cjerdonek authored on Nov. 17, 2020, 7:20 a.m. | | Additions are marked like this |
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 generalized triangle, it has two vertices outside the plane. As a polygon in the plane, it is a pentagon with four right angles.  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. Three of these cases include a self-intersecting configuration, which is also shown.  DefinitionsFollowing 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 polygonsBefore 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. 
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 polygonsThis 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—
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
| 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.  DefinitionsFollowing 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 polygonsBefore 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.
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 polygonsThis 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—
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
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