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Stereohomology by Alston Chen, 2007

MEANING OF ELEMENTARY MATRICES IN PROJECTIVE

GEOMETRY AND ITS APPLICATIONS

How to represent some of the commonly used geometric transformations in

computer graphics with elementary matrices

Alston Zi-qiang CHEN

Department of Polymer Science, East China University of Science & Technology, Shanghai, 200237,P R China

stereohomology@126.com, stereohomology@gmail.com

Abstract:

By taking advantage of an extension of Desargues theorem, and the extended Desarguesian conﬁguration

thereof, the analytical deﬁnition for the concept of stereohomology geometric transformation in projective

geometry is proposed in this work. A series of such commonly used geometric transformations in computer

graphics as central projection, parallel projection, centrosymmetry, reﬂection and translation transformation,

which can be included in stereohomology, are thus analytically deﬁned in projective geometry. It has been

proved that, the transformation matrices of stereohomology are actually equivalent to the existing concept in

numerical analysis: elementary matrices. Consequently, the current work not only presents a new represen-

tation method for the geometric transformations mentioned here, but also depicts the meaning of elementary

matrices in projective geometry. Some of the concepts, speculations and rules are also proposed for stereoho-

mology’s applications in computer graphics, especially in 3D reconstructing objects from multiple views.

Keywords:

elementary matrix, homogeneous coordinates, Desargues theorem, perspective projection, 3D reconstruction,

projective geometry.

1 INTRODUCTION

As is well known that, such geometric transformations

as translation transformation, perspective projection

in computer graphics, can be represented as transfor-

mation matrices only through the homogeneous co-

ordinate representation. Since it is very convenient

for us to represent different geometric transformations

and their compound operations in matrices by this

method, homogeneous coordinate has been widely ac-

cepted in computer graphics for many years.

Yet it is surprising to see that presently there is

no rigorous deﬁnition for the geometric transforma-

tions represented by matrices based on homogeneous

coordinates. Though ideally, a geometric transforma-

tion, and therefore its transformation matrix, should

be deﬁned and identifed reference coordinate system

independently, the current representing method and

the deﬁnitions in computer graphics, at least for geo-

metric transformation matrices, are not reference co-

ordinate system independent. Actually, the current

deﬁnitions for these commonly used geometric trans-

formations and their transformation matrices, are not

perfect, and even not theoretically rigorous, which can

be seen from the following aspects:

First, by using homogeneous coordinates, we as-

sume that the geometric transformations are deﬁned

in projective space not in Euclidean space. As we

know, in most of the current computer graphics text-

books, the geometric transformations, for example,

central projection, and parallel projection, are actually

deﬁned in Euclidean space by using the concepts of

measure, distance directly or implicitly in Euclidean

geometry, which will be inapplicable in projective ge-

ometry.

Second, conventionally, the geometric transfor-

mation matrices can also be established for per-

spective projection, translation transformation and so

forth, based on homogeneous coordinates representa-

tion and specially selected reference coordinate sys-

tems. While mainly due to the limitations of the tradi-

tional deﬁnitions of either the geometric transforma-

tions or their transformation matrices, there is no sim-

ple or straightforward method for establishing trans-

formation matrices reference-coordinate-systeminde-

pendently.

Since most of the geometric transformations

can be represented by transformation matrices only

through homogeneous coordinates, an improved rep-

resentation method for geometric transformations bet-

ter be based on homogeneous coordinates and projec-

tive geometry.

Stereohomology by Alston Chen, 2007

MEANING OF ELEMENTARY MATRICES IN PROJECTIVE GEOMETRY AND ITS APPLICATIONS

If all the geometric transformation can be repre-

sented by square matrices, we have the following con-

clusion:

Theorem 0. The different transformation matrices in

different reference coordinate systems of the same ge-

ometric transformation are similar matrices to each

other.

Proof. Suppose a geometric transformation T

projective space, which transforms an arbitrary point

or hyperplane X into Y ; and the homogeneous coor-

dinates of X and Y in reference coordinate systems

(I) and (II) are (x), (x)

′

, (y), (y)

′

respectively; and

the coordinate transformation matrix from the refer-

ence coordinate system (I) to (II) is a nonsingular

matrix T , i.e., (x)

′

= T (x), (y)

′

= T (y); Suppose

the transformation matrices of T

in different refer-

ence coordinate systems (I) and (II) are A and B

respectively, i.e., (y) = A(x), (y)

′

= B(x)

′

;. Then

we have:

(y)

′

= T (y) = T A (x) = T A T

−1

(x)

′

(1)

Since X is arbitrary, then B = T A T

−1

, B and A

are similar to each other.

