Generalized hyperbolic triangle

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This article defines and describes hyperbolic triangles whose vertices are allowed to lie outside the hyperbolic plane and its boundary. We call these triangles generalized hyperbolic triangles because their representation as polygons can have more than three sides. By "outside the plane," we mean that from the perspective of the hyperboloid model of the hyperbolic plane, the point corresponds to a vector in Lorentz space with real as opposed to imaginary norm. We explain the meaning of this further below. When viewed as polygons, 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, the polygons can be — 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 five possible cases. (There are also some self-intersecting cases, which we don't currently show.) This figure shows the five possible cases for the shape associated to a generalized hyperbolic triangle. 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. 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 defines and describes hyperbolic triangles whose vertices are allowed to lie outside the hyperbolic plane and its boundary. We call these triangles generalized hyperbolic triangles because their representation as polygons can have more than three sides. By "outside the plane," we mean that from the perspective of the hyperboloid model of the hyperbolic plane, the point corresponds to a vector in Lorentz space with real as opposed to imaginary norm. We explain the meaning of this further below. When viewed as polygons, 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, the polygons can be — 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 five possible cases. (There are also some self-intersecting cases, which we don't currently show.) This figure shows the five possible cases for the shape associated to a generalized hyperbolic triangle. 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, 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.

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

By "outside the plane," we mean that from the perspective of the hyperboloid model of the hyperbolic plane, the point corresponds to a vector in Lorentz space with real as opposed to imaginary norm. We explain the meaning of this further below.

When viewed as polygons, 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, the polygons can be — 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 five possible cases. (There are also some self-intersecting cases, which we don't currently show.)

This figure shows the five possible cases for the shape associated to a generalized hyperbolic triangle.

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.

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.

Definitions

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

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, 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.

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 e43c5ad34e2281cd32ba74bf1344d08715d86b45

Definitions

Interpretation as polygons

References

Definitions

Interpretation as polygons

List of possible polygons

References