Research Article | Open Access
Orthogonal Projections Based on Hyperbolic and Spherical -Simplex
Orthogonal projection along a geodesic to the chosen k-plane is introduced using edge and Gram matrix of an n-simplex in hyperbolic or spherical n-space. The distance from a point to k-plane is obtained by the orthogonal projection. It is also given the perpendicular foot from a point to k-plane of hyperbolic and spherical n-space.
One of the fundamental notions in geometry is orthogonal projection and also studied extensively through the long history of mathematics and physics. There are many applications of orthogonal projection. The concept of orthogonal projection plays an important role in the scattering theory, the theory of many-body resonance, and different branches of theoretical and mathematical physics.
Let be -dimensional vector space equipped with the scalar product which is defined byIf the restriction of scalar product on a subspace of is positive definite, then the subspace is called space-like; if it is positive semi-definite and degenerate, then is called light-like; if contains a time-like vector of , then is called time-like.
is called de Sitter -space. The -dimensional unit pseudohyperbolic space is defined aswhich has two connected components, each of which can be considered as a model for the -dimensional hyperbolic space . Throughout this paper we consider hyperbolic -spaceHence, each pair of points , in satisfy . The hyperbolic distance for is defined by . Since each determines a time-like hyperplane of , we have hyperplane of .
Let be -dimensional vector space equipped with the scalar product which is defined byThe -dimensional unit spherical space is given by The spherical distance between and is given by .
We consider that is a vector subspace spanned by the vectors in . By using Lemma 27 of , one can easily see that is -dimensional time-like subspace and is -dimensional time-like subspace of . Consequently, for , the hyperplane intersects at the time-like -plane of . One can define the same tools for spherical -space.
Let be a hyperbolic or spherical simplex with vertices and let be the face opposite to vertex . Then, according to the first section of , we have the edge matrix and Gram matrix of . Let and be the determinant and th-minor of ; then the unit outer normal vector of is given by where is the curvature of space.
The intersection of with -dimensional time-like subspace is called -dimensional plane of . Similarly, a -plane of spherical space is given by the same way.
When a geodesic is drawn orthogonally from a point to a -plane; its intersection with the -plane is known as perpendicular foot on that -plane in or . The length of a geodesic segment bounded by a point and its perpendicular foot is called the distance between that point and -plane. The distance between a vertex and its any opposite -face is called -face altitude of an -simplex.
The orthogonal projection to -plane in Euclidean space is well-known (see [4–7]). The orthogonal projection to -plane in Euclidean space is given in . The orthogonal projection taking a point in and mapping it to its perpendicular foot on a hyperplane are studied in [3, 9], respectively. The distance between a point and a hyperbolic (spherical) hyperplane is introduced in . The altitude of -face of hyperbolic -simplex is given in .
The orthogonal projection taking a point along a geodesic and mapping to its perpendicular foot, where geodesic meets orthogonally the chosen -plane of projection, has not been studied. The aim of this paper is to study such orthogonal projections according to the edge matrix of a simplex in or .
Let be the determinant of submatrix of and let be the determinant of submatrix of . Suppose that and are the determinant of submatrices and , respectively.
Lemma 1. Let be a hyperbolic or spherical simplex with the edge matrix and Gram matrix . Let and be th minor of and , respectively. Then and , where .
Proof. It can be seen from .
Let , , and , , be , , types submatrices of and , respectively. Suppose that , , and diagonal matrix are partitioned as , , and , respectively.
Concerning Lemma 1 along with Schur complement of a symmetric matrix, we have the following lemma.
Lemma 2. Let and be Schur complements of the submatrices and . Then
Proof. It is obvious that , are symmetric and , are invertible. Since the inverse of Schur complement of in is the submatrix of , we haveSimilarly, for the Schur complement of , we obtainThen we have and, by the same way, we get Thus, we obtain the desired results using Lemma 1.
2. Orthogonal Projection to -Plane of Based on a Hyperbolic -Simplex
If are vertices of any hyperbolic simplex , then is a basis of . Let be a subspace spanned by , and let be the unit outer normal to , . Hence is another basis of .
Let be a -plane which contains a -face with vertices of . Then the set is a basis of the -dimensional subspace of in . Since are vertices of , the subset of can be extended bases and of . As a consequence, we see that is a basis of -dimensional subspace .
Theorem 3. Let be a point and let be a -plane in . Then the orthogonal projection of to is given by
Proof. For a point , by [10, Theorem 3.11], there is a point such that . Therefore, we can writeThen, we have , .
Taking we obtainBy Lemma 2 and of , we see that and this implies thatSubstituting (17) into (13), we obtainBy , there exists a unique such that . Since is the orthogonal projection of to , we havewhich completes the proof.
In case of the orthogonal projection to a hyperplane, we obtain and . Substituting these equalities into the statement of Theorem 3, we reach the result of [3, Theorem 4.1] and [14, Proposition 2.2], as follows: where is the unit normal of in . This result is also a generalization of Theorem 3 [3, 14].
Theorem 4. Let be a point and let be a -plane in . Then, where is the distance between and .
