Generalized hyperbolic triangle

This article is about polygons in the hyperbolic plane that can be represented as triangles in the real projective plane \(\mathbf{RP}^{2}\). Informally, these can be thought of as hyperbolic triangles whose vertices are allowed to lie outside the hyperbolic plane (e.g. when using the Klein model or Poincaré disk model of the hyperbolic plane). Since there doesn't seem to be a common name for these polygons, we call them generalized hyperbolic triangles for the purposes of this encyclopedia.

The polygons that are generalized hyperbolic triangles are, roughly speaking, the polygons one can get by replacing one or more vertices of a standard hyperbolic triangle with (for each replaced vertex) an additional side and two right-angled vertices. In addition to standard triangles with three sides, these polygons include—quadrilaterals with at least two right angles, pentagons with at least four right angles, and hexagons with six right angles. The figure below shows all of the possibilities grouped into five cases. Four of these cases include a self-intersecting configuration, which is shown in the bottom row of the figure.

Viewing these polygons as generalized triangles is useful because it lets certain statements about triangles be stated also for these polygons. The hyperbolic law of cosines is an example of such a statement.

Preliminaries

We first review some background related to the hyperboloid and Klein (aka projective) models of the hyperbolic plane. Our discussion and notation loosely follows [Th, ch. 2.3].

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 \(Q(x) = x_1^2 + x_2^2 - x_3^2\). Given a vector \(v\in \mathbf{E}^{2,1}\), the quantity \(Q(v)\) is either positive, negative, or zero. The related quantity \(\sqrt{Q(v)}\) is the length, or norm, of \(v\) and is either positive real, positive imaginary, or zero, respectively.

Let \(\mathbf{RP}^{2}\) denote the space of 1-dimensional vector subspaces in \(\mathbf{E}^{2,1}\) (aka the real projective plane). Thus, points in the real projective plane correspond to 1-dimensional subspaces. Similarly, define a line in \(\mathbf{RP}^{2}\) to be the image in \(\mathbf{RP}^{2}\) of a 2-dimensional subspace in \(\mathbf{E}^{2,1}\). Given a point \(p\in \mathbf{RP}^{2}\), we can choose a non-zero vector \(v\in \mathbf{E}^{2,1}\) that lies in the corresponding 1-dimensional subspace and ask whether \(Q(v)\) is positive, negative, or zero. The answer doesn't depend on the \(v\) that was chosen, so the answer is a property of the point \(p\) in \(\mathbf{RP}^{2}\). Thus, points in \(\mathbf{RP}^{2}\) fall naturally into three categories depending on whether \(Q(v)\) is positive, negative, or zero.

The subset of \(\mathbf{RP}^{2}\) corresponding to vectors \(v\) with \(Q(v)<0\) (positive imaginary length) corresponds to the hyperbolic plane (e.g. in the hyperboloid and Klein models of the hyperbolic plane). Call this subset \(H\subset \mathbf{RP}^{2}\).

The subset corresponding to vectors of length zero equals \(\partial H=\overline{H} - H\subset \mathbf{RP}^{2}\). These are the points on the boundary of \(H\) and are also known as ideal points.

Finally, the subset corresponding to vectors \(v\) with \(Q(v)>0\) (positive real length) equals \(\mathbf{RP}^{2} - \overline{H}\). These are the points outside the hyperbolic plane and its boundary and are sometimes called ultra ideal points. Call this subset \(U\). We have then that \(\mathbf{RP}^{2}\) is the disjoint union of \(H\), \(\partial H\), and \(U\).

We now describe how to interpret a point in \(U\) geometrically (see also [Th, p. 71]). Consider a point \(p\) in \(U\). Choose a non-zero vector \(v\in \mathbf{E}^{2,1}\) that lies in the corresponding 1-dimensional subspace. Consider the plane \(v^\perp\) in \(\mathbf{E}^{2,1}\) orthogonal to \(v\). This projects to a line in \(\mathbf{RP}^{2}\). The line's intersection with \(H\) is a geodesic line in \(H\) we denote \(p^\perp\). Thus, points outside the hyperbolic plane can be interpreted as geodesics inside the plane.

