so good is not incredibly high polygon counts or lots of extravagant lighting and particle effects.

It can also vary as the scene or camera is changed, causing one polygon to " win " the z test, then another, and so on.

* A polygon encloses a region ( called its interior ) and so it always has a measurable area.

The original, fully detailed version contained over 250, 000 polygons although it was designed so that layers of detail could easily be removed to reduce the polygon count.

A related theorem is CPCFC, in which triangles is replaced with figures so that the theorem applies to any polygon or polyhedron.

The first through fourth coronas of the central polygon consist entirely of congruent copies of P, so its Heesch number is at least four.

It is easy to deduce the area of a disk from basic principles: the area of a regular polygon is half its apothem times its perimeter, and a regular polygon becomes a circle as the number of sides increases, so the area of a disk is half its radius times its circumference ( i. e. r × 2πr ).