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射影几何

Theorems

• The Fundamental Theorem of Projective Geometry Any four planar non-collinear points (a quadrangle) can be sent to any quadrangle via a projectivity, that is a sequence of perspectivities.

Wildberger's key theorems

• Pappus's hexagon theorem Two sets of collinear points yield a third set of collinear points.
• Pascal's theorem The same but where all six points are on a conic section, with a pair of lines being a degenerate case.
• Desargues' theorem Two triangles are in perspective axially if and only if they are in perspective centrally.

Concepts

Incidence geometry

Perspective maps lines to lines, conics to conics.

Do parallel lines going one way and going the other way meet at the same point at infinity? Or two different points at infinity? This is answered by considering a line as a vector space. So going around we get to the same line. But we have a different orientation. So the notion of orientation of a point becomes relevant. The line of infinity consists of oriented points.

Line geometry

• Map lines to lines. Projective geometry additionally maps zero to zero. And infinity to infinity? And do the lines have an orientation? And is that orientation preserved?
• Linear equations are intersections of hyperplanes.
• Projective geometry is linear algebra.
• Projective geometry can be identified with linear algebra, with all (invertible) linear transformations. That is why it is considered the most basic geometry in the Erlangen program. However, I am relating the affine geometry with a free monoid. The affine geometry can be thought of as a movie screen, and each point on the screen can be imagined as a line (a beam of light) extending outside of the screen to a projector. So there is always an extra dimension. Projective geometry has a "zero".
• In projective geometry, vectors are points and bivectors are lines.
• Projective geometry transforms conics into conics.
• Fundamental theorems of affine and projective geometry
  • Fundamental Theorem of Affine geometry. let {$X,X'$} be two finite dimensional affine spaces over two fields {$K,K'$} of same dimension {$d\geq 2$}, and let {$f:X\to X'$} be a bijection that sends collinear points to collinear points, i.e. such that for all {$a,b,c\in X$} that are collinear, {$f(a),f(b),f(c)$} are collinear too. Then {$f$} is a semi-affine isomorphism.
  • This means that there is a field isomorphism {$\sigma:K\to K'$} such that for any point {$a\in X$} the map induced by {$f_a: X_a\to X'_{f(a)}$} is a {$\sigma$}-semi-linear isomorphism.
  • Fundamental Theorem of Projective geometry. let {$P(X),P(X')$} be two finite dimensional projective spaces over two fields {$K,K'$} of same dimension {$d\geq 2$}, and let {$f:P(X)\to P(X')$} be a bijection that sends collinear points to collinear points. Then {$f$} is a semi-linear isomorphism.
• Projective geometry: Tiesė perkelta į kitą tiesę išsaugoja keturių taškų dvigubą santykį (cross ratio).

Conics

• Consider geometrically how to use the conics, the point at infinity, etc. to imagine Nothing, Something, Anything, Everything as stages in going beyond oneself into oneself.
• Investigate: What happens to the shape of a circle when we move the tip of the cone? Suppose the circle is a shaded area. In what sense is the parabola a circle which touches infinity? In what sense is the directrix a focus? Does the parabola extend to the other side, reaching up to the directrix? Is a hyperbola an inverted ellipse, with the shading on either side of the curves, and the middle between them unshaded? What is happening to the perspective in all of these cases?
• Think of the two foci of a conic as the source (start of all) and the sink (end of all). When are they the same point? (in the case of a circle?)

Notes

• Any projective space is of the form {$\mathbb{KP}^n$} for some skew field {$\mathbb{K}$} except when n=2. Then it holds if the Axiom of Desargues holds in the space, but there are also exceptional spaces. This fact may relate to the fact that semisimple Lie algebras are related to {$\mathbb{R}$}, {$\mathbb{C}$} or {$\mathbb{H}$} - the classical Lie algebras - with the exceptions being very limited.
• If we have the real plane and think of it projectively, then there is a line at infinity, encircling infinity, where the points on that line have orientation (given by the orientation of intersecting lines). But if we think of the plane as the complex numbers, then perhaps instead of a line it is sufficient to consider a single complex number as the point of infinity, in that it comes with a sense of angle.
• Why is projective geometry related to lines, sphere, projection, point at infinity?
• Harmonic pencil: Look for what it would mean for a ratio to be {$i$} and the product to be -1. The answer is {$e^{\frac{\pi}{4}i}$} and {$e^{\frac{3\pi}{4}i}$}, or {$e^{-\frac{\pi}{4}i}$} and {$e^{-\frac{3\pi}{4}i}$}
• {$\mathrm{Aut}_{H\circ f}(\mathbb{P}^1_\mathbb{C})\simeq \mathrm{PSL}(2,\mathbb{C})$}
• {$\mathrm{Aut}_{H\circ f}(\mathbb{U})\simeq \mathrm{PSL}(2,\mathbb{R})$}
• {$\mathrm{Aut}_{H\circ f}(\mathbb{C})\simeq \mathrm{P}\Delta(2,\mathbb{R}) \equiv$} upper-triangular elements of {$\mathrm{PSL}(2,\mathbb{C})$}
• Desargues theorem in geometry corresponds to the associative property in algebra.
• A projective space is best understood from one dimension higher. And it is understood in terms of breaking down into smaller affine spaces. And these dimensions higher and lower are related to extending the chain of dimensions as given by Lie algebras. And the reversal of counting is related to the reversal of the orientation of lines etc. as a line is considered as a circle.
• Homogeneous coordinates add a variable Z that makes each term of maximal power N by contributing the needed power of Z.
• Spinor requires double rotation. Projective sphere is the opposite: half a rotation gets you back where you were.
• Sextactic points on a simple closed curve.
• Projective Geometry. A Short Introduction
• Projective Geometry: From Foundations to Applications Beutelspacher and Rosenbaum