Proof. Since , the result follows Theorem 3.
Corollary 5. Let be a point and let be a hyperplane of determined by . Then the distance between and is given by
Corollary 6. Let be a hyperbolic simplex with vertices . Then the perpendicular foot from to -face is given by where are vertices of -face .
If we replace by and use , we see that If we consider the last equation in the proof of Theorem 3, we see that ; that result is a generalization of [11, Proposition 4] to the -face of a hyperbolic -simplex. Since is the diagonal th-entry of , the altitude from to -face with vertices is given by where is the distance between and -face .
By , for -face , we have the following corollary.
Corollary 7. Let be a hyperbolic simplex with vertices . Then the perpendicular foot from to -face is given by where are vertices of .
Using for , we obtain the following known result [11, Proposition 4].
Corollary 8. Let be a hyperbolic simplex with vertices . Then the altitude from to -face is given by where are vertices of .
3. Orthogonal Projections to a -Plane of Based on a Spherical -Simplex
Let be with vertices . Then is a basis of . If is the subspace spanned by , then is another basis of where is the unit outer normal to for .
Let be a -plane which contains a face with vertices . Then the set is a basis of the -dimensional subspace in . As a consequence, we have a basis of -dimensional subspace .
Theorem 9. Let be a point and let be a -plane in . Then the orthogonal projection of to is given by
Proof. By [10, Theorem 3.11], for , there is a such that . Therefore, we can writeThen, we haveTaking we findBy Lemma 2 and of , we see that This implies thatSubstituting (35) into (30), we obtainBy , there exists a unique such that . Since is the orthogonal projection of to , we have which completes the proof.
By Theorem 9, we have where is the unit normal of the in .
Theorem 10. Let be a point and let be a -plane in . Then where is the distance between and .
By taking instead of and using in (36), we obtain where is a vertex of . Hence, we have the following corollary.
Corollary 11. Let be a spherical -simplex with vertices ; then the perpendicular foot from to -face is given by where are vertices of .
Let be the altitude from the vertex to the -face with vertices for . Then is given by By equality in , the th-entry of the Schur complement satisfies .
Let be the -face with vertices of . Then, we have
The proof of the following corollary is obtained by using the above equations.
Corollary 12. Let be a spherical simplex with vertices . Then the perpendicular foot from to face is given by where are vertices of .
Corollary 13. Let be a spherical simplex with vertices . Then the altitude from to -face is given by where are vertices of .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
- B. O'Neill, Semi-Riemannian Geometry, Academic Press, New York, NY, USA, 1983.
- B. Karlıga and A. T. Yakut, “Vertex angles of a simplex in hyperbolic space Hn,” Geometriae Dedicata, vol. 120, pp. 49–58, 2006.
- S. Izumiya and F. Tari, “Projections of hypersurfaces in the hyperbolic space to hyperhorospheres and hyperplanes,” Revista Matemática Iberoamericana, vol. 24, no. 3, pp. 895–920, 2008.
- J. H. Rieger, “Families of maps from the plane to the plane,” Journal of the London Mathematical Society. Second Series, vol. 36, no. 2, pp. 351–369, 1987.
- V. I. Arnol'd, “Singularities of systems of rays,” Russian Mathematical Surveys, vol. 38, no. 2, pp. 87–176, 1983.
- J. W. Bruce, “Generic geometry, transversality and projections,” Journal of the London Mathematical Society, vol. 49, no. 1, pp. 183–194, 1994.
- J. W. Bruce, P. J. Giblin, and F. Tari, “Families of surfaces: height functions and projections to planes,” Mathematica Scandinavica, vol. 82, no. 2, pp. 165–185, 1998.
- B. Karol, Multidimensional Analytic Geometry, vol. 50, Instytut Matematyczay Polaskiej Akad. Nauk. To., Warsaw, Poland, 1969, translation from the Polish by Halina Spalinska.
- L. J. Alías, T. Kurose, and G. Solanes, “Hadamard-type theorems for hypersurfaces in hyperbolic spaces,” Differential Geometry and its Applications, vol. 24, no. 5, pp. 492–502, 2006.
- E. B. Vinberg, Geometry II, vol. 29 of Encyclopaedia of Mathematical Sciences, Springer, New York, NY, USA, 1993.
- A. D. Mednykh and M. G. Pashkevich, “Elementary formulas for a hyperbolic tetrahedron,” Siberian Mathematical Journal, vol. 47, no. 4, pp. 687–695, 2006.
- B. Karliğa, “Edge matrix of hyperbolic simplices,” Geometriae Dedicata, vol. 109, pp. 1–6, 2004.
- R. A. Brualdi and H. Schneider, “Determinantal identities: gauss, schur, cauchy, sylvester, kronecker, jacobi, binet, laplace, muir, and cayley,” Linear Algebra and its Applications, vol. 52-53, pp. 769–791, 1983.
- A. Ushijima, “The tilt formula for generalized simplices in hyperbolic space,” Discrete & Computational Geometry, vol. 28, no. 1, pp. 19–27, 2002.
Copyright © 2015 Murat Savas et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.