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# Geometry

Question | Answer |
---|---|

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