# Book: Geometries

Understand

• what is meant by affine, projective, conformal and symplectic geometry
• the relationship between the discrete and continuous case of projective geometry

Geometries

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.

Affine geometry

• The medians of a triangle are coincident. They meet at the point (a + b + C) /3
• Ceva's theorem.
• Menelaus's theorem.

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'.

Group A(Rn)

An affine transformation or affinity of Rn is one of the form Translation (Ta) composed with Linear transformation (L).

• If A, B and C are collinear, so are their images under any affine map.
• A translation of a linear subspace of Rn is called an affine subspace. For example, any line or plane in R3 is an affine subspace. Affine transformations map affine subspaces to affine subspaces.
• Parallel lines are mapped to parallel lines.

Projective geometry

The set of linear transformations, without restriction. It can be identified with the vector space.

• Pappus's theorem
• Pascal's theorem
• Desargues theorem

Conformal geometry

Symplectic geometry

Symplectic establishes the distinction between inside and outside (orientation).

Similarity geometry

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.

Euclidean geometry

Group I(Rn)

Transformations

Analyze Wikipedia category: Transformations and the article Wikipedia: Transformations.

Požiūrio

• reflection: atskyrimas - vienybė - požiūrio atskyrimas nuo visumos
• shear: išvertimas - dvejybė
• rotation: išplėtimas - trejybė
• dilation: atsisakymas - ketverybė
• squeeze: apvertimas - penkerybė
• translation: įsisavinimas - šešerybė
• sudūrimas - septynerybė
• suvedimas grandine, pasiklydusiu vaiku - nulybė?

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.

• The (affine) whole is (projectively) recopied.
• The (affine) whole is (conformally) rescaled.
• The (projective) multiple is (conformally) rescaled.
• The (conformal) set is (symplectically) redistributed.
• The (projective) multiple is (symplectically) redistributed.
• The (affine) whole is (symplectically) redistributed.

Transformations in cinematography

• Shear: sideshot
Parsiųstas iš http://www.ms.lt/sodas/Book/Geometries
Puslapis paskutinį kartą pakeistas 2018 lapkričio 11 d., 17:37