##### 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 (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 , are called weighted sums. Similar expressions can be defined for *locations* but some restrictions apply: A difference of *locations* is a *direction* called . The following holds

- .
- .

The last expression can be generalized to include all weighted sums . 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 . This element will be denoted by . A *direction* can be added to a *location* to produce another *location*. Indeed, if then for any we can define . is called the *translation* of the location .

Affine subsets which always includes locations produced by weighted sums of its elements are called *spans*. The set

of elements is the smallest span containing . It is denoted by . The set is called *independent* if removal of any of its elements results in a smaller span. If is spanned by an independent set, then 1 less of set’s cardinality is called the dimension of the span and is denoted by . 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 are independent, the set is called the simplex with vertices and is denoted by . is called a segment. is called a triangle. is called a tetrahedron. The weighted sum is called the center of . The special case of 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 , the set of differences is called its associated span of directions. Affine spans are called parallel if its associated spans of directions coincide. If is the entire Affine space and is a fixed line of directions, then the lines for which is the line of directions are mutually parallel. This subset is again an affine space which we call the quotient of and denote by . Note that .

## Scalar Product, Norm and Distance

For any pair of directions , their *Scalar Product* is a numer denoted by . The rules are the same as those for the ordinary multiplications of numbers. In particular, abbreviates to . The expression is called the norm and is denoted by . For a fixed independent set the squared norm of a weighted sum is a homogeneous polynomial of degree 2. For a convenient selection of the independent set that polynomial can be written as . A selection of signs is called the signature of the norm and is denoted by a pair . We consider constant function a norm with signature . A line can have norm, a plane – norms, a space – norms and a 4d space – norms. The signature of a norm restricted to a subspace has one or both of the counters decremented. Norms with signature are called Euclidean and those with signature are called Minkowskian. Directions with zero norm are called *isotropic*. Euclidean norm has no isotropic directions other than . In Minkowskian norm the set of isotropic directions is called the *isotropic cone*. When the norm is Euclidean, the expression is called the distance separating . 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.