Polytopes

Polytope{K,Dim,T}

We say that a geometry is a K-polytope when it is a collection of "flat" sides that constitue a K-dimensional subspace. They are called polygon and polyhedron respectively for 2D (K=2) and 3D (K=3) subspaces, embedded in a Dim-dimensional space. The parameter K is also known as the rank or parametric dimension of the polytope: https://en.wikipedia.org/wiki/Abstract_polytope.

The term polytope expresses a particular combinatorial structure. A polyhedron, for example, can be decomposed into faces. Each face can then be decomposed into edges, and edges into vertices. Some conventions act as a mapping between vertices and higher dimensional features (edges, faces, cells...), removing the need to store all features.

Additionally, the following property must hold in order for a geometry to be considered a polytope: the boundary of a (K+1)-polytope is a collection of K-polytopes, which may have (K-1)-polytopes in common. See https://en.wikipedia.org/wiki/Polytope.

Notes

• Type aliases are Polygon, Polyhedron.

Polygon{Dim,T}

A polygon is a 2-polytope, i.e. a polytope with parametric dimension 2.

Polyhedron{Dim,T}

A polyhedron is a 3-polytope, i.e. a polytope with parametric dimension 3.

Segment(p1, p2)

An oriented line segment with end points p1, p2. The segment can be called as s(t) with t between 0 and 1 to interpolate linearly between its endpoints.

See also Line.

Ngon(p1, p2, ..., pN)

A N-gon is a polygon with N vertices p1, p2, ..., pN oriented counter-clockwise (CCW). In this case the number of vertices is fixed and known at compile time. Examples of N-gon are Triangle (N=3), Quadrangle (N=4), Pentagon (N=5), etc.

Notes

• Although the number of vertices N is known at compile time, we use abstract vectors to store the list of vertices. This design allows constructing N-gon from views of global vectors without expensive memory allocations.

• Type aliases are Triangle, Quadrangle, Pentagon, Hexagon, Heptagon, Octagon, Nonagon, Decagon.

Chain(p1, p2, ..., pn)

A polygonal chain from a sequence of points p1, p2, ..., pn. See https://en.wikipedia.org/wiki/Polygonal_chain.

PolyArea(outer, [inner1, inner2, ..., innerk]; fix=true)

A polygonal area with outer chain, and optional inner chains inner1, inner2, ..., innerk.

Chains can be a vector of Point or a vector of tuples with coordinates for convenience.

The option fix tries to correct issues with polygons in the real world, including issues with:

• orientation - Most algorithms assume that the outer ring is oriented counter-clockwise (CCW) and that all inner rings are oriented clockwise (CW).

• degeneracy - Sometimes data is shared with degenerate rings (i.e. only 2 vertices).

Tetrahedron(p1, p2, p3, p4)

A tetrahedron with points p1, p2, p3, p4.

Hexahedron(p1, p2, ..., p8)

A hexahedron with points p1, p2, ..., p8.

Pyramid(p1, p2, p3, p4, p5)

A pyramid with points p1, p2, p3, p4, p5.