Theorem 0 indicates that, to deﬁne a geometric

transformation reference coordinate system indepen-

dently, we better take advantage of the inherent char-

acteristics of the geometric transformation matrix,

i.e., properties of the transformation matrix regard-

ing eigen value(s), eigenvector(s), and the geometric

meaning thereof, rather than the traditional method

which seems a little empiristic, e.g., the zero matrix

entries and their indexes in a matrix play a key role in

identifying a geometric transformation, which are not

intrinsic when reference coordinate system changes.

Unless otherwise speciﬁed, the current work will

be discussed only in the real ﬁeld: R, and a geomet-

ric transformations will be a point (not hyperplane)

transformation discussed in one reference coordinate

system. Therefore, T can be used to represent both a

transformation and the transformation matrix thereof.

Symbols like (s), (x),(x)

, (π) and so on, represent the

homogeneous coordinate column vectors for the point

S, X, X

, and hyperplane π and so on, respectively.

2 STEREOHOMOLOGY

2.1 Extension of Desargues theorem &

Desarguesian conﬁguration

Theorem 1 (Extended Desargues theorem

[1]

). In n-

dimensional projective space (2 6 n ∈ Z

), if the

homogeneous coordinate vectors of the n + 1 points

, X

, ..., X

, X

n+1

, are linearly independent, any

n homogeneous coordinate vectors of the set of n+1

points Y

, Y

, ..., Y

, Y

n+1

, are also linearly inde-

pendent, and there exists a ﬁxed point S, the homo-

geneous coordinate vector of which is linearly depen-

dent to that of any two of the corresponding points:

and Y

, ( i = 1, · · · , n+1 ). Then the rank of the

homogeneous coordinate vectors set of the C

n+1

in-

tersection points of line pairs X

and Y

, which

will here be deﬁned as S

def

== X

∩ Y

(i 6= j, i, j = 1,· · ·, n+1), should be n.

Proof. Since ∃ 0 6= λ

, λ

′

∈ R ( i = 1,· · · , n+1 ),

which make that the homogeneous coordinate vector

of S can be linearly expressed as:

(s) = λ

(x)

+ λ

′

(y)

= λ

(x)

+ λ

′

(y)

= λ

n+1

(x)

n+1

+ λ

′

n+1

(y)

n+1

(2)

(s), (x)

, and (y)

, are the homogeneous coordinate

vectors of S, X

, Y

respectively , 0 6= λ

, λ

′

∈ R

(∀ i = 1,· · · ,n+1). Since all the vectors are homo-

geneous coordinate vectors, the linear expression of

(s) in Eq.(2) can thus be simpliﬁed as:

(s) = (x)

+ (y)

= (x)

+ (y)

= (x)

n+1

+ (y)

n+1

(3)

For n=2, the current statement is equivalent to the

conventional Desargues theorem in projective geom-

etry.

[2]

When n > 3 , since C

n+1

> n + 1, in the

n+1

intersection points, there are n+1 different in-

tersection points of the C

n+1

which can be expressed

as:











(s)

1,2

=(x)

−(x)

= (y)

− (y)

(s)

2,3

=(x)

−(x)

= (y)

− (y)

(s)

k,k+1

=(x)

−(x)

k+1

= (y)

k+1

− (y)

(s)

n,n+1

=(x)

−(x)

n+1

= (y)

n+1

− (y

)

(s)

n+1,1

=(x)

n+1

−(x)

= (y)

− (y)

n+1

{ (4)

Summing all the n + 1 different homogeneous

coordinate vectors together, it will be straightfor-

ward to ﬁnd that the n + 1 different vectors are

linearly dependent. So the rank of the vector set

Stereohomology by Alston Chen, 2007

MEANING OF ELEMENTARY MATRICES IN PROJECTIVE GEOMETRY AND ITS APPLICATIONS

is less than n + 1. Since the rank of the matrix

[(x)

(x)

· · · (x)

