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# Geometry

Distance for spherical geometry d(A, B) := ∠AOB · R
Polar triangle A'=Pol(BC) and ∠AOA' ≤ π/2
Bipolar theorem If A'B'C'=Pol(ABC), then ABC=Pol(A'B'C') Angles are π − a, π − b, π − c and sides are π − α, π − β, π − γ
Thales theorem The base angles of a spherical isosceles triangle are equal
Area of a spherical triangle (α + β + γ − π)R^2
Sum of angles of a spherical triangle π < α + β + γ ≤ 3π
Sum of sides of a spherical triangle 0 < a + b + c ≤ 2π
Isometries of spherical geometry Rotation, reflection and glide reflection
Homothety A map which is a scalar multiplication
Pappus' theorem Let a and b be lines with points A1,A2,A3 and B1,B2,B3. Pi=AjBknAkBj. Then the points Pi, Pj and Pk are collinear.
Pascal's theorem If A, B, C, D, E, F lie on a conic then the points AB ∩ DE, BC ∩ EF, CD ∩ F A are collinear.
Desargues' theorem If the lines joining the vertices of triangles A1A2A3 and B1B2B3 intersect at one point S, then the intersection points Pi = AjAk ∩ BjBk are collinear.
Distance for Klein model d(A, B) = 1/2|ln|[A, B, X, Y ]||
Parabolic Mobius transformation Conjugate to z → z+1
Elliptic Mobius transformation Conjugate to z → az, |a| = 1, two similar fixpoints
Hyperbolic Mobius transformation Conjugate to z → az, |a| =/ 1 and a ∈ R, two different fixpoints
Loxodromic Mobius transformation Conjugate to z → az, two different fixpoints
Inversion Takes A to A' lying on the ray OA s.t. |OA|·|OA'|=r^2
Distance for Poincare disc model d(A, B)=|ln|[A, B, X, Y ]||
Isometries of Poincare disc model Mobius transformations, inversions, reflections preserving the disc
Every isometry of the Poincare disc model can be written as az+b/cz+d or az~+b/cz~+d
Sum of angles in a hyperbolic triangle Less than pi
Distance for upper half-plane model d(A, B)=|ln|[A, B, X, Y ]||
Every isometry of the upper half-plane model can be written as az+b/cz+d or a(-z~)+b/c(-z~)+d
Area of hyperbolic triangle π−(α+β+γ)
Area of n-gon (n−2)π−sum(αi)
Distance for two-sheet hyperboloid model d(A, B) = 1/2|ln|[A, B, X, Y ]||
Reflection in hyperboloid model ra : x → x−2((x,a)/(a,a))a
Elliptic isometry 1 fixpoint in H^2
Parabolic isometry 1 fixpoint in ∂H^2
Hyperbolic isometry 2 fixpoints in ∂H2
Created by: Marge-Homer