1. Introduction
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 R1n+1 be (n+1)-dimensional vector space equipped with the scalar product 〈,〉 which is defined by(1)〈x,y〉=-x1y1+∑i=2n+1xiyi.If the restriction of scalar product on a subspace W of R1n+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 R1n+1, then W is called time-like.

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1
n
=
{
x
∈
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1
n
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1
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〈
x
,
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is called de Sitter n-space. The n-dimensional unit pseudohyperbolic space is defined as(2)H0n=x∈R1n+1∣〈x,x〉=-1,which has two connected components, each of which can be considered as a model for the n-dimensional hyperbolic space Hn. Throughout this paper we consider hyperbolic n-space(3)Hn={x∈H0n∣x1>0}.Hence, each pair of points pi, pj in Hn satisfy 〈pi,pj〉<0. The hyperbolic distance for p,q∈Hn is defined by arccosh(-〈p,q〉). Since each e∈S1n determines a time-like hyperplane of R1n+1, we have hyperplane e⊥∩Hn of Hn.

Let Rn+1 be (n+1)-dimensional vector space equipped with the scalar product 〈,〉E which is defined by(4)x,yE=∑i=1n+1xiyi.The n-dimensional unit spherical space Sn is given by (5)Sn={x∈Rn+1∣x,xE=1}.The spherical distance ds(p,q) between p and q is given by arccos(〈p,q〉E).

We consider that W is a vector subspace spanned by the vectors e1,e2,…,en-k in S1n. By using Lemma 27 of [1], one can easily see that W is (n-k)-dimensional time-like subspace and V=e1⊥∩e2⊥∩⋯∩en-k⊥ is (k+1)-dimensional time-like subspace of R1n+1. Consequently, for i=1,2,…,n-k, the hyperplane ei⊥∩Hn intersects at the time-like k-plane V∩Hn of Hn. One can define the same tools for spherical n-space.

Let Δ be a hyperbolic or spherical n-simplex with vertices p1,…,pn+1 and let Δi be the face opposite to vertex pi. Then, according to the first section of [2], we have the edge matrix M and Gram matrix G of Δ. Let |M| and Mij be the determinant and ijth-minor of M; then the unit outer normal vector of Δi is given by (6)ei=-ϵ∑j=1n+1MijpjMii|M|, i=1,…,n+1,where ϵ is the curvature of space.

The intersection of Hn with (k+1)-dimensional time-like subspace is called k-dimensional plane of Hn [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 Hn or Sn. 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 [4–7]). The orthogonal projection to k-plane in Euclidean space is given in [8]. The orthogonal projection taking a point in Hn 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 [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 Hn or Sn.

Let mk+1 be the determinant of submatrix M(k+1,…,n+1) of M and let gk+1 be the determinant of submatrix G(k+1,…,n+1) of G. Suppose that mij and gij are the determinant of submatrices M1⋯k+1i1⋯k+1ji,j=k+2,…,n+1 and G1⋯k+1i1⋯k+1ji,j=k+2,…,n+1, respectively.

Lemma 1.
Let Δ be a hyperbolic or spherical n-simplex with the edge matrix M and Gram matrix G. Let Mii and Gii be ith minor of M and G, respectively. Then M-1=TGT and G-1=TMT, where T=Gii/ϵ|G|δij=Mii/ϵ|M|δij.

Proof.
It can be seen from [12].

Let M11, M12, M22 and G11, G12, G22 be (k+1)×(k+1), (k+1)×(n-k), (n-k)×(n-k) types submatrices of M and G, respectively. Suppose that M, G, and diagonal matrix T are partitioned as M11M12M12M22, G11G12G12G22, and T1100T22, respectively.

Concerning Lemma 1 along with Schur complement of a symmetric matrix, we have the following lemma.

Lemma 2.
Let SMii and SGii be Schur complements of the submatrices Mii and Gii. Then (7)Mii-1=TiiSGjjTii, Gii-1=TiiSMjjTii i≠j; i,j=1,2.

