I should use the term 6 specifications which link my 4 geometries.
Erlangen program identifies a geometry with the group of transformations which do not change it. I want to identify a geometry with the monoid of actions that can be taken within it. It is a free monoid in the case of affine geometry but becomes a group with the introduction of inverses, transformations.
The usual three-way classification of conic sections into ellipses, hyperbolas and parabolas is an affine classification. For example, any two ellipses are related by an affine transformation.
As in the case of an isometry, an affine transformation is determined by the image of any n + 1 independent points (ones which do not lie in an (n - 1)-dimensional affine subspace). In the case of an affine transformation, any n + 1 independent points can be mapped to any n + 1 independent points. In particular, in R2 there is a unique affine transformation taking a triangle ABC into a triangle A'B'C'.
An affine transformation or affinity of Rn is one of the form Translation (Ta) composed with Linear transformation (L).
The set of linear transformations, without restriction. It can be identified with the vector space.
Symplectic establishes the distinction between inside and outside (orientation).
A similarity transformation or similitude is an affine map which preserves angles.
A similarity transformation can be written Translation (Ta) composed with positive scalar (lambda) x Orthogonal (linear) transformation. Since it preserves angles, all vectors must be stretched by the same amount, lambda.
The well-known theorem of Pythagoras can be proved by "similar triangle" methods.
I am looking for 6 transformations of perspective which link the structure of one geometry with the dynamics of another geometry. I think these 6 transformations relate to ways of interpreting multiplication, to restructurings of sequences, hierarchies and networks, and to the axioms of set theory which define sets.
Transformations in cinematography