Triangles in \(\mathbf{RP}^{2}\)

Define a triangle in \(\mathbf{RP}^{2}\) to be a triple \(T=(p_0, p_1, p_2)\) of points not all on the same line. We call the three \(p_i\) the vertices of \(T\). For convenience, view the indices as elements of \(\mathbb{Z}/3\mathbb{Z}\) so that incrementing the index by one cycles through (e.g. \(p_{2+1}=p_0\)). Requiring the points not to lie on the same line ensures that the triangle isn't degenerate. If \(v_0\), \(v_1\), and \(v_2\) are vectors in \(\mathbf{E}^{2,1}\) representing the three \(p_i\), this condition is equivalent to the \(v_i\) being linearly independent. Additionally, if we consider the basis of vectors \(w_0\), \(w_1\), and \(w_2\) dual to the \(v_i\), then the \(w_i\) define a new triangle \((q_0, q_1, q_2)\) in \(\mathbf{RP}^{2}\) that we call the triangle \(T^*\) dual to \(T\).

It is convenient to define a shorthand for referring to different types of triangles. The shorthand is a pair of nonnegative integers \((r,s)\) we call the signature. Let \(T=(p_0, p_1, p_2)\) be a triangle in \(\mathbf{RP}^{2}\). Let \(r\) be the number of \(p_i\) in \(\overline{H}\). The remaining \(p_i\) are in \(U\). There are \(3-r\) of them, and they correspond to geodesics in \(H\). Let \(s\) be the number of pairs of these geodesics that intersect in \(\overline{H}\). We define the signature of \(T\) to be \((r,s)\). The signature is a concept introduced only for the purpose of this article.

For example, consider a triangle \(T\) defined by three points in \(\overline{H}\). This has signature \((3,0)\). As another example, consider the dual triangle \(T^*\) whose three vertices in \(\mathbf{RP}^{2}\) correspond to the three sides of the \(T\) we just defined. The triangle \(T^*\) has signature \((0,3)\). More generally, if \((r,s)\) is the signature of a triangle \(T\), then the signature of the dual \(T^*\) is \((s,r)\). We will list more examples of triangles and their signatures below.

Realizing triangles as polygons

In this section, we describe a way to construct from any triangle \(T\) in \(\mathbf{RP}^{2}\) a canonical polygon \(P\) in \(H\), possibly with some ideal vertices.

To describe the construction briefly, for each vertex of \(T\), if the vertex is inside the hyperbolic plane, then use the point in the plane as a vertex of \(P\). If the vertex is outside the plane, then use the geodesic line inside the plane corresponding to that vertex as the basis for an additional side of \(P\). Then, form a polygon from those points and sides, connecting them with perpendicular segments as needed. We also describe the construction more formally in the following paragraphs.

Let \((p_0, p_1, p_2)\) be the triangle \(T\). For each \(i\)—

If \(p_i\in H\), then include the point \(p_i\) as a vertex of \(P\).
If \(p_i\in \overline{H}\) and \(p_{i+1}\in \overline{H}\), then \(p_i\) and \(p_{i+1}\) are points or ideal points. Include the geodesic segment connecting \(p_i\) and \(p_{i+1}\) as a side of \(P\).
If \(p_i\in U\), then \(p_i\) corresponds to a geodesic \(p_i^\perp\) in \(H\). In this case, do the following steps (see the figure below for an illustration of this case):
- Consider the point \(p_{i-1}\) adjacent to \(p_i\). If \(p_{i-1}\in \overline{H}\), then include the perpendicular segment from \(p_{i-1}\) to \(p_i^\perp\) as a side \(s\) of \(P\). Similarly, if \(p_{i-1}\in U\), then include the common perpendicular segment between \(p_{i-1}^\perp\) and \(p_i^\perp\) as a side \(s\) of \(P\). In each case, let \(x_0^i\) denote the intersection of \(s\) with \(p_i^\perp\).
- Do the same for the other point \(p_{i+1}\) adjacent to \(p_i\), and let \(x_1^i\) denote that point of intersection.
- Include \(x_0^i\) and \(x_1^i\) and the geodesic segment between them as vertices and a side of \(P\).

