Polytopes

Polytope{K,M,CRS}

We say that a geometry is a K-polytope when it is a collection of "flat" sides that constitute a K-dimensional subspace. They are called chain, polygon and polyhedron respectively for 1D (K=1), 2D (K=2) and 3D (K=3) subspaces. 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 Chain, Polygon, Polyhedron.

Chain{M,CRS}

A chain is a 1-polytope, i.e. a polytope with parametric dimension 1. See https://en.wikipedia.org/wiki/Polygonal_chain.

See also Segment, Rope, Ring.

Polygon{M,CRS}

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

See also Ngon and PolyArea.

Polyhedron{M,CRS}

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

See also Tetrahedron, Hexahedron and Pyramid.

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 Rope, Ring, Line.

Rope(p1, p2, ..., pn)

An open polygonal chain from a sequence of points p1, p2, ..., pn.

See also Chain and Ring.

Ring(p1, p2, ..., pn)

A closed polygonal chain from a sequence of points p1, p2, ..., pn.

See also Chain and Rope.

Ngon(p₁, p₂, ..., pₙ)

A N-gon is a polygon with N ≥ 3 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.

PolyArea(outer)
PolyArea([outer, inner₁, inner₂, ..., innerₖ])

A polygonal area with outer ring, and optional inner rings inner₁, inner₂, ..., innerₖ.

Rings can be a vector of Point or a vector of tuples with coordinates for convenience, in which case the first point should not be repeated at the end of the vector.

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.