Generalized hyperbolic triangle

cjerdonek authored on Dec. 7, 2020, 4:42 a.m.

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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. This picture shows an example of a generalized hyperbolic triangle in the Poincaré disk model of the hyperbolic plane. As a polygon in the plane, it is a pentagon with four right angles. As a generalized triangle, it has one vertex in the plane and two outside (all three shown in bold). The two outside are shown schematically as the centers of the geodesic arcs corresponding to two of the sides. The polygons that are generalized hyperbolic triangles are, roughly speaking, the 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—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 below. 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). 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 \(U\). Define a line in \(\mathbf{RP}^{2}\) to be the image in \(\mathbf{RP}^{2}\) of a 2-dimensional subspace in \(\mathbf{E}^{2,1}\). 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. For convenience, view the indices as elements of \(\mathbb{Z}/3\mathbb{Z}\) so that adding 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 will be helpful to have a shorthand for talking about different types of triangles in \(\mathbf{RP}^{2}\). Given a triangle \(T=(p_0, p_1, p_2)\), let \(u\) be the number of \(p_i\) in \(U\), and let \(v\) be the number of \(p_i\) in \(\overline{H}\) (so that \(u + v = 3\)). Define \((u, v)\) to be the signature of \(T\). For example, a triangle in the hyperbolic plane (three points in \(H\)) corresponds to a triangle in \(\mathbf{RP}^{2}\) with signature \((0, 3)\). Realizing triangles as polygons Given a triangle \(T\) in \(\mathbf{RP}^{2}\), we define in this section a way to construct from \(T\) a canonical polygon \(P\) in \(H\), possibly with some ideal vertices. 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.	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. This picture shows an example of a generalized hyperbolic triangle in the Poincaré disk model of the hyperbolic plane. As a polygon in the plane, it is a pentagon with four right angles. As a generalized triangle, it has one vertex in the plane and two outside (all three shown in bold). The two outside are shown schematically as the centers of the geodesic arcs corresponding to two of the sides. The polygons that are generalized hyperbolic triangles are, roughly speaking, the 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—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. 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). 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 \(U\). Define a line in \(\mathbf{RP}^{2}\) to be the image in \(\mathbf{RP}^{2}\) of a 2-dimensional subspace in \(\mathbf{E}^{2,1}\). 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. For convenience, view the indices as elements of \(\mathbb{Z}/3\mathbb{Z}\) so that adding 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 will be helpful to have a shorthand for talking about different types of triangles in \(\mathbf{RP}^{2}\). Given a triangle \(T=(p_0, p_1, p_2)\), let \(u\) be the number of \(p_i\) in \(U\), and let \(v\) be the number of \(p_i\) in \(\overline{H}\) (so that \(u + v = 3\)). Define \((u, v)\) to be the signature of \(T\). For example, a triangle in the hyperbolic plane (three points in \(H\)) corresponds to a triangle in \(\mathbf{RP}^{2}\) with signature \((0, 3)\). Realizing triangles as polygons Given a triangle \(T\) in \(\mathbf{RP}^{2}\), we define in this section a way to construct from \(T\) a canonical polygon \(P\) in \(H\), possibly with some ideal vertices. 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.

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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.

This picture shows an example of a generalized hyperbolic triangle in the Poincaré disk model of the hyperbolic plane. As a polygon in the plane, it is a pentagon with four right angles. As a generalized triangle, it has one vertex in the plane and two outside (all three shown in bold). The two outside are shown schematically as the centers of the geodesic arcs corresponding to two of the sides.

The polygons that are generalized hyperbolic triangles are, roughly speaking, the 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—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 below.

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). 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 \(U\).

Define a line in \(\mathbf{RP}^{2}\) to be the image in \(\mathbf{RP}^{2}\) of a 2-dimensional subspace in \(\mathbf{E}^{2,1}\). 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. For convenience, view the indices as elements of \(\mathbb{Z}/3\mathbb{Z}\) so that adding 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 will be helpful to have a shorthand for talking about different types of triangles in \(\mathbf{RP}^{2}\). Given a triangle \(T=(p_0, p_1, p_2)\), let \(u\) be the number of \(p_i\) in \(U\), and let \(v\) be the number of \(p_i\) in \(\overline{H}\) (so that \(u + v = 3\)). Define \((u, v)\) to be the signature of \(T\). For example, a triangle in the hyperbolic plane (three points in \(H\)) corresponds to a triangle in \(\mathbf{RP}^{2}\) with signature \((0, 3)\).

Realizing triangles as polygons

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

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].

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.

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

Realizing triangles as polygons

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 \(P\).

List of possible polygons

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.

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

References

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

cjerdonek / Generalized hyperbolic triangle 19a3992c82baf9fe064f18981b9ec791b90cc737

Preliminaries

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

Realizing triangles as polygons

List of possible polygons

References

Preliminaries

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

Realizing triangles as polygons

List of possible polygons

References