Smooth manifolds allow any number of differentiations at all points {differential geometry}. Differential geometries use metrics and covariant derivatives. Riemannian geometry and pseudo-Riemannian geometry are differential geometries.
Smooth manifolds have global differentiable functions {differential topology}. Differential topology uses no metric and no covariant derivatives.
One point B can lie between two other points A and C {intermediacy}: ABC. All intermediate points B make a segment. All intermediate points B, plus points A and C, make an interval. All points past A in direction AC make a ray. An interval and its two rays make a line. Two rays with a common point make an angle.
Non-metric geometry {ordered geometry} can use intermediacy. Ordered geometry can be the basis of affine geometry and its sub-geometries: Euclidean geometry (parabolic geometry), absolute geometry, and hyperbolic geometry, because straight lines, parabolas, and hyperbolas are open figures and have between-ness. Ordered geometry cannot be the basis of projective geometry, because circles and ellipses are closed figures and so do not have between-ness.
Algebras {geometric algebra} can represent real and complex vector-space non-coordinate classical and relativistic geometries. Geometric-algebra elements (vectors) have dimension (grade), scalar amount (magnitude), space orientation/angle (direction), and relative direction {direction sense} (up or down, inside or outside, or positive or negative). Two vectors have a scalar dot product (inner product) {interior product}, bivector cross product (outer product), and inner product plus outer product (geometric product) (multivector). Grassmann and Clifford algebras generalize geometric algebra.
Hypernumber algebras {Clifford algebra} [1878] have 2^n dimensions for n number components. Dimensions represent reflections and rotations. Rotations are reflection combinations. Reflections convert right to left, or vice versa. Clifford algebras model spinors.
In generalized geometric algebras {Grassmann algebra}, the basis elements are the unit-magnitude dimensions, which can be any number and can be non-orthogonal. Elements are dimension linear combinations and have grade, magnitude, direction, and direction sense.
Operations are reflections. Elements add to make a new element. Elements multiply to make an element of one higher dimension (wedge product) {Grassmann product, algebra}. Parallel vectors are commutative. Perpendicular vectors are anti-commutative. Elements are associative for addition and multiplication. Grassmann algebra [1844 and 1862] is Clifford algebra in which two successive reflections cancel, rather than making rotation, and so there are no rotations and no need for metric or perpendicularity.
Non-metric geometry {projective geometry}| can require that parallel lines meet at infinity {elliptic parallel axiom}: at a vanishing point on the horizon line. In the projective plane, two lines intersect at one point {elliptic incidence property}, so, for a line and a point not on the line, all lines through the point intersect the line once. In projective space, two planes intersect at one line.
Projective geometry involves constructions with straightedge only (no compass). Straight lines remain straight. Projective geometry is not about circles, angles, and parallels. All conic sections are equivalent.
Projective geometry has no circles, no angles, no measurements (no metric), no parallels, and no between-ness (no intermediacy).
Collinearity is invariant in projective geometry. Therefore, projective geometry has invariant point and line structures (incidence structure) (incidence relation) that make generalized planes. The Fano plane is the simplest 2-dimensional finite incidence structure.
Cross ratio is relation of projective harmonic conjugates. Cross ratios are invariant in projective geometry. Cross ratio is ratio of distance ratios. For four distinct collinear points A, B, C, and D in sequence, the cross ratio is (AC/BC)*(BD/AD) = (medium1/short)*(medium2/long) = (AC/BC) / (AD/BD) = (medium1/short)/(long/medium2) = (AC*BD)/(BC*AD) = (13*24)/(23*14) = (medium1*medium2)/(short*long). For a line, if three points and the cross ratio are known, the fourth point is known.
Projective geometry has surface curve categories.
Projective geometry includes non-Riemannian geometry, single elliptic geometry, double elliptic geometry, and hyperbolic geometry.
Spheres {analytic surface} are the only constant-positive-curvature closed surfaces with no singularities.
Ratios {cross ratio} can be distance from line-segment point to line-segment end divided by distance from line-segment point to other line-segment end. Cross ratio can apply in non-metric geometries. Cross ratio is relation of projective harmonic conjugates. Cross ratios are invariant in projective geometry. Cross ratio is ratio of distance ratios. For four distinct collinear points A, B, C, and D in sequence, the cross ratio is (AC/BC)*(BD/AD) = (medium1/short)*(medium2/long) = (AC/BC) / (AD/BD) = (medium1/short)/(long/medium2) = (AC*BD)/(BC*AD) = (13*24)/(23*14) = (medium1*medium2)/(short*long). For a line, if three points and the cross ratio are known, the fourth point is known.
