Affine Geometry

By | 2019-12-02
Formal Definition

Affine theory deals with 2 sets: One which elements are called locations and another, which elements are called directions. The directions form an additive group. Every additive group carries a derived composition between its elements and scalars. It is defined recursively as 1\vec{A}\stackrel{\text{\tiny def}}{=}\vec{A} \mbox{ and } (\alpha + \beta)\vec{A} \stackrel{\text{\tiny def}}{=} \alpha \vec{A}+\beta \vec{A} (composition symbol implied). Greek letters will be reserved for scalar names and the upper case letters for location names and arrow-ed upper case letters for direction names. In both cases, lower-case indicates a set-of-elements.
Directional expressions \alpha \vec{A}+\beta \vec{B}+..., are called weighted sums. Similar expressions can be defined for locations but some restrictions apply: A difference of locations B - A is a direction called \mbox{from } A \mbox{ to } B. The following holds

  • B - A = \vec{0} \mbox{ means } A=B
  • (B-A)+(C-B)=C-A
  • A-B=-(B-A).
  • \alpha A - \alpha B \stackrel{\text{\tiny def}}{=} \alpha (A-B).

The last expression can be generalized to include all weighted sums \alpha A+\beta B+... \mbox{ s.t. }\alpha +\beta +...=0. On the other hand, if the sum of weights is nonzero, this expression is not defined. However, in that case we postulate existence of a unique element C \mbox{ s.t. } \alpha A+\beta B+...-(\alpha +\beta +...)C =\vec{0}. This element will be denoted by \frac{\alpha A+\beta B+...}{\alpha +\beta +...}. A direction can be added to a location to produce another location. Indeed, if \vec{D}=B-A then for any C we can define C+\vec{D} \stackrel{\text{\tiny def}}{=} C + B - A. C+\vec{D} is called the translation of the location C \mbox{ in the direction } \vec{D}.

Affine subsets which always includes locations produced by weighted sums of its elements are called spans. The set
of elements \alpha A+\beta B+... \mbox{ s.t. } \alpha +\beta +...=1 is the smallest span containing A, B,... . It is denoted by span(A,B,...). The set A, B,... is called independent if removal of any of its elements results in a smaller span. If s is spanned by an independent set, then 1 less of set’s cardinality is called the dimension of the span and is denoted by dim(s). Spans of dimension 1 are called lines, those of dimension 2 – planes and those of dimension 1-less than that of the entire Affine Space – hyperplanes. If A, B, ... are independent, the set \alpha A+\beta B+... \mbox{ s.t. } \alpha +\beta +...=1, \alpha \ge 0, \beta \ge 0, ... is called the simplex with vertices A, B, ... and is denoted by [A, B, ...]. [A,B] is called a segment. [A,B,C] is called a triangle. [A,B,C,D] is called a tetrahedron. The weighted sum \frac {A_{1}+A_{2}+...+A_{n}}{n} is called the center of [A_{1},...A_{n}]. The special case of \frac{A+B}{2} is called the mid-point. The concept of independence and span carries over verbatim to the set of directions simply by removing the restrictions on the sum of weights. For any span a, the set of differences a - a is called its associated span of directions. Affine spans b,c are called parallel if its associated spans of directions coincide. If u is the entire Affine space and \vec{d} is a fixed line of directions, then the lines for which \vec{d} is the line of directions are mutually parallel. This subset is again an affine space which we call the quotient of u \mbox{ by } \vec{d} and denote by u / \vec{d}. Note that dim(u /  \vec{d}} = dim(u) - 1.

Scalar Product, Norm and Distance

For any pair of directions \vec{U},\vec{V}, their Scalar Product is a numer denoted by \vec{U}\centerdot\vec{V}. The rules are the same as those for the ordinary multiplications of numbers. In particular, \vec{U} \centerdot \vec{U} abbreviates to \vec{U}^2. The expression \sqrt{\vec{U}^{2}} is called the norm and is denoted by |\vec{U}|. For a fixed independent set \vec{S},\vec{T},... the squared norm of a weighted sum \sigma \vec{S}+\tau \vec{T}+... is a homogeneous polynomial of degree 2. For a convenient selection of the independent set that polynomial can be written as \pm \sigma ^2 \pm \tau ^2 .... A selection of signs is called the signature of the norm and is denoted by a pair (\mu ,\nu ) \mbox{ where } \mu \mbox{ counts the pluses, } \nu \mbox{ counts the minuses}. We consider constant 0 function a norm with signature (0,0). A line can have (1,0) norm, a plane – (2,0),(1,1) norms, a space – (3,0),(2,1) norms and a 4d space – (4,0),(3,1),(2,2) norms. The signature of a norm restricted to a subspace has one or both of the counters decremented. Norms with signature (\mu ,0) are called Euclidean and those with signature (\mu , 1) are called Minkowskian. Directions with zero norm are called isotropic. Euclidean norm has no isotropic directions other than \vec{0}. In Minkowskian norm the set of isotropic directions is called the isotropic cone. When the norm is Euclidean, the expression |A-B| is called the distance separating A, B. If the norm is Minkowskian that expression is called the interval. In Euclidean space restrictions of the norm to a subspace is also Euclidean. In Minkowskian space that restriction is Euclidean if the subspace does not contain nonzero isotropic directions and is Minkowskian otherwise.

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