Projective geometry

• Projective geometry relates one plane (upon which the projection is made) with another plane (where the "eye" is, the zero where all the lines come from). And thus the line through the eye which is parallel to the plane needs to be added. Thus we can have homogeneous coordinates. And we have the decomposition of projective space into a sum of affine spaces of each dimension. Projective geometry is the space of one-dimensional subspaces, and they all include zero, thus they are the lines which go through zero. Or the hyperplanes which go through zero.
• projective geometry - no constant term - replace with additional dimension - thus get lines going through zero point ; otherwise in linear equations have to deal with a constant term - relate this to the kinds of variables
• "viewing line" y=1 thus [x/y: 1] and "viewing plane" z=1 thus [x/z:y/z:1]
• [1:2:0] is a point that is a "direction" (two directions)
• A vector subspace needs to contain zero. How is this related to projective geometry? Vector spaces: Two different coordinate systems agree on the origin 0.
• Projective geometry: way of embedding a 1-dimensional subspace in a 2-dimensional space or a 3-dimensional space. (Lower dimensions embedded in higher dimensions.) Vector spaces must include 0. So that is a big restriction on projective geometry that distinguishes it from affine geometry?
• https://en.m.wikipedia.org/wiki/Homography Homography two approaches to projective geometry with fields or without
• A projective space may be constructed as the set of the lines of a vector space over a given field (the above definition is based on this version); this construction facilitates the definition of projective coordinates and allows using the tools of linear algebra for the study of homographies. The alternative approach consists in defining the projective space through a set of axioms, which do not involve explicitly any field (incidence geometry, see also synthetic geometry); in this context, collineations are easier to define than homographies, and homographies are defined as specific collineations, thus called "projective collineations".
• Given any field F,2 one can construct the n-dimensional projective space Pn(F) as the space of lines through the origin in Fn+1. Equivalently, points in Pn(F) are equivalence classes of nonzero points in Fn+1 modulo multiplication by nonzero scalars.
• Sylvain Poirier: Some key ideas, probably you know, but just in case:

The (n+p-1)-dimensional projective space associated with a quadratic space with signature (n,p), is divided by its (n+p-2)-dimensional surface (images of null vectors), which is a conformal space with signature (n-1,p-1), into 2 curved spaces: one with signature (n-1,p) and positive curvature, the other with dimension (n,p-1) and negative curvature. Just by changing convention, the one with signature (n-1,p) and positive curvature can also seen as a space with signature (p,n-1) and negative curvature; and similarly for the other.

• Cross-ratio can be written as ratio of ratios: {$$\frac{\frac{AC}{CB}}{\frac{AD}{DB}}$$}

Basic theorems: Triple Quad formula and Pythagorean theorem

• Cross law
• Spread law
• Triple spread formula
• Sum and Product: Introduction to Projective Geometry via Tic-Tac-Toe Grids #SoME2
• What is the relationship between dilation and the homogeneous coordinates for projective geometry?
• Robin Hartshorne. Foundations of projective geometry.
• Going beyond yourself you have a point (a symmetry breaking) (on the circle). It becomes external reference point. But outside it is compact, and in a coordinate system. Projective geometry gives us such a point at infinity.
• Projective line over {$F_1$} has two points. The second points is infinity. So what does it mean to say {$0=1=\infty$}?
• Note also that infinity is the flip side of zero - they make a pair. They are alternate ways of linking together the positive and negative values.
• Could there be biprojective, triprojective, and so on, N-projective geometry for N perspectives?