Orthogonal Projections Based on Hyperbolic and Spherical n-Simplex

In this paper, 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 foots from a point to k-plane of hyperbolic and spherical n-space.


Introduction
One of the fundamental notions in geometry is orthogonal projection and also studies 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 R n+1 1 be (n + 1)−dimensional vector space equipped with the scalar product , which is defined by If the restriction of scalar product on a subspace W of R 1 n+1 is positive definite, then the subspace W is called space-like; if it is positive semi-definite and degenerate, then W is called light-like; if W contains a time-like vector of R 1 n+1 , then W is called time-like.
S n 1 = {x ∈ R 1 n+1 | x, x = 1} is called de Sitter n-space. The n−dimensional unit pseudo-hyperbolic space is defined as which has two connected components, each of which can be considered as a model for the n-dimensional hyperbolic space H n . Throughout this paper we consider hyperbolic n−space Hence, each pair of points p i , p j in H n satisfy p i , p j < 0. The hyperbolic distance for p, q ∈ H n is defined by arccosh(− p, q ). Since each e ∈ S n 1 determines a time-like hyperplane of R n+1 1 , we have hyperplane e ⊥ ∩ H n of H n .
Let R n+1 be (n+ 1)−dimensional vector space equipped with the scalar product , E which is defined by The n-dimensional unit spherical space S n is given by The spherical distance d s (p, q) between p and q is given by arccos( p, q E ).
We consider W is a vector subspace spanned by the vectors e 1 , e 2 , ..., e n−k in S n 1 . By using Lemma 27 of [1], one can easily see that W is (n− k)−dimensional time-like subspace and Consequently, for i = 1, 2, ..., n − k, the hyperplane e ⊥ i ∩ H n intersect at the time-like k−plane V ∩ H n of H n . One can define the same tools for spherical n−space.
Let △ be a hyperbolic or spherical n−simplex with vertices p 1 , ..., p n+1 , and △ i be the face opposite to vertex p i . Then, according to the first section of [2], we have the edge matrix M and Gram matrix G of △. Let |M | and M ij be the determinant and ijth−minor of M , then the unit outer normal vector of △ i is given by where ǫ is the curvature of space.
The intersection of H n with (k + 1)−dimensional time-like subspace is called k−dimensional plane of H n [3]. Similarly, a k−plane of spherical space is given by the same way.
When a geodesic is drawn orthogonally from a point to a k−plane, its intersection with the k−plane is known as perpendicular foot on that k−plane in H n or S n . The length of a geodesic segment bounded by a point and its perpendicular foot is called the distance between that point and k−plane. The distance between a vertex and its any opposite k−face is called k−face altitude of an n−simplex.
The orthogonal projection to 2−plane in Euclidean space is well-known (see [4], [5], [6], [7]). The orthogonal projection to k−plane in Euclidean space is given in [8]. The orthogonal projection taking a point in H n and mapping it to its perpendicular foot on a hyperplane are studied in [3] and [9], respectively. The distance between a point and a hyperbolic(spherical) hyperplane is introduced in [10]. The altitude of (n − 1)−face of hyperbolic n−simplex is given in [11].
The orthogonal projection taking a point along a geodesic and mapping to its perpendicular foot, where geodesic meets orthogonally the chosen k−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 H n or S n .
Let m k+1 be the determinant of sub-matrix M (k + 1, ..., n + 1) of M and g k+1 be the determinant of sub-matrix G(k + 1, ..., n + 1) of G. Suppose that m j i and g j i be the determinant of sub-matrix M , respectively. Then Proof It can be seen from [12].
Proof It is obvious that M, G are symmetric and M ii , G ii are invertible. Since the inverse of Schur complement of M 11 in M is the sub-matrix of M −1 , we have Similarly for the Schur complement of M 22 , we obtain by the same way, we get Thus, we obtain the desired results using Lemma 1.1. Proof For a point p ∈ H n , by [10, Theorem 3.11], there is a pointṗ ∈ W such that pṗ ∈ W ⊥ . Therefore, we can write Then, we have − p, e t = n+1 s=k+2 λ s e s , e t , t = k + 2, ..., n + 1. Taking we obtain By Lemma 1.2 and the equation (5) of [14], we see that , and this implies Substituting (3) into (1), we obtaiṅ By [10], there exists a unique σ(p) ∈ W h such that σ(p) = cṗ. Sinceṗ is the orthogonal projection of p to W , we have where e j is the unit normal of W j in R n+1

1
. This result is also a generalization of Theorem 2.1 [3,13]. where ξ(p, W h ) is the distance between p and W h .
As an immediate consequence of Theorem 2.2, we obtain the following known result[10, Section 4].

Corollary 2.3 Let p be a point and W h
j be a hyperplane of H n determined by e j . Then the distance ξ(p, W h j ) between p and W h j is given by By taking p j instead of p in (4) and using p j , e t = − |M | M tt δ jt , we obtainṗ If we replace p by p j and use p j , e t = − |M | M tt δ jt , we see that If we consider the last equation in the proof of Theorem 2.1, we see that cosh ξ(p j , W h ) = 1 + m j j m k+1 , that result is a generalization of [11,Proposition 4] to the k−face W h of a hyperbolic n−simplex. Since m j j m k+1 is the diagonal jjth−entry of S M 11 = [a ij ], the altitude from p j to k−face W h with vertices p 1 , ..., p k+1 is given by 3 Orthogonal Projections to a k−plane of S n based on a Spherical n−simplex Let △ be with vertices p 1 , ..., p n+1 . Then {p 1 , ..., p n+1 } is a basis of R n+1 . If W j is the subspace spanned by {p 1 , ...,p j , ..., p n+1 }, then {e 1 , ..., e n+1 } is another basis of R n+1 where e j is the unit outer normal to W j for j = 1, ..., n + 1.
By Lemma 1.2 and the equation (5) of [14], we see that .