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The boundary representation for a solid object, its surface, is a polygon mesh of vertices, edges and faces. Its topology is captured by the graph of the connections between faces. Its topology is captured by the graph of the connections between faces.
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.
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.
In solid modeling and computer-aided design, boundary representation (often abbreviated B-rep or BREP) is a method for representing a 3D shape [1] by defining the limits of its volume. A solid is represented as a collection of connected surface elements, which define the boundary between interior and exterior points.
Every body of constant width is a convex set, its boundary crossed at most twice by any line, and if the line crosses perpendicularly it does so at both crossings, separated by the width. By Barbier's theorem , the body's perimeter is exactly π times its width, but its area depends on its shape, with the Reuleaux triangle having the smallest ...
The expression thus derived from a fundamental polygon of a surface turns out to be the sole relation in a presentation of the fundamental group of the surface with the polygon edge labels as generators. This is a consequence of the Seifert–van Kampen theorem. Gluing edges of polygons is a special kind of quotient space process. The quotient ...
A Johnson solid is a convex polyhedron whose faces are all regular polygons. [2] Here, a polyhedron is said to be convex if the shortest path between any two of its vertices lies either within its interior or on its boundary, none of its faces are coplanar (meaning they do not share the same plane, and do not "lie flat"), and none of its edges are colinear (meaning they are not segments of the ...