Elementary Matrix Representation of Some Commonly Used Geometric Transformations in Computer Graphics and Its Applications The Authors1 anonymous Abstract By using an extension of Desargues theorem, and the extended Desarguesian configuration thereof, the analytical definition for the 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 from this point of view. It has been proved that, the transformation matrices of stereohomologyare actually equivalent to the existing concept in numerical analysis: elementary matrices. Based on the meaning of elementary matrices in projective geometry thus obtained, a novel approach of 3D reconstructing objects from multiple views is proposed together with some of the concepts, principles and rules for its applications in computer graphics. Keywords: Elementary matrix; Homogeneous coordinates; Desargues theorem; Perspective projection; 3D reconstruction; Projective geometry. 1. Introduction for example, central projection, and parallel projection, are ac- tually defined in Euclidean space by using the concepts of mea- As is well known that, such geometric transformations as trans- sure or distance directly or implicitly which will be inapplicable lation transformation, perspective projection in computergraph- in projective space. ics, can be represented as square transformation matrices only Second, conventionally, the geometric transformation ma- through the homogeneous coordinate representation. trices can also be established for perspective projection, trans- Yet it is surprising to see that presently there is no rigor- lation transformation and so forth, based on homogeneous co- ous definition for the geometric transformations represented by ordinates representation in special reference coordinate system- square matrices based on homogeneous coordinates. Ideally,a s. While mainly due to the limitations of the traditional rep- geometric transformation and its transformation matrix, should resentations, there is actually no simple and straightforward be defined and identified reference coordinate system indepen- reference-coordinate-system independent method for transfor- dently, while the conventional representation method and the mation matrix establishing. definitions in computer graphics, at least for geometric trans- Since most of the geometric transformations can be repre- formation matrices, are not in such a way. sented by square transformation matrices only through homo- Actually, the conventionaldefinitions for the commonlyused geneous coordinates, an improved representation for geometric geometric transformations and their transformation matrices, transformations better be based on homogeneouscoordinates in are not perfect, or even not theoretically rigorous in the fol- projective space. lowing two aspects: For geometric transformations which can be represented by First, by using homogeneous coordinates, we assume that square matrices, we have the following conclusion: the geometric transformations are defined in projective space not in Euclidean space. As we know, in most of the curren- Theorem 0. The different transformation matrices in different t computer graphics textbooks, the geometric transformations, reference coordinate systems of the same geometric transfor- mation are similar matrices. T ∗Corresponding author. Tel.:+xxxxxx/Fax.:+xxxxxxx Proof. Suppose a geometric transformation 0 in projective s- Email address: anonymous (The Authors ) pace, which transforms an arbitrary point or hyperplane X into 1Corresponding author is anonymous. Y; and the homogeneous coordinates of X and Y in reference Preprint submitted to Computer Graphics Forum June 16, 2011 Elementary Matrix Representation of Some Commonly Used Geometric Transformations 2 coordinate systems (I) and (II) are (x), (x)′,(y), (y)′ respec- 2.2. Definition of stereohomology tively; and the coordinate transformation matrix from the ref- Definition 2 (Generalized projective transformation). If the rank erence coordinate system (I)to(II) is a nonsingular matrix T, of the n+1 dimensional coefficient matrix (ti, j) of the following = = i.e., (x)′ T(x), (y)′ T(y); Suppose the transformation matri- transformation in n dimensional projective space is equal to or ces of T0 in different reference coordinate systems (I) and(II) greater than n, then the transformation in projective space is = = are A and B respectively, i.e., (y) A(x), (y)′ B(x)′;. Then called a generalized projective transformation ( 0 , ρ R): we have: ∈ = + + + ρx1′ t1,1 x1 t1,2x2 t1,n+1 xn+1 = = = 1 ··· (y)′ T (y) T A (x) TAT− (x)′ (1) = + + + ρx2′ t2,1 x1 t2,2x2 t2,n+1 xn+1 ··· (3) Since X is arbitrary, then B = TAT 1, B and A are similar to − . .. each other. . = + + + ρxn′ +1 tn+1,1 x1 tn+1,2x2 tn+1,n+1 xn+1 Theorem 0 indicates that, in order to define a geometric ··· transformation reference coordinate system independently, we Definition 3 (invariant point). In projective space, if the homo- can