(x)

n+1

] is n+1, which is a non-

singular matrix, and after column elementary transfor-

mations, which can be transformed into another non-

singular matrix:

[(x)

−(x)

, (x)

−(x)

, · · · , (x)

−(x)

n+1

, (x)

n+1

]

then vectors

(x)

−(x)

, (x)

−(x)

, · · · , (x)

−(x)

n+1

are linearly independent, the rank of which is n. It

can also be proved that, ∀ n > 2, the homogeneous

coordinate vectors of other intersection points can all

be linearly expressed by the vectors in

(x)

− (x)

, (x)

− (x)

, · · · , (x)

− (x)

n+1

Then the rank of all the C

n+1

intersection point

vector set should be n.

Similar to the Desargues theorem and its inverse,

the inverse of Theorem 1 is true either.

Deﬁnition 1 (Extended Desarguesian conﬁguration).

If a set of 2n+3 points, X

, X

, · · · , X

, X

n+1

, S,

, Y

, · · · , Y

, Y

n+1

, meet the constraints in both

Theorem 1 and its inverse, then the conﬁguration that

consists of all these points is an Extended Desargue-

sian conﬁguration, denoted as

· · · X

n+1

-S-Y

· · · Y

n+1

(5)

2.2 Deﬁnition of Stereohomology

Deﬁnition 2 (Generalized projective transformation).

If the rank of the n + 1 dimensional coefﬁcient ma-

trix (t

i,j

) of the following transformation in n dimen-

sional projective space is equal to or greater than n,

then the transformation in projective space is called a

generalized projective transformation ( 0 6= ρ ∈ R):











ρx

′

= t

1,1

+ t

1,2

+ · · · + t

1,n+1

n+1

ρx

′

= t

2,1

+ t

2,2

+ · · · + t

2,n+1

n+1

ρx

′

n+1

n+1,1

n+1,2

+ · · · +t

n+1,n+1

n+1

(6)

Deﬁnition 3 (invariant point). In projective space, if

the homogeneous coordinate vector of a point is an

eigenvector of a generalized projective transformation

T , then the point is called an invariant point of T .

Deﬁnition 4 (General Invariant hyperplane). Suppose

X is any point in hyperplane π, if its corresponding

point X

′

through a generalized projective transforma-

tion matrix T , is also in π, then π is a general invari-

ant hyperplane of T .

Deﬁnition 5 (Invariant hyperplane). Suppose X is

any point in the general invariant hyperplane π of T ,

if X’s corresponding point X

′

coincides with X, then

π is an invariant hyperplane of T .

Deﬁnition 6 (null vector). If a generalized projective

transformation T is singular, then there exists a point,

denoted as X, of which the homogeneous coordinate

vector, (x), will be transformed into (0, · · · , 0)

⊤

, to

which there is no corresponding geometric point in

projective space, such a column homogeneous coor-

dinate (x) is called a null vector for T .

Deﬁnition 7 (linear independence). In projective

space, if the homogeneous coordinate vectors of a

series different points, or of a series of hyperplanes,

are linear independent, then these points, or these hy-

perplanes are linearly independent. If the rank of the

vector set is less than or especially, equal to 2, these

points or these hyperplanes are collinear.

Lemma 1 (existence of “stereohomology center”). In

n-dimensional projective space, if a generalized pro-

jective transformation T has an invariant hyperplane

π, then there exists a unique ﬁxed point, which is

collinear with any point and its image point through

the transformation T ; and when T is nonsingular,

the ﬁxed point is an invariant point of T ; when T is

singular, the ﬁxed point is the null vector of T .

Proof. Denote the generalized projective transforma-

tion matrix as T , and its invariant hyperplane as π.

Suppose the homogeneous coordinate vector of any

point X (denote its corresponding through T point

as Y ) can be linearly expressed by that of an arbitrary

invariant point H in π(the homogeneous coordinate

(h): ρ

· (h)=T (h), 0 6= ρ

∈ R is the eigenvalue for

all the points in π ) and that of a ﬁxed point C ( the

corresponding homogeneous coordinate: (c)) (Here,

H ∈ π, while ﬁxed point C /∈ π ) as:

(x) = (h) + ω · (c) ω ∈ R (7)

therefore, Y ’s homogeneous coordinate can be:

(y) =T · (x) = ρ

· (h) + ω · T · (c)

∈ R, and ω ∈ R (8)

From “Eq.(8) − Eq.(7) × ρ

”, we have:

(y) − ρ

· (x) = ω · (T · (c) − ρ

· (c))

which indicates that there exist a ﬁxed point (denoted

as S, the uniqueness of which can be proved through

proof by contradiction), the homogeneous coordinate

of which can be expressed as

(s) = T · (c) − ρ

· (c),

and which is collinear to X and the corresponding

point Y .