If three line segments meet at point, they have one point each such that the three lines defined by line-segment endpoints intersect the three lines defined by the three pairs of points at three points on same line {Desargue's theorem} {Desargue theorem}.
For line segments, distance from a point to one end divided by distance from the point to other end {harmonic section} {harmonic ratio} equals negative of distance from one point {ideal point, line} on line-segment extension to one end divided by distance from ideal point to other end: PE1 / PE2 = - IE1 / IE2. For all other line-segment points, ratios {anharmonic ratio} are not harmonic. Point position determines ideal-point position {harmonic conjugate, endpoint}. If ideal point is at infinity, line-segment point is at midpoint. Projections keep harmonic ratio and relative positions the same.
Points lie on lines, lines include points, and points and lines have Cartesan point and line products {incidence structure} {incidence relation}. Projective geometry preserves incidence structure. Incidence structure is independent of intermediacy.
Mapping can project sphere onto plane and maintain constant bearings {rhumb line, loxodrome} {loxodrome}.
A point is not on a line. In Euclidean geometry, parallel lines do not meet {parallel axiom}|, and only one line through the point is parallel to the line. In elliptic geometry, parallel great circles meet at two points, and only one line through the point is parallel to the line. Parallel great circles meet at poles of sphere. Parallel lines can make obtuse angles, as on imaginary-radius spheres, or acute angles, as on real-radius spheres. In hyperbolic geometry, parallel great circles do not meet, and many lines through the point are parallel to the line. In projective geometry, parallel lines meet at infinity.
Three sides of a conic-inscribed hexagon intersect at three points, which lie on same line {Pascal's theorem} {Pascal theorem}.
Projective geometry (Karl von Staudt) can use line-segment cross ratio and length analog {algebra of throws}. Lines (ray) can have cross-ratio coordinates 0 for one line-segment end, 1 for line-segment middle, and infinity for other line-segment end. Line segments can transform into arrays, and vice versa.
Non-metric geometry {affine geometry} can use parallel or cylindrical geometric projection, with projection center at infinity {ideal point at infinity}. Only one line goes through two points. Through a point not on a line, only one line is parallel to the line. Straight lines stay straight and parallel lines stay parallel, but lengths and angles can alter {non-Riemannian geometry}, so transformations and translations preserve parallel lines and distance ratios, but not necessarily congruence. Affine geometry is projective geometry with one line or plane as the points at infinity, which transformations leave invariant. (Projective geometry has only one projective plane.) Affine geometry is Euclidean geometry without congruence.
Ratios of separations are invariant in affine geometry. For three distinct collinear points A, B, and C in sequence, length ratio is AB/BC = medium1/medium2. Alternatively, AC/BC = long/medium2. Alternatively, AB/AC = medium2/long. For a line, if two points and the ratio of their separations are known, the third point is known.
Space can have no curvature {Euclidean geometry}| {parabolic space} {parabolic metric geometry}. Geodesics are straight lines. Parabolic metric geometry preserves angle size and similar figures. Each line has one distinct real ideal point, and no line is perpendicular to itself. As conic sections, parabolas have no asymptotes and one distinct real ideal point.
postulates
Euclidean geometry has five assumptions {postulate}. Straight lines can go from any point to any other point. Finite straight lines can extend to any length and remain straight and in same direction. Circles have a center point and finite straight radius with one end in center. Two perpendicular straight lines intersect to make equal angles {right angle}. One and only one straight line through a point not on another straight line does not intersect second straight line {parallel postulate, Euclid}.
First two postulates do not state that lines are unique lines but use lines as if they are unique. Euclid's use of superposition without postulation is questionable.
axioms
Euclidean geometry has five facts that depend on reason or common sense {axioms, Euclid}. Things equal to same thing are equal to each other. If equal things add to equal things, sums are equal. If equal things subtract from equal things, differences are equal. Things that exactly overlap {coincide} are equal. Wholes are greater than or equal to any parts.
definitions
Euclidean geometry relies on definitions. Points are infinitely small space volumes. Two points define a line. Continuous infinite point sets can define lines. Lines have finite line segments. Lines have two infinite rays starting at a line point. Angles have two straight rays starting at vertex point.
