Geometry

Simplex

A simplex, plural: simplices, is a generalization of the notion of the simplest possible polytope in a given space. for example:

• a 0-simplex is a point,
• a 1-simplex is a line segment,
• a 2-simplex is a triangle,
• a 3-simplex is a tetrahedron,
• a 4-simplex is a 5-cell,
• etc…

Synthetic Geometry

Affine Space

A generalization of some properties of Euclidean space that allows for independence of concepts of distance and measure of angles, keeping only the properties related to parallelism and ration of length of parallel line segments.

In an affine space, there is no distinguished point that serves an an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. There are instead displacement vectors, a.k.a. translation vectors or simply translations, between two points of space. Thus it makes sense to subtract two points of the space, given a translation vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a displacement vector to a point of an affine space, resulting in a new point translated from the starting point by that vector.

Any vector space may be viewed as an affine space; this amounts to forgetting the special role played by the zero vector. In this case, the elements of the vector space may be viewed either as points of the affine space or as displacement vectors or translations. When considered as a point, the zero vector is called the origin. Adding a fixed vector to the element of a linear subspace of an vector space produces an affine subspace. One commonly says that this affine subspace has been obtained by translating (away from the origin) the linear subspace by the translation vector. In finite dimensions, such an affine subspace is the solution set of an inhomogeneous linear system. The displacement vectors for that affine space are the solutions of the corresponding homogeneous linear system, which is a linear subspace. Linear subspaces, in contrasts, always contain the origin of the vector space.

The dimension of an affine space is defined as the dimension of the vector space of its translations. An affine space of dimension one is an affine line. An affine space of dimension two is an affine plane. An affine subspace of $n-1$ in an affine space or vector space of dimension $n$ is an affine hyperplane.

Euclidean Geometry

Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he describes in his textbook on geometry, “The Elements”. The fundamental space in which this geometry is performed is know as the Euclidean space which is a good approximation of physical space over short distances, as reality (the physical space) is non-Euclidean. Euclidean geometry can operate in any Euclidean space of any non-negative integer dimension.

Affine Transform

A geometric transform that preserves lines and parallelism but not necessarily distances and angles.

Coordinate Systems

Number line

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Cartesian Coordinate System

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Polar Coordinate System

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Barycentric Coordinate System

A barycentric coordinate system is a coordinate system in which the location of a point is specified by a reference to a simplex. the barycentric coordinates of a point can be interpreted as masses place at the vertices of the simplex, such that the point is the center of mass (or barycenter) of these masses. These masses can be zero or negative; they are all positive if-and-only-if the point is inside the simplex.