Proof.
It is obvious that M, G are symmetric and Mii, Gii are invertible. Since the inverse of Schur complement of M11 in M is the submatrix of M-1, we have(8)M-1=M11-1+M11-1M12SM11-1M21M11-1-M11-1M12SM11-1-SM11-1M21M11-1SM11-1.Similarly, for the Schur complement of M22, we obtain(9)M-1=SM22-1-SM22-1M12M22-1-M22-1M21SM22-1M22-1+M22-1M21SM22-1M12M22-1.Then we have (10)M-1 =SM22-1-M11-1M12SM11-1-M22-1M21SM22-1SM11-1,and, by the same way, we get (11)G-1=SG22-1-G11-1G12SG11-1-G22-1G21SG22-1SG11-1.Thus, we obtain the desired results using Lemma 1.

2. Orthogonal Projection to <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M150"><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:math></inline-formula>-Plane of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M151"><mml:mrow><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula> Based on a Hyperbolic <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M152"><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:math></inline-formula>-Simplex
If p1,p2,…,pn+1 are vertices of any hyperbolic n-simplex Δ, then {p1,p2,…,pn+1} is a basis of R1n+1. Let Wj be a subspace spanned by {p1,p2,…,p^j,…,pn+1}, and let ej be the unit outer normal to Wj, j=1,…,n+1. Hence {e1,e2,…,en+1} is another basis of R1n+1.

Let Wh be a k-plane which contains a k-face with vertices p1,p2,…,pk+1 of Δ. Then the set {p1,p2,…,pk+1} is a basis of the (k+1)-dimensional subspace of W in R1n+1. Since p1,p2,…,pk+1 are vertices of Δ, the subset {p1,p2,…,pk+1} of R1n+1 can be extended bases {p1,p2,…,pn+1} and {p1,…,pk+1,ek+2,…,en+1} of R1n+1. As a consequence, we see that {ek+2,…,en+1} is a basis of (n-k)-dimensional subspace W⊥.

Theorem 3.
Let p be a point and let Wh be a k-plane in Hn. Then the orthogonal projection σ(p) of p to Wh is given by (12)σp=p+∑s,t=k+2n+1MssMttmts〈p,et〉es|M|mk+1 ·1-∑s,t=k+2n+1MssMttmts〈p,et〉〈p,es〉|M|mk+1-1.

Proof.
For a point p∈Hn, by [10, Theorem 3.11], there is a point p˙∈W such that pp˙→∈W⊥. Therefore, we can write(13)pp˙→=∑s=k+2n+1λses.Then, we have -〈p,et〉=∑s=k+2n+1λs〈es,et〉, t=k+2,…,n+1.

Taking (14)G22=Gk+2⋯n+1k+2⋯n+1=〈es,et〉s,t=k+2,…,n+1,L=λk+2⋯λn+1,we obtain(15)L=-G22-1〈p,et〉.By Lemma 2 and (5) of [13], we see that (16)G22-1=MiiMjjmji-|M|mk+1i,j=k+2,…,n+1,and this implies that(17)λs=∑t=k+2n+1MssMttmts〈p,et〉|M|mk+1, s=k+2,…,n+1.Substituting (17) into (13), we obtain(18)p˙=p+∑s,t=k+2n+1MssMttmts〈p,et〉es|M|mk+1.By [10], there exists a unique σ(p)∈Wh such that σ(p)=cp˙. Since p˙ is the orthogonal projection of p to W, we have(19)c=1-∑s,t=k+2n+1MssMttmts〈p,et〉〈p,es〉|M|mk+1-1which completes the proof.

In case of the orthogonal projection to a hyperplane, we obtain mjj=|M| and mk+1=Mjj. 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: (20)σ(p)=p-p,ejej1+p,ej2, j=1,…,n+1,where ej is the unit normal of Wj in R1n+1. This result is also a generalization of Theorem 3 [3, 14].

