cjerdonek / Generalized hyperbolic triangle c515493fe1662616597ecc8ba7ae6130f9c8cb4d

This article discusses triangles in the hyperbolic plane whose vertices are permitted to correspond to vectors not just with imaginary norm but also real norm when viewed using the Minkowski model of the hyperbolic plane. We call these generalized hyperbolic triangles. Speaking more loosely, these are hyperbolic triangles whose vertices are allowed to lie outside the hyperbolic plane, including outside the boundary.

These generalized triangles correspond to the hyperbolic polygons one can get when one is allowed to replace each vertex of a hyperbolic triangle in the usual sense with an additional side and two right-angled vertices. The resulting polygons include-- quadrilaterals with at least two right angles, pentagons with at least four right angles, and all-right hexagons.

This article discusses hyperbolic triangles whose vertices are allowed to lie outside the hyperbolic plane and its boundary. We call these generalized hyperbolic triangles because their representation as a polygon can have more than three sides.

(By a point "outside the plane," we mean more formally that the point corresponds in the hyperboloid model of the hyperbolic plane to a vector in Lorentz space with real norm as opposed to imaginary. We discuss this further below.)

Roughly speaking, generalized hyperbolic triangles are polygons one can get by replacing one or more vertices of a normal hyperbolic triangle with an extra side and two right-angled vertices. In addition to normal triangles with three sides, these polygons include (1) quadrilaterals with at least two right angles, (2) pentagons with at least four right angles, and (3) all-right hexagons.

Definition

Following a notation similar to [Th, p. 64], let \(\mathbf{E}^{2,1}\) denote the three-dimensional Lorentz space (aka Minkowski space) with indefinite metric \(ds^2 = dx_1^2 + dx_2^2 - dx_3^2\) and associated quadratic form \(Q(x) = x_1^2 + x_2^2 - x_3^2\). The set of vectors \(v\in \mathbf{E}^{2,1}\) with \(Q(v)=-1\) is a 2-sheeted hyperboloid. Let \(H\) denote the upper sheet (i.e. \(x_3>0\)) of this hyperboloid. This is the hyperboloid in the hyperboloid model of the hyperbolic plane.

Define a generalized hyperbolic triangle to be a triple \((v_1, v_2, v_3)\) of vectors in \(\mathbf{E}^{2,1}\) that are linearly independent. The linear independence ensures that the triangles aren't degenerate (e.g. all points lying on the same line). In the following section, we describe how to interpret a generalized hyperbolic triangle as a polygon in the plane. We also call the polygons that can result generalized hyperbolic triangles.

Interpretation as polygons

Before describing how to interpret generalized hyperbolic triangles as polygons, we first describe how to interpret a single vector in \(\mathbf{E}^{2,1}\). See [Th, pp. 76-82] for a more detailed treatment of some of this material.

Thus, consider any non-zero vector \(v\in \mathbf{E}^{2,1}\). Then \(Q(v)\) is either positive, negative, or zero. The related quantity \(\sqrt{Q(v)}\) is known as the length, or norm, of \(v\) and is either real, imaginary, or zero, respectively. If \(Q(v)<0\) (imaginary length \(v\)), then the 1-dimensional subspace spanned by \(v\) intersects \(H\) in a single point, and \(v\) can be interpreted as that point in \(H\). If \(Q(v)>0\) (real length \(v\)), the 1-dimensional subspace spanned by \(v\) doesn't intersect \(H\), and we are in the case of \(v\) lying outside the plane. In this case, consider the 2-dimensional plane \(v^\perp\) in \(\mathbf{E}^{2,1}\) orthogonal to \(v\). The intersection of \(v^\perp\) with \(H\) forms a geodesic in \(H\). We will use this geodesic below when addressing triangles. Finally, if \(Q(v)=0\) (zero length \(v\)), then \(v\) corresponds to an ideal point, or point on the boundary of \(H\).

Consider now a triple \((v_1, v_2, v_3)\) of vectors in \(\mathbf{E}^{2,1}\). To interpret this triple as a polygon, apply the interpretation above for each \(v_i\) separately. Namely, if \(Q(v)\leq 0\), the vector corresponds to a point in \(H\) or on the boundary of \(H\). If \(Q(v)>0\), the vector corresponds to a geodesic in \(H\). Connect the resulting three points and/or geodesics to form a polgyon, as follows. To connect a point with an adjacent point, use the unique geodesic segment connecting the two points. To connect a geodesic with an adjacent geodesic, if they don't already intersect, use the unique geodesic segment that intersects both geodesics perpendicularly. To connect a geodesic with an adjacent point, if they don't already intersect, use the unique geodesic segment from the point to the geodesic that intersects the geodesic perpendicularly.

References

• [Th] William P. Thurston, Three-dimensional Geometry and Topology, Vol. 1, edited by Silvio Levy, Princeton Mathematical Series, 35, 1997.