Generalized hyperbolic triangle

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This article is about polygons in the hyperbolic plane that can be represented as triangles in the real projective plane \(\mathbf{RP}^{2}\) (e.g. when using the projective model of the hyperbolic plane). Informally, these can be thought of as hyperbolic triangles whose vertices are allowed to lie outside 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 picture below shows an example of a generalized hyperbolic triangle and its representation as a triangle in \(\mathbf{RP}^{2}\). As a polygon in the plane, it is a pentagon with four right angles. As a generalized triangle, it has two vertices outside the plane. 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 all right angles. The figure below shows the possibilities grouped into five cases. Four of these cases include a self-intersecting configuration, which is also shown. Definitions 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. The surface \(H\) 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. Requiring linear independence ensures that the triangles aren't degenerate (e.g. all points lying on the same line). It will be helpful to have a shorthand for talking about different types of generalized hyperbolic triangles. Thus, given a generalized hyperbolic triangle \(T = (v_1, v_2, v_3)\), let \(p\) (for "positive") be the number of vectors \(v_i\) with \(Q(v_i)>0\), and let \(q\) be the remaining number with \(Q(v_i)\leq 0\) (so that \(p + q = 3\)). We say that \((p, q)\) is the signature of \(T\). For example, a triple of vectors representing three points in the hyperbolic plane has signature \((0, 3)\). In the next 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 \(\mathbf{E}^{2,1}\) in terms of the hyperbolic plane. 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 \(H\). Finally, connect the resulting three points and/or geodesics to form a polygon, 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. This figure shows the construction of the polygon corresponding to a generalized hyperbolic triangle \((v_1, v_2, v_3)\) when \(v_1\) and \(v_2\) have imaginary length and \(v_3\) has real length (i.e. one vertex lies outside the hyperbolic plane). The resulting polygon is a quadrilateral with two right angles. The figure above illustrates the interpretation described above for a generalized hyperbolic triangle \((v_1, v_2, v_3)\) with \(v_1\) and \(v_2\) of imaginary length and \(v_3\) of real length. List of possible polygons This section lists the combinatorial possibilities for polygons that can arise as generalized hyperbolic triangles. There are five possibilities in all, or eight 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 each of these cases, including the self-intersecting cases. Each row of the figure depicts, using bold vertices for vectors with imaginary length and complete geodesic arcs for vectors with real length, both of the polygon's two representations as a generalized triangle. These generalized triangles are dual to each other, and the signature of the generalized triangle is shown beneath the corresponding depiction. This figure shows the possible polygons in the plane that can arise as a generalized hyperbolic triangle. In each row, the pictures on the left and right depict the dual generalized triangles that give rise to the same polygon. The signature of the corresponding generalized triangle 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 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.

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This article is about polygons in the hyperbolic plane that can be represented as triangles in the real projective plane \(\mathbf{RP}^{2}\) (e.g. when using the projective model of the hyperbolic plane). Informally, these can be thought of as hyperbolic triangles whose vertices are allowed to lie outside 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 picture below shows an example of a generalized hyperbolic triangle and its representation as a triangle in \(\mathbf{RP}^{2}\). As a polygon in the plane, it is a pentagon with four right angles. As a generalized triangle, it has two vertices outside the plane.

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 all right angles. The figure below shows the possibilities grouped into five cases. Four of these cases include a self-intersecting configuration, which is also shown.

Definitions

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. The surface \(H\) 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. Requiring linear independence ensures that the triangles aren't degenerate (e.g. all points lying on the same line).

It will be helpful to have a shorthand for talking about different types of generalized hyperbolic triangles. Thus, given a generalized hyperbolic triangle \(T = (v_1, v_2, v_3)\), let \(p\) (for "positive") be the number of vectors \(v_i\) with \(Q(v_i)>0\), and let \(q\) be the remaining number with \(Q(v_i)\leq 0\) (so that \(p + q = 3\)). We say that \((p, q)\) is the signature of \(T\). For example, a triple of vectors representing three points in the hyperbolic plane has signature \((0, 3)\).

In the next 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 \(\mathbf{E}^{2,1}\) in terms of the hyperbolic plane. 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 \(H\).

Finally, connect the resulting three points and/or geodesics to form a polygon, 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.

This figure shows the construction of the polygon corresponding to a generalized hyperbolic triangle \((v_1, v_2, v_3)\) when \(v_1\) and \(v_2\) have imaginary length and \(v_3\) has real length (i.e. one vertex lies outside the hyperbolic plane). The resulting polygon is a quadrilateral with two right angles.

The figure above illustrates the interpretation described above for a generalized hyperbolic triangle \((v_1, v_2, v_3)\) with \(v_1\) and \(v_2\) of imaginary length and \(v_3\) of real length.

List of possible polygons

This section lists the combinatorial possibilities for polygons that can arise as generalized hyperbolic triangles. There are five possibilities in all, or eight 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 each of these cases, including the self-intersecting cases. Each row of the figure depicts, using bold vertices for vectors with imaginary length and complete geodesic arcs for vectors with real length, both of the polygon's two representations as a generalized triangle. These generalized triangles are dual to each other, and the signature of the generalized triangle is shown beneath the corresponding depiction.

This figure shows the possible polygons in the plane that can arise as a generalized hyperbolic triangle. In each row, the pictures on the left and right depict the dual generalized triangles that give rise to the same polygon. The signature of the corresponding generalized triangle 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 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.

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.

cjerdonek / Generalized hyperbolic triangle 1a436cac81ac842c8dd05d19211e23e744a78042

Definitions

Interpretation as polygons

List of possible polygons

References

Preliminaries

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

Realizing triangles as polygons

List of possible polygons

References