page_title: Schwarz triangle function
section_title: Hyperboloid and Klein models
hierarchical_section_title: Schwarz triangle function / Hyperboloid and Klein models
caption_attribution_description: English: Poincaré disk, hemispherical and hyperboloid models are related by central projection from −1. Klein disk model is vertical projection from hemispheric model. Poincaré half-plane model here projected from the hemisphere model by rays from left end of Poincaré disk model.
context_page_description: In complex analysis, the Schwarz triangle function or Schwarz s-function is a function that conformally maps the upper half plane to a triangle in the upper half plane having lines or circular arcs for edges. Let πα, πβ, and πγ be the interior angles at the vertices of the triangle. If any of α, β, and γ are greater than zero, then the Schwarz triangle function can be given in terms of hypergeometric functions as:
  where a = /2, b = /2, c = 1-α, a' = a - c + 1 = /2, b' = b - c + 1 = /2, and c' = 2 - c = 1+α. This mapping has singular points at z=0, 1, and ∞, corresponding to the vertices of the triangle with angles πα, πγ, and πβ respectively. At these singular points, and
  . This formula can be derived using the Schwarzian derivative.
  This function can be used to map the upper half-plane to a spherical triangle on the Riemann sphere if α + β + γ > 1, or a hyperbolic triangle on the Poincaré disk if α + β + γ < 1. When α + β + γ = 1, then the triangle is a Euclidean triangle with straight edges: a=0, and the formula reduces to that given by the Schwarz–Christoffel transformation.
context_section_description: In this section two different models are given for hyperbolic geometry on the unit disk or equivalently the upper half plane.
  The group G = SU(1,1) is formed of matrices
  with
  It is a subgroup of Gc = SL(2,C), the group of complex 2 × 2 matrices with determinant 1.
  The group Gc acts by Möbius transformations on the extended complex plane. The subgroup G acts as automorphisms of the unit disk D and the subgroup G₁ = SL(2,R) acts as automorphisms of the upper half plane. If
  then
  since the Möbius transformation corresponding M is the Cayley transform carrying the upper half plane onto the unit disk and the real line onto the unit circle.
  The Lie algebra  of SU(1,1) consists of matrices
  with x real. Note that X² = (|w|² – x²) I and
  The hyperboloid  in  is defined by two conditions. The first is that det X = 1 or equivalently Tr X² = –2. By definition this condition is preserved under conjugation by G. Since G is connected it leaves the two components with x > 0 and x < 0 invariant. The second condition is that x > 0. For brevity, write X = (x,w).
  The group G acts transitively on D and  and the points 0 and (1,0) have stabiliser K consisting of matrices
  with |ζ| = 1. Polar decomposition on D implies the Cartan decomposition G = KAK where A is the group of matrices
  Both spaces can therefore be identified with the homogeneous space G/K and there is a G-equivariant map f of   onto D sending (1,0) to 0. To work out the formula for this map and its inverse it suffices to compute g(1,0) and
  g(0) where g is as above. Thus g(0) = β/α and
  so that
  recovering the formula
  Conversely if z = iw/(x + 1), then |z|² = (x – 1)/(x + 1), giving the inverse formula
  This correspondence extends to one between geometric properties of D and . Without entering into the correspondence of G-invariant Riemannian metrics, each geodesic circle in D corresponds to the intersection of 2-planes through the origin, given by equations Tr XY = 0, with  . Indeed, this is obvious for rays arg z = θ through the origin in D—which correspond to the 2-planes arg w = θ—and follows in general by G-equivariance.
  The Klein model is obtained by using the map F(x,w) = w/x as the correspondence between  and D. Identifying this disk with (1,v) with |v| < 1, intersections of 2-planes with  correspond to intersections of the same 2-planes with this disk and so give straight lines. The Poincaré-Klein map given by
  thus gives a diffeomorphism from the unit disk onto itself such that Poincaré geodesic circles are carried into straight lines. This diffeomorphism does not preserve angles but preserves orientation and, like all diffeomorphisms, takes smooth curves through a point making an angle less than π (measured anticlockwise) into a similar pair of curves. In the limiting case, when the angle is π, the curves are tangent and this again is preserved under a diffeomorphism. The map K yields the Klein model of hyperbolic geometry. The map extends to a homeomorphism of the unit disk onto itself which is the identity on the unit circle. Thus by continuity the map K extends to the endpoints of geodesics, so carries the arc of the circle in the disc cutting the unit circle orthogonally at two given points on to the straight line segment joining those two points.
  (Note that on the unit circle the radial derivative of K vanishes, so that the condition on angles no longer applies there.)
caption: Geometric relations between Poincaré disk model, hyperboloid model and Klein model
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