Schwarz triangle function: Difference between revisions

Content deleted Content added
Tessellation by Schwarz triangles: actually removing moved content
Line 94:
 
This correspondence extends to one between geometric properties of ''D'' and <math>\mathfrak{H}</math>. Without entering into the correspondence of ''G''-invariant [[Riemannian metric]]s,{{efn|1=The Poincaré metric on the disk corresponds to the restriction of the ''G''-invariant pseudo-Riemannian metric ''dx''<sup>2</sup> – ''dw''<sup>2</sup> to the hyperboloid.}} each geodesic circle in ''D'' corresponds to the intersection of 2-planes through the origin, given by equations Tr ''XY'' = 0, with <math>\mathfrak{H}</math>. 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 [[Beltrami-Klein model]] is obtained by using the map ''F''(''x'',''w'') = ''w''/''x'' as the correspondence between <math>\mathfrak{H}</math> and ''D''. Identifying this disk with (1,''v'') with |''v''| < 1, intersections of 2-planes with <math>\mathfrak{H}</math> correspond to intersections of the same 2-planes with this disk and so give straight lines. The Poincaré-Klein map given by
 
:<math>K(z) = iF \circ f^{-1}(z) = \frac{2z}{1 + |z|{}^2}</math>
 
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 {{pi}} (measured anticlockwise) into a similar pair of curves.{{efn|1=The condition on tangent vectors '''x''', '''y''' is given by det ('''x''','''y''') ≥ 0 and is preserved because the determinant of the Jacobian is positive.}} In the limiting case, when the angle is {{pi}}, the curves are tangent and this again is preserved under a diffeomorphism. The map ''K'' yields the ''Beltrami-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. The use of the Beltrami-Klein model corresponds to [[projective geometry]] and [[cross ratio]]s. (Note that on the unit circle the radial derivative of ''K'' vanishes, so that the condition on angles no longer applies there.)<ref>{{harvnb|Iversen|1992}}</ref><ref>{{harvnb|Magnus|1974}}</ref>
 
The group ''G''<sub>1</sub> = SL(2,'''R''') is formed of real matrices
 
:<math> g = \begin{pmatrix} a & b \\ c & d\end{pmatrix}
</math>
 
with <math>ad -bc =1.</math> The action is by Möbius transformations on the upper half-plane. The Lie algebra of ''G''<sub>1</sub> is <math>\mathfrak{g}_1</math>, the space of 2 x 2 real matrices of trace zero,
 
:<math> X = \begin{pmatrix} x & y-t \\ y+t & -x\end{pmatrix}.</math>
 
By transport of construction — conjugating by ''C'' — the symmetric bilinear form Tr ad(''X'')ad(''Y'') = 4 Tr ''XY'' is invariant under conjugation and has signature (2,1). As above ''X''<sup>2</sup> = (''x''<sup>2</sup> + ''y''<sup>2</sup> – ''t''<sup>2</sup>) ''I'' and
 
:<math> \det X = t^2 - x^2 - y^2 =-\tfrac12 \operatorname{Tr} X^2.</math>
 
By [[Sylvester's law of inertia]],<ref>{{harvnb|Ratcliffe|2019|pages=52–56}}</ref> the [[Killing form]] (or [[Cartan-Killing form]]) is, up to equivalence, the unique symmetric bilinear form of signature (2,1) with corresponding quadratic form –''x''<sup>2</sup> –''y''<sup>2</sup> +''t''<sup>2</sup>; and ''G'' / {±''I''} or equivalently ''G''<sub>1</sub> / {±''I''} can be identified with SO(2,1). <!--The group SO(2,1) acts transitively on the two components of the non-zero part of the light cone ''x''<sup>2</sup> + ''y''<sup>2</sup> = ''t''<sup>2</sup>. On the interior of the two time-like components, ''x''<sup>2</sup> + ''y''<sup>2</sup> < ''t''<sup>2</sup> with ''t'' strictly positive or negative, it is a disjoint union of orbits, namely the hyperboloids ''x''<sup>2</sup> + ''y''<sup>2</sup> + ''a''<sup>2</sup> = ''t''<sup>2</sup>. For fixed ''a'', say ''a'' = 1, the Beltrami-Klein model gives a equivariant homeomorphism ''f'' of the hyperboloid ''x''<sup>2</sup> + ''y''<sup>2</sup> + 1 = ''t''<sup>2</sup> onto the open unit disk, ''f''(''x'',''y'',''t'') = (''x'' + ''iy'')/''t''. Compactifying the hyperboloid by adding a circle at infinity — the rays of the light cone — ''f'' extends to a homeomorphism onto the closed unit disk.-->
 
==Conformal mapping of Schwarz triangles==