Only Euclidean geometry, single elliptic geometry, double elliptic geometry, and hyperbolic geometry permit rigid figure motions.
In two dimensions, geometry can have saddle shape {hyperbolic geometry}| {Lobachevskian geometry, hyperbolic}. If space has hyperbolic geometry, infinitely many lines through a point not on a line are parallel to the line and do not intersect the line. If a shape becomes larger or smaller, shape changes, so figures cannot be similar.
Triangle angles add to less than pi. Pi minus angle sum varies directly with triangle area A: C * A = pi - (a + b + c), where C is Gaussian curvature. Gaussian curvature C = -1 / R^2, so for hyperbolic space, curvature radius is imaginary {pseudo-radius}.
As conic sections, hyperbolas have two asymptotes and two distinct real ideal points.
Hyperbolic surfaces can map conformally to discs in Euclidean planes {conformal model} {Poincaré model} {Poincaré disc}. Hyperbolic-surface straight lines are circle arcs that intersect disc edge at right angles. Hyperbolic-surface angles are the same as disc angles.
Hyperbolic surfaces can map non-conformally to discs in Euclidean planes {Klein representation}. Straight lines in hyperbolic surfaces are straight lines in discs. Angles in hyperbolic surfaces are not the same as angles in discs.
In discs, distances closer to edge contract more, and disc has bound. In hyperbolic surfaces, space has no boundary distances and does not contract. Distance between points A and B is 0.5 * ln((QA / QB) * (PB / PA)), where Q is closer to B and on disc and P is closer to A and on disc. Projective model is conformal model but with distances from center multiplied by (2 * R^2) / (R^2 + r^2), where R is disc radius, and r is distance of point from center. Conformal disc can project vertically onto hemisphere above disc {hemispheric representation}.
Lines can project from conformal circle {equatorial disc} to sphere south pole and to northern hemisphere {stereographic projection, geometry}.
Hyperbolic surface can project to Poincaré half-plane. Metric is (dx^2 + dy^2)/y^2.
Hyperbolic surface can project to surface with constant negative curvature {pseudo-sphere} {pseudosphere}. Distances in pseudo-sphere and Euclidean space are equal. To construct pseudo-sphere, rotate tractrix around its asymptote. To construct tractrix, use zero-rest-mass rod with large mass at one end resting on surface with friction and with other end on asymptote, then slide end along asymptote to trace curve with other end. Metrics 4 * (dx^2 + dy^2)/(1 - x^2 - y^2)^2 {Poincaré metric} can have constant negative curvature.
Special-relativity Minkowskian geometry is a hyperbolic space.
Surface geometries {intrinsic geometry} need not use, or depend on, surrounding space or coordinates. Properties {intrinsic property} do not change if coordinates change. Equations {intrinsic equation} can use no coordinates. Intrinsic equations typically use only curvature radius and arc length.
Geometry {metric geometry}| can use magnitudes and measures. Euclidean geometry has a metric. Topology is non-metric geometry. In two dimensions, all metric geometries are projective geometry augmented by conic. In three dimensions, all metric geometries are projective geometry augmented by quadric {absolute curve}. Metric-geometry determinant must equal positive one or negative one.
Separation is invariant in Euclidean geometry. For two distinct collinear points A and B in sequence, separation is AB. For a line, if one point and the separation are known, the second point is known.
Geometry {non-Euclidean geometry}| can be about curved spaces. Curved spaces can be hyperbolic spaces or spherical spaces.
On spheres, great circles intersect at two points. So that two points determine a line, assume that opposite sphere points are identical {single elliptic geometry}, so two great circles intersect at only one point. For every great circle, through any point outside the great circle, no lines are parallel to the great circle. In triangles, angle sum is greater than 180 degrees. As conic sections, ellipses have no asymptotes and two imaginary ideal points.
On spheres, great circles intersect at two points. Two points do not determine one line. For every great circle, through any point outside the great circle, no lines are parallel to the great circle {spherical geometry} {double elliptic geometry}. In triangles, angle sum is greater than 180 degrees. As conic sections, circles and ellipses have no asymptotes and two imaginary ideal points.
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Date Modified: 2022.0225