Stereohomology by Alston Chen, 2007

MEANING OF ELEMENTARY MATRICES IN PROJECTIVE GEOMETRY AND ITS APPLICATIONS

Since any line through S is transformed into itself,

then S is an invariant point when T is nonsingular or

S is the null vector when T is singular(s /∈ π).

Deﬁnition 8 (Stereohomology). In n-dimensional

projective space, a generalized projective transforma-

tion is called a stereohomology, if (1) the coefﬁcient

matrix thereof has a stereohomology hyperplane (de-

noted as π ), the homogeneous coordinate vector of

any point on which is an eigenvector of the coefﬁ-

cient matrix, and the rank of the vector set with all

these eigenvectors is n; and (2) there exists a ﬁxed

point ( called stereohomology center, denoted as S ),

the homogeneous coordinate vector is linearly depen-

dent to those of any point in the n-dimensional projec-

tive space, and its corresponding points transformed

by the generalized projective transformation.

According to Lemma 1, Deﬁnition 8 can actually

be simpliﬁed as: a generalized projective transforma-

tion with an invariant hyperplane.

Deﬁnition 9. (Elementary homology) An elementary

homology geometric transformation is a stereohomol-

ogy, of which the stereohomology center is not on the

stereohomology hyperplane, i.e., the inner product of

the homogeneous coordinate vectors of the stereoho-

mology center and the stereohomology hyperplane, is

not equal to zero.

Deﬁnition 10. (Elementary perspective) An elemen-

tary perspective is a stereohomology, of which the

stereohomology center is on the stereohomology hy-

perplane, i.e., the inner product of the homogeneous

coordinate vectors of the stereohomology center and

the stereohomology hyperplane, is zero.

Lemma 2 (Existence & uniqueness theorem). For the

Extended Desarguesian Conﬁguration (Eq.(5) in Def-

inition 1), there exists a unique generalized projec-

tive transformation, denoted as T , which transforms

, X

,· · · X

, X

n+1

and S into Y

, Y

,· · · Y

n+1

and S(or null) respectively.

Proof. The proof is similar to that of the fundamen-

tal theorem for projective transformation in projective

geometry, and an analytical solution for T ’s transfor-

mation matrix can also be presented, though it is too

complicated in form for any direct application.

For instance, in three dimensional projective

space, suppose the extended Desarguesian conﬁgu-

ration is ABCD-S-A

′

, and the correspond-

ing homogeneous coordinate vectors of the points in

ABCD-S-A

′

are

A : (a

, a

) A

′

: (a

′

, a

′

, a

′

, a

′

)

B : (b

, b

) B

′

: (b

′

, b

′

, b

′

, b

′

)

C : (c

, c

) C

′

: (c

′

, c

′

, c

′

, c

′

)

D : (d

, d

) D

′

: (d

′

, d

′

, d

′

, d

′

)

S : (s

, s

)

respectively; denote the following determinants as:

∆



, ∆



∆



, ∆



∆

′



′



, ∆

′



′



∆

′



′



, ∆

′



′



then the ﬁnal form of the transformation matrix T

in 3-dimensional projective space, which transforms

A, B, C, D, into A

′

, B

′

, C

′

, D

′

, and S into S

(or null) in the extended Desarguesian conﬁguration

(Eq. (5) in Deﬁnition 1), respectively, will be obtained

in the following form, which can easily be extended

to n-dimensional projective space (∀ 0 6= k ∈ R):

= k ×







′

∆

′

∆

′

∆

′

∆

′

∆

′

∆

′

∆

′

∆

′

∆

′

∆

′

∆

′

∆

′

∆

′

∆

′

∆

′

∆

′













∆







−1

(9)

Stereohomology by Alston Chen, 2007

MEANING OF ELEMENTARY MATRICES IN PROJECTIVE GEOMETRY AND ITS APPLICATIONS

Certainly, to obtain such a unique geometric trans-

formation and its transformation matrix (Eq.(9)) from

the coordinate information of an Extended Desargue-

sian Conﬁguration rather than such geometric con-

straints as projection center and projection plane for a

central projection, which may be much simple and di-

rect for the transformation matrix determination, the

real objective is to ﬁnd the relationship between the

transformation matrix derived from the Extended De-

sarguesian Conﬁguration and stereohomology trans-

formation matrix, then ﬁnally ﬁnd a general and sim-

ple analytical form for stereohomology transforma-

tion matrix which is straightforward for application.

