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Farey sunburst of order 6, with 1 interior (red) and 96 boundary (green) points giving an area of 1 + 96 / 2 − 1 = 48 [1]. In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary.
A supporting plane is a plane that intersects the boundary of D but not the interior of D. One defines the width of the body as before. If the width of D is the same in all directions, then one says that the body is of constant width and calls its boundary a surface of constant width, and the body itself is referred to as a spheroform.
Boundary representation has also been extended to allow special, non-solid model types called non-manifold models. As described by Braid, normal solids found in nature have the property that, at every point on the boundary, a small enough sphere around the point is divided into two pieces, one inside and one outside the object.
In elementary geometry, a face is a polygon [note 1] on the boundary of a polyhedron. [3] [4] Other names for a polygonal face include polyhedron side and Euclidean plane tile. For example, any of the six squares that bound a cube is a face of the cube. Sometimes "face" is also used to refer to the 2-dimensional features of a 4-polytope.
A simple polygon is the boundary of a region of the plane that is called a solid polygon. The interior of a solid polygon is its body, also known as a polygonal region or polygonal area. In contexts where one is concerned only with simple and solid polygons, a polygon may refer only to a simple polygon or to a solid polygon.
It is relatively simple to prove that the Jordan curve theorem holds for every Jordan polygon (Lemma 1), and every Jordan curve can be approximated arbitrarily well by a Jordan polygon (Lemma 2). A Jordan polygon is a polygonal chain, the boundary of a bounded connected open set, call it the open polygon, and its closure, the