Schwarz triangle function: Difference between revisions

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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''² –''y''² +''t''²; and ''G'' / {±''I''} or equivalently ''G''₁ / {±''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''² + ''y''² = ''t''². On the interior of the two time-like components, ''x''² + ''y''² < ''t''² with ''t'' strictly positive or negative, it is a disjoint union of orbits, namely the hyperboloids ''x''² + ''y''² + ''a''² = ''t''². For fixed ''a'', say ''a'' = 1, the Beltrami-Klein model gives a equivariant homeomorphism ''f'' of the hyperboloid ''x''² + ''y''² + 1 = ''t''² 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.-->

==Convex polygons==

[[File:Triheptagonal tiling.svg|thumb|upright|A uniform tiling in the Poincaré disk model by regular convex septagons and triangles]]

[[File:Uniform tiling 73-t1 klein.png|thumb|upright|The same tiling in the Beltrami-Klein model]]

In this section the main results on convexity of hyperbolic polygons are deduced from the corresponding results for Euclidean polygons by considering the relation between Poincaré's disk model and the Klein model. A polygon in the unit disk or upper half plane is made up of a collection of a finite set of vertices joined by geodesics, such that none of the geodesics intersect. In the Klein model this corresponds to the same picture in the Euclidean model with straight lines between the vertices. In the Euclidean model the polygon has an interior and exterior (by an elementary version of the [[Jordan curve theorem]]), so, since this is preserved under homeomorphism, the same is true in the Poincaré picture.

As a consequence at each vertex there is a well-defined notion of interior angle.

In the Euclidean plane a polygon with all its angles less than {{pi}} is convex, i.e. the straight line joining interior points of the polygon also lies in the interior of the polygon. Since the Poincaré-Klein map preserves the property that angles are less than {{pi}}, a hyperbolic polygon with interior angles less than {{pi}} is carried onto a Euclidean polygon with the same property; the Euclidean polygon is therefore convex and hence, since hyperbolic geodesics are carried onto straight lines, so is the hyperbolic polygon. By a continuity argument, geodesics between points on the sides also lie in the closure of the polygon.

A similar convexity result holds for polygons which have some of their vertices on the boundary of the disk or the upper half plane. In fact each such polygon is an increasing union of polygons with angles less than {{pi}}. Indeed, take points on the edges at each ideal vertex tending to the two edges joining those points to the ideal point with the geodesic joining them. Since two interior points of the original polygon will lie in the interior of one of these smaller polygons, each of which is convex, the original polygon must also be convex.<ref>{{harvnb|Magnus|1974|page=37}}</ref>

==Tessellation by Schwarz triangles==