The following lemmas and theorems will help answer

this question.

Lemma 3 (Existence of an n-dimensional

Eigenspace). The obtained n+1 dimensional

transformation matrix of generalized projective

transformation T in Lemma 2, has an eigenvalue

with geometric multiplicity of n, i.e., there exists

an n-dimensional Eigenspace of the transformation

matrix, and a hyperplance of the transformation T .

Proof. Here only gives the proof when the transfor-

mation matrix is nonsingular, i.e., not only all the ho-

mogeneous coordinate vectors (x)

are linearly inde-

pendent, but all the vectors (y)

are linearly indepen-

dent. And the symbol T here will be used to repre-

sent both the geometric transformation and the trans-

formation matrix.

To prove the current statement, we need to con-

struct and use the properties of the minimal polyno-

mial of the n+1-dimensinal transformation matrix T .

First, since the homogeneous coordinate vector of

S can be linearly expressed as Eq. (2), and since

all the n+1 homogeneous coordinate vectors of either

(x)

or (y)

( i = 1,· · · , n+1 ) are linearly independent,

∃ 0 6= µ

∈ R (i = 1,· · · , n+1 ), which make (s) can

be linearly expressed as:

(s) =

n+1

i=1

(x)

(10)

Since (x)

and (y)

are the corresponding points

through the geometric transformation T (∀ i =

1, · · · , n + 1), i.e., there exist a series of 0 6= ρ

∈

R (i = 1, . . .,n+1), which satisfy:

(y)

= T (x)

(i = 1, · · · , n+1) (11)

and ρ

(s) = T (s) (12)

combine all the above results in Eq.(2, 10, 11, 12)

together, we will obtain the linear expression of (s)

by (y)

in two different forms:

n+1

i=1

− 1

(s) =

n+1

i=1

′

) (13)

and

(s) =

n+1

i=1

(x)

n+1

i=1

) (14)

Since the linear expression of (s) by (y)

should

be unique once the homogeneous coordinate vectors

are selected, comparing the corresponding factors be-

fore each (y)

, we can obtain:

′

= const. (∀ i = 1, · · · , n+1) (15)

Since according to Eq. (12)

(s) =

T · (s) (16)

and according to Eq.(2)

(s) = λ

(x)

+ λ

′

)

= λ

(x)

′

T · (x)

′

· (x)

(17)

∀ i = 1, · · · , n+1

Then from Eq.(17) we can obtain:

(T − ρ



T +

′



(x)

= 0 (18)

∀ i = 1, · · · , n+1

Since ρ

is an eigenvalue of the transformation

matrix T , and T 6= I,

(t − ρ

)



t +

′



(19)

is the minimal characteristic polynomial of the trans-

formation matrix T . Consequently,

−

′

= const. (20)

is also an eigenvalue of transformation matrix T , and

then it can be proved that the geometric multiplicity

thereof is n.

Then prove that all the homogeneous coordinate

vectors of the C

n+1

different intersection points, de-

noted as (s)

i,j

, (i 6= j, i,j=1,· · · ,n+1), are the associ-

ated eigenvectors of the eigenvalue Eq. (20).

According to Eq. (2), which can be rewritten as:

(s) = λ

(x)

+ λ

′

(y)

(i=1, · · · , n+1)

and Eq. (4), which can be rewritten as (i, j =1,· · · ,

n+1):

(s)

i,j

= λ

(x)

− λ

(x)

= λ

′

(y)

− λ

′

(y)

(21)

and

T · (s)

i,j

= T · (λ

(x)

− λ

(x)

)

= λ

(y)

− λ

(y)

(22)

Comparing the linear expression of (s)

i,j

in Eq. (21)

and T ·(s)

i,j

in Eq. (22) by (y)

and (y)

, considering

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