take advantage of the inherentcharacteristics of the geomet- geneous coordinate vector of a point is an eigenvector of a gen- ric transformation matrix, i.e., properties of the transformation eralized projective transformation T , then the pointis called an matrix regarding eigen value(s), eigenvector(s), and the geo- invariant point of T . metric meanings thereof. Definition 4 (General Invariant hyperplane). Suppose X is any Unless otherwise specified, the current work will be dis- π R point in hyperplane , if its corresponding point X′ through a cussed only in the real field: , and a geometric transforma- generalized projective transformation matrix T , is also in π, tions will be a point (not hyperplane) transformation discussed then π is a general invariant hyperplane of T . in one reference coordinate system. Therefore, T can be used to represent both a transformation and the transformation ma- Definition 5 (Invariant hyperplane). Suppose X is any point in trix thereof. Symbols like (s), (x),(x)i,(π) and so on, represent the general invariant hyperplane π of T , if X’s corresponding the homogeneous coordinate column vectors for the point S , X, point X′ coincides with X, then π is an invariant hyperplane of Xi, and hyperplane π and so on, respectively. T . Definition 6 (null vector). If a generalized projective transfor- 2. Stereohomology mation T is singular, then there exists a point, denoted as X, of which the homogeneous coordinate vector, (x), will be trans- 2.1. Extension of Desargues theorem & Desarguesian configu- formed into (0, , 0)⊤, to which there is no corresponding ge- ration ometric point in··· projective space, such a column homogeneous Theorem 1 (Extended Desargues theorem [1, 2]). In n- dimen- coordinate (x) is called a null vector for T . sional projective space (2 6 n Z+), if the homogeneous coor- ∈ Lemma 1 (existence of “stereohomology center”). In n- dimen- dinate vectors of the n+1 points X1, X2, ..., Xn, Xn+1, are linearly independent, any n homogeneous coordinate vectors of the n+1 sional projective space, if a generalized projective transforma- tion T has an invariant hyperplane π, then there exists a unique points Y1, Y2, ..., Yn, Yn+1, are also linearly independent, and there exists a fixed point S, the homogeneous coordinate vector fixed point, which is collinear with any point and its image point T T of which is linearly dependent to those of any two of the corre- through ; and when is nonsingular, the fixed point is an in- variant point of T ; when T is singular, the fixed point is the sponding point pairs: Xi and Yi, ( i = 1, , n+1). Then the ··· 2 null vector of T . rank of the homogeneous coordinate vectors of the Cn+1 inter- section points of line pairs XiX j and YiY j, which will here be Proof. Denote the generalized projective transformation matrix def defined as S i j =S ji == XiX j YiY j (i , j, i, j = 1, , n + 1), as T , and its invariant hyperplane as π. Suppose the homoge- should be n. ∩ ··· neous coordinate vector of any point X (denote its correspond- ing point through T as Y) can be linearly expressed by that of Proof. See [1]. an arbitrary invariant point H π(ρ (h)=T (h), 0 , ρ R is ∈ H · H ∈ Similar to the Desargues theorem and its inverse, the inverse the eigenvalue for all the points in π ) and that of a fixed point < of Theorem 1 also is true. C π as: (x) = (h) + ω (c) ω R (4) + · ∈ Definition 1 (Extended Desarguesian configuration). If 2n therefore, Y’s homogeneous coordinate can be: 3 points, X , X , , X , X , S , Y , Y , , Y , Y , meet 1 2 ··· n n+1 1 2 ··· n n+1 the constraints in both Theorem 1 and its inverse, then the con- (y) =T (x) = ρ (h) + ω T (c) · H · · · figuration that consists of all these points is an Extended Desar- ρ R, and ω R (5) guesian configuration, denoted as H ∈ ∈ From “Eq.(5) Eq.(4) ρH ”, we have: X1 X2 Xn Xn+1-S -Y1Y2 Yn Yn+1 (2) − × ··· ··· (y) ρ (x) = ω (T (c) ρ (c)) − H · · · − H · Elementary Matrix Representation of Some Commonly Used Geometric Transformations 3 which indicates that there exist a unique fixed point, denoted as S below(the uniqueness can be proved through proof by contra- 1 diction): a1∆1 b1∆2 c1∆3 d1∆4 − = (s) T (c) ρH (c), · − · a2∆1 b2∆2 c2∆3 d2∆4 which is collinear to X and the corresponding Y. (6) Since any line through S is transformed into itself, then S is × a ∆ b ∆ c ∆ d ∆ 3 1 3 2 3 3 3 4 an invariant point when T is nonsingular or S is the null vector when T is singular ( S < π in this case). a4∆1 b4∆2 c4∆3 d4∆4 Definition 7 (Stereohomology). In n- dimensional projective The square transformation matrix of stereohomologyshould space, a generalized projective transformation is called a stere- be obtained via more simple and direct method rather than that ohomology, if (1) the coefficient matrix thereof has a stereoho- in (Eq.(6)) from the coordinate information of an Extended De- mology hyperplane (denoted as π ), the homogeneous coordi- sarguesian Configuration.