Theorem 4.
Let p be a point and let Wh be a k-plane in Hn. Then, (21)coshξp,Wh =1-∑s,t=k+2n+1MssMttmtsp,etp,esMmk+1,where ξ(p,Wh) is the distance between p and Wh.

Proof.
Since 〈p,σ(p)〉=-coshξ(p,Wh), the result follows Theorem 3.

As an immediate consequence of Theorem 4, we obtain the following known result [10, Section 4].

Corollary 5.
Let p be a point and let Wjh be a hyperplane of Hn determined by ej. Then the distance ξ(p,Wjh) between p and Wjh is given by (22)coshξ(p,Wjh)=1+p,ej2.

By taking pj instead of p in (18) and using 〈pj,et〉=-|M|/Mttδjt, we obtain (23)p˙j=pj+∑s=k+2n+1Mss|M|mjsmk+1es,p˙j,p˙j=-1-mjjmk+1,where pj is a vertex of Δ. The proof of the following corollary is obvious from Theorem 3.

Corollary 6.
Let Δ be a hyperbolic simplex with vertices p1,…,pn+1. Then the perpendicular foot from pj to k-face Wh is given by (24)σpj=pj+∑s=k+2n+1(Mss/Mmjs/mk+1)es1+(mjj/mk+1), j=k+2,…,n+1,where p1,…,pk+1 are vertices of k-face Wh.

If we replace p by pj and use 〈pj,et〉=-|M|/Mttδjt, we see that (25)σpj,pj=-1+mjjmk+1.If we consider the last equation in the proof of Theorem 3, we see that coshξ(pj,Wh)=1+(mjj/mk+1); that result is a generalization of [11, Proposition 4] to the k-face Wh of a hyperbolic n-simplex. Since mjj/mk+1 is the diagonal jjth-entry of SM11=aij, the altitude from pj to k-face Wh with vertices p1,…,pk+1 is given by (26)coshξpj,Wh=1+ajj,where ξ(pj,Wh) is the distance between pj and k-face Wh.

By p˙j=pj+M/Mjjej, for (n-1)-face Wjh, we have the following corollary.

Corollary 7.
Let Δ be a hyperbolic simplex with vertices p1,…,pn+1. Then the perpendicular foot from pj to (n-1)-face Wjh is given by (27)σpj=pj+|M|/Mjjej1+(M/Mjj), j=1,…,n+1,where p1,…,p^j,…,pn+1 are vertices of Wjh.

Using Gjj=-(GMjj)/|M| for j=1,…,n+1, we obtain the following known result [11, Proposition 4].

Corollary 8.
Let Δ be a hyperbolic simplex with vertices p1,…,pn+1. Then the altitude ξ(pj,Wjh) from pj to (n-1)-face Wjh is given by (28)coshξ(pj,Wjh)=1+MMjj, j=1,…,n+1,where p1,…,p^j,…,pn+1 are vertices of Wjh.

3. Orthogonal Projections to a <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M290"><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:math></inline-formula>-Plane of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M291"><mml:mrow><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula> Based on a Spherical <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M292"><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:math></inline-formula>-Simplex
Let Δ be with vertices p1,…,pn+1. Then {p1,…,pn+1} is a basis of Rn+1. If Wj is the subspace spanned by {p1,…,p^j,…,pn+1}, then {e1,…,en+1} is another basis of Rn+1 where ej is the unit outer normal to Wj for j=1,…,n+1.

Let Ws be a k-plane which contains a k-face with vertices p1,p2,…,pk+1. Then the set {p1,p2,…,pk+1} is a basis of the (k+1)-dimensional subspace W in Rn+1. As a consequence, we have a basis {ek+2,…,en+1} of (n-k)-dimensional subspace W⊥.

