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仿射几何学

读物

• W: Affine geometry

John O'Connor: Affine Theorems

• 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.
• Affine = local (no infinity).
• Affine geometry is agnostic regarding coordinate system. So it doesn't distinguish if we flip all the positive and minus choices.
• Affine transformation extends a linear transformation by a column (the translation) and a row (of zeroes) and a diagonal element (of 1). Thus it is similar to a Lie algebra (Dn ?) extending An.

Path geometry

• Affine geometry: Point + Vector = Point. Vector + Vector = Vector. Point - Point = Vector. But we can't add two points because we don't have any origin for them to reference.
• Unions of spaces.

Affine geometry

• Allowing only positive "coefficients" is related to positive definiteness, convexity.
• Does not assume Euclid's third and fourth axioms.
• Different coordinate systems don't agree on any origin.
• Dual ways of defining a geometry: Affine geometry can be developed in two ways that are essentially equivalent.
  • In synthetic geometry, an affine space is a set of points to which is associated a set of lines, which satisfy some axioms (such as Playfair's axiom).
  • Affine geometry can also be developed on the basis of linear algebra. In this context an affine space is a set of points equipped with a set of transformations (that is bijective mappings), the translations, which forms a vector space (over a given field, commonly the real numbers), and such that for any given ordered pair of points there is a unique translation sending the first point to the second; the composition of two translations is their sum in the vector space of the translations.
• In traditional geometry, affine geometry is considered to be a study between Euclidean geometry and projective geometry. On the one hand, affine geometry is Euclidean geometry with congruence left out; on the other hand, affine geometry may be obtained from projective geometry by the designation of a particular line or plane to represent the points at infinity.[16] In affine geometry, there is no metric structure but the parallel postulate does hold. Affine geometry provides the basis for Euclidean structure when perpendicular lines are defined, or the basis for Minkowski geometry through the notion of hyperbolic orthogonality.[17] In this viewpoint, an affine transformation geometry is a group of projective transformations that do not permute finite points with points at infinity.
• https://en.m.wikipedia.org/wiki/Affine_geometry triangle area pyramid volume
• https://en.m.wikipedia.org/wiki/Motive_(algebraic_geometry) related to the connection between affine and projective space
• Tiesė perkelta į kitą tiesę išsaugoja trijų taškų paprastą santykį (ratio).
• Affine varieties correspond to prime ideals and as such are irreducible. So they are the building blocks of the closed subsets of the Zariski topology.
• Affine geometry - free monoid - without negative sign (subtraction) - lattice of steps - such as Young tableaux as paths on Pascal's triangle.
• Sylvain Poirier: Affine representations of that quadric are classified by the choice of

the horizon, or equivalently the polar point of that horizon (the point representing in the projective space the direction orthogonal to that hyperplane). So there are 3 possibilities. The null one sees it as a paraboloid and gives it an affine geometry. The 2 others, with the different signs, see it as a quadric whose center is the polar point, and give it 2 different curved geometries

• Have a larger automorphism group which includes not just the invertible matrices GL(n,F) but also the translations. Thus this extends the nxn matrices to be (n+1)x(n+1) matrices where the bottom row is (0,0,...,0,1).
• Affine transformations preserve points, lines, parallel lines, conics, polynomial functions.
• They do not preserve circles, angles, lengths because of shear, translation in one direction.
• Think of odd orthogonal groups as having an external zero which, if included as a set of extra dimensions, establishes an affine geometry.
• Affine geometry forgets the origin. Is there a related forgetful functor?
• If n is odd, O(n) is the internal direct product of SO(n) and {±I}. Here is it possible to think of SO(n) as having an affine geometry and take the origin at {±I}?
• How is geometry related to (odd real, even real, complex, quaternionic) inner products?
• https://en.m.wikipedia.org/wiki/Affine_transformation