# Orthogonal Projections Based on Hyperbolic and  Sphericaln-Simplex

Authors: Murat Savas; Baki Karliga; Atakan T. Yakut
Journal: Advances in Mathematical Physics
 (2015)

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## Abstract

Orthogonal projection along a geodesic to the chosenk-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.


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## Body

## 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.LetR1n+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 calledspace-like; if it is positive semi-definite and degenerate, then W is calledlight-like; if W contains a time-like vector of R1n+1, then W is calledtime-like.S
1
n
=
{
x
∈
R
1
n
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1
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〈
x
,
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} is calledde Sittern-space. The n-dimensionalunit 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 spaceHn. Throughout this paper we considerhyperbolicn-space(3)Hn={x∈H0n∣x1>0}.Hence, each pair of points pi, pj in Hn satisfy 〈pi,pj〉<0. Thehyperbolic 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.LetRn+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-dimensionalunit spherical spaceSn 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 thatW 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.LetΔ 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 ofHn 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 ak-plane; its intersection with the k-plane is known asperpendicular foot on that k-plane in Hn or Sn. The length of a geodesic segment bounded by a point and its perpendicular foot is calledthe distance between that point andk-plane. The distance between a vertex and its any opposite k-face is called k-face altitude of an n-simplex.The orthogonal projection to2-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 chosenk-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.Letmk+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.
LetΔ 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].LetM11, 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 Lemma1 along with Schur complement of a symmetric matrix, we have the following lemma.Lemma 2.
LetSMii and SGii be Schur complements of the submatrices Mii and Gii. Then (7)Mii-1=TiiSGjjTii,Gii-1=TiiSMjjTiii≠j;i,j=1,2.Proof.
It is obvious thatM, 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 tok-Plane of Hn Based on a Hyperbolic n-Simplex

Ifp1,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.LetWh 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.
Letp 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 pointp∈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 obtainmjj=|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.
Letp be a point and let Wh be a k-plane in Hn. Then, (21)cosh⁡ξp,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)〉=-cosh⁡ξ(p,Wh), the result follows Theorem 3.As an immediate consequence of Theorem4, we obtain the following known result [10, Section 4].Corollary 5.
Letp 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)cosh⁡ξ(p,Wjh)=1+p,ej2.By takingpj 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.
LetΔ 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 replacep 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 cosh⁡ξ(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)cosh⁡ξpj,Wh=1+ajj,where ξ(pj,Wh) is the distance between pj and k-face Wh.Byp˙j=pj+M/Mjjej, for (n-1)-face Wjh, we have the following corollary.Corollary 7.
LetΔ 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.UsingGjj=-(GMjj)/|M| for j=1,…,n+1, we obtain the following known result [11, Proposition 4].Corollary 8.
LetΔ 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)cosh⁡ξ(pj,Wjh)=1+MMjj,j=1,…,n+1,where p1,…,p^j,…,pn+1 are vertices of Wjh.


## 3. Orthogonal Projections to ak-Plane of Sn Based on a Spherical n-Simplex

LetΔ 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.LetWs 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.
Letp 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 Theorem9, we have (38)σp=p-p,ejEej1-p,ejE2,where ej is the unit normal of the Wj in Rn+1.Theorem 10.
Letp be a point and let Ws be a k-plane in Sn. Then (39)cos⁡θp,Ws=1-∑s,t=k+2n+1‍MssMttmtsp,etEp,esEMmk+1,where θ(p,Ws) is the distance between p and Ws.By takingpj 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.
LetΔ 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.Letθ(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)cos⁡θ(pj,Ws)=1-mjjmk+1.By equality (5) in [13], the jjth-entry of the Schur complement SM11=bij satisfies bjj=mjj/mk+1.LetWjs 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.
LetΔ 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.
LetΔ 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)cos⁡θ(pj,Wjs)=1-MMjj,j=1,…,n+1,where p1,…,p^j,…,pn+1 are vertices of Wjs.



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*Source: 808250-2015-03-05.xml*