Theorem 9.
Let p be a point and let Ws be a k-plane in Sn. Then the orthogonal projection σ(p) of p to Ws is given by (29)σp=p-∑s,t=k+2n+1MssMttmtsp,etEes|M|mk+1 ·1-∑s,t=k+2n+1MssMttmtsp,etEp,esE|M|mk+1-1.

Proof.
By [10, Theorem 3.11], for p∈Sn, there is a p˙∈W such that pp˙→∈W⊥. Therefore, we can write(30)pp˙→=∑s=k+2n+1λses.Then, we have(31)〈p,pp˙→〉E=∑s=k+2n+1λses,etE.Taking (32)G22=Gk+2⋯n+1k+2⋯n+1=es,etEs,t=k+2,…,n+1,L=λk+2⋯λn+1,we find(33)L=-G22-1p,etE.By Lemma 2 and (5) of [13], we see that (34)G22-1=MiiMjjmji|M|mk+1i,j=k+2,…,n+1.This implies that(35)λs=-∑t=k+2n+1MssMttmts〈p,et〉E|M|mk+1, s=k+2,…,n+1.Substituting (35) into (30), we obtain(36)p˙=p-∑s,t=k+2n+1MssMttmtsp,etEes|M|mk+1.By [10], there exists a unique σ(p)∈Ws such that σ(p)=cp˙. Since p˙ is the orthogonal projection of p to W, we have (37)c=1-∑s,t=k+2n+1MssMttmtsp,etEp,esEMmk+1-1which completes the proof.

By Theorem 9, we have (38)σp=p-p,ejEej1-p,ejE2,where ej is the unit normal of the Wj in Rn+1.

Theorem 10.
Let p be a point and let Ws be a k-plane in Sn. Then (39)cosθp,Ws =1-∑s,t=k+2n+1MssMttmtsp,etEp,esEMmk+1,where θ(p,Ws) is the distance between p and Ws.

By taking pj instead of p and using 〈pj,ej〉E=-|M|/Mjj in (36), we obtain (40)p˙j=pj+∑s=k+2n+1Mss|M|mjsmk+1es,p˙j,p˙jE=1-mjjmk+1, j=k+2,…,n+1,where pj is a vertex of Δ. Hence, we have the following corollary.

Corollary 11.
Let Δ be a spherical n-simplex with vertices p1,…,pn+1; then the perpendicular foot from pj to k-face Ws is given by (41)σpj=pj+∑s=k+2n+1Mss|M|mjsmk+1es ·1-mjjmk+1-1, j=k+2,…,n+1,where p1,…,pk+1 are vertices of Ws.

Let θ(pj,Ws) be the altitude from the vertex pj to the k-face Ws with vertices p1,…,pk+1 for j=k+2,…,n+1. Then θ(pj,Ws) is given by (42)cosθ(pj,Ws)=1-mjjmk+1.By equality (5) in [13], the jjth-entry of the Schur complement SM11=bij satisfies bjj=mjj/mk+1.

Let Wjs be the (n-1)-face with vertices p1,…,p^j,…,pn+1 of Δ. Then, we have (43)p˙j=pj+|M|Mjjej,pj,ejE=-|M|Mjj, j=1,…,n+1.

The proof of the following corollary is obtained by using the above equations.

Corollary 12.
Let Δ be a spherical simplex with vertices p1,…,pn+1. Then the perpendicular foot from pj to (n-1)-face Wjs is given by (44)σ(pj)=pj+|M|/Mjjej1-|M|/Mjj, j=1,…,n+1,where p1,…,p^j,…,pn+1 are vertices of Wjs.

Corollary 13.
Let Δ be a spherical simplex with vertices p1,…,pn+1. Then the altitude θ(pj,Wjs) from pj to (n-1)-face Wjs is given by (45)cosθ(pj,Wjs)=1-MMjj, j=1,…,n+1,where p1,…,p^j,…,pn+1 are vertices of Wjs.