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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.
The boundaries are defined by lines, conics, polylines, surface curves, or b spline curves; ISO 10303-514 Advanced boundary representation, a solid defining a volume with possible voids that is composed by advanced faces; ISO 10303-509 Manifold surface, a non intersecting area in 3D that is composed by advanced faces
In solid modeling and computer-aided design, the Euler operators modify the graph of connections to add or remove details of a mesh while preserving its topology. They are named by Baumgart [1] after the Euler–Poincaré characteristic. He chose a set of operators sufficient to create useful meshes, some lose information and so are not invertible.
In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object; [1] a three-dimensional solid bounded exclusively by faces is a polyhedron. A face can be finite like a polygon or circle, or infinite like a half-plane or plane. [2]
This equation, stated by Euler in 1758, [2] is known as Euler's polyhedron formula. [3] It corresponds to the Euler characteristic of the sphere (i.e. χ = 2 {\displaystyle \ \chi =2\ } ), and applies identically to spherical polyhedra .
Another important class of simple polygons are the star-shaped polygons, the polygons that have a point (interior or on their boundary) from which every point is visible. [ 2 ] A monotone polygon , with respect to a straight line L {\displaystyle L} , is a polygon for which every line perpendicular to L {\displaystyle L} intersects the interior ...
A Seifert surface of a knot is however a manifold with boundary, the boundary being the knot, i.e. homeomorphic to the unit circle. The genus of such a surface is defined to be the genus of the two-manifold, which is obtained by gluing the unit disk along the boundary.
In geometry, an edge is a particular type of line segment joining two vertices in a polygon, polyhedron, or higher-dimensional polytope. [1] In a polygon, an edge is a line segment on the boundary, [2] and is often called a polygon side. In a polyhedron or more generally a polytope, an edge is a line segment where two faces (or polyhedron sides ...