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.
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Date Modified: 2022.0224