This figure illustrates part of the construction of a polygon from a triangle in \(\mathbf{RP}^{2}\). It illustrates the two sub-cases in the case of \(p_1\in U\), namely \(p_0\in \overline{H}\) and \(p_2\in U\).

The figure below illustrates the construction above for a triangle \(T=(p_0, p_1, p_2)\) in \(\mathbf{RP}^{2}\) with \(p_0\) and \(p_1\) in \(H\) and and \(p_2\) in \(U\). In particular, \(T\) has signature \((1, 2)\).

This figure shows the construction of the polygon in \(H\) corresponding to a triangle \((p_0, p_1, p_2)\) in \(\mathbf{RP}^{2}\) with \(p_0\) and \(p_1\) in \(H\) and and \(p_2\) in \(U\). The resulting polygon is a quadrilateral with two right angles.

If \(T\) is a triangle in \(\mathbf{RP}^{2}\) and \(P\) is the hyperbolic polygon that results from the construction above, we say that \(P\) is the realization of \(T\) in \(H\). In the reverse direction, we say that \(T\) is a representation of \(P\) in \(\mathbf{RP}^{2}\).

This brings us to our formal definition of a generalized hyperbolic triangle. Define a generalized hyperbolic triangle to be a hyperbolic polygon that can be represented as a triangle in \(\mathbf{RP}^{2}\).

If a triangle \(T\) in \(\mathbf{RP}^{2}\) is a representation of a hyperbolic polygon \(P\), it is the case that the dual triangle \(T^*\) is also a representation of \(P\). Thus, every generalized hyperbolic triangle has two representations. We call these representations dual representations.

List of possible polygons

This section lists the combinatorial possibilities for generalized hyperbolic triangles. There are five possibilities in all, or nine if one permits the polygon to have self-intersections. They are—

Case 1. triangle (three sides).
Case 2*. quadrilateral (four sides) with exactly one pair of adjacent right angles.
Case 3*. quadrilateral (four sides) with one pair of opposite right angles.
Case 4*. pentagon (five sides). This case has at least four right angles.
Case 5*. hexagon (six sides). This case has all right angles.

These cases are mutually exclusive. This is also why "exactly one" was added to case 2. Without saying "exactly one," a Lambert quadrilateral (quadrilateral with three right angles) would be in cases both 2 and 3.

Cases 2, 3, 4, and 5 each include an additional configuration where the polygon has self-intersections. This is indicated by the asterisk (*).

The figure below illustrates all five of these cases, including the self-intersecting configuration for each. Each row corresponds to either the self-intersecting or non-self-intersecting configuration of the case of the same number, depending on whether or not the number has a "b" after it, respectively. For example, the row numbered "2" depicts the non-self-intersecting configuration for Case 2, and row "2b" shows the self-intersecting configuration.

The left and right sides of each row both depict the same generalized hyperbolic triangle, along with one of its representations as a triangle \(T\) in \(\mathbf{RP}^{2}\). The representations on the left and right are the two dual representations, with the signature of the triangle \(T\) shown below. On each side, the triangle \(T\) is depicted using bold vertices and complete geodesic arcs using a thinner line. The bold vertices correspond to the vertices of \(T\) that lie in \(\overline{H}\), and the geodesic arcs correspond to the vertices of \(T\) that lie in \(U\) (i.e. outside \(\overline{H}\)).

This figure shows all nine combinatorial possibilities for generalized hyperbolic triangles. In each row, the pictures on the left and right depict the two dual representations of the generalized triangle as a triangle in \(\mathbf{RP}^{2}\). The signature of the representation is shown below.

References

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