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Matrix Math ©Anthony Steed 1999 Overview ■ To revise • Vectors • Matrices ■ New stuff • Homogenous co-ordinates • 3D transformations as matrices Vectors and Matrices ■ Matrix is an array of numbers with dimensions M (rows) by N (columns) • by matrix • element 2,3 is (3) ■ Vector can be considered a x M matrix  0 − − 2  1 −1  −5 0    • v = ( x y z) Types of Matrix ■ Identity matrices - I 1 0  ■ 0  Diagonal 1 0 0 0  0 1 0 0 0  0 −1 0 0 ■ 0 0 0 0 −  0 0 0  Symmetric a b c  b d e  c e f    • Diagonal matrices are (of course) symmetric • Identity matrices are (of course) diagonal Anthony AnthonySteed: Steed: EQ EQneeds needsfixing fixing Operation on Matrices ■ Addition • Done elementwise a b   p q a + p b + q  + =  c d   r s   c + r d + s ■ Transpose • “Flip” (M by N becomes N by M) T 1 9 1 6  8 =  7  3  3     Operations on Matrices ■ Multiplication • Only possible to multiply of dimensions – x1 by y1 and x2 by y2 iff y1 = x2 • resulting matrix is x1 by y2 – e.g Matrix A is by and Matrix by by • resulting matrix is by – Just because A x B is possible doesn’t mean B x A is possible! Matrix Multiplication Order ■ ■ A is n by k , B is k by m C = A x B defined by k ■ c = ∑a b BxA ijnot necessarily equal il ljto AxB l =1 *   * * * * *  *   *     * *    *   =              * * * * *          * * * * *    *      =             * * * * *       * * * * *      * =           Example Multiplications 1   1 − 1 =    − 1           −  0     −  0  =   − − 1        Inverse ■ If A x B = I and B x A = I then A = B-1 and B = A-1 Anthony AnthonySteed: Steed: EQ EQneeds needsfixing fixing 3D Transforms ■ In 3-space vectors are transformed by by matrices a b c  ( x y z )  d e f  = ( xa + yd + zg xb + ye + zf xc + yf + zi ) g h i  ■ Example? 10 Scale ■ ■ Scale uses a diagonal matrix  a 0  b  = ( xa yb zc ) ( ) x y z Scale by along x and -2 along z 0 c   2 0  ( 5)   = ( − 10)  0 − 2 11 Rotation ■ Rotation about z axis Y  cos(θ ) sin(θ )   - sin(θ ) cos(θ )    0 1 ■  Note z values remain the same whilst x and y change ( cos( θ ) x - sin ( θ ) y θ sin ( θ ) x + cos( θ ) y ) ( x y) X 12 Rotation X, Y and Scale ■ About X 1 0   cos(θ ) - sin(θ )    sin( θ ) cos( θ )   ■ About Y ■ Scale (should look familiar)  a 0  b 0 0 c    cos(θ ) - sin(θ )     sin(θ ) cos(θ )    13 Homogenous Points ■ ■ Add 1D, but constrain that to be equal to (x,y,z,1) Homogeneity means that any point in 3-space can be represented by an infinite variety of homogenous 4D points • (2 1) = (4 2) = (3 4.5 1.5) ■ Why? • 4D allows as to include 3D translation in matrix form 14 Homogenous Vectors ■ Vectors != Points ■ Remember points can not be added ■ If A and B are points A-B is a vector ■ Vectors have form (x y z 0) ■ Addition makes sense 15 Translation in Homogenous Form ■  0 0  0 ( x y z 1)  0  = ( x + a y + b z + c 1) Note that the homogenous is preserved (* * * 1), and aside from the translation  a component   b c 1 the matrix is I 16 Putting it Together ■ ■  R1 R  R R5  R R8 R is rotation and scale components T T2  T is translation component R3 R6 R9 T3 0 0 0  1 17 Order Matters ■ Composition order of transforms matters • Remember that basic vectors change so “direction” of translations changed 1  ( X Y Z 1)  00 0  1  ( X Y Z 1)  00 2  0 0 0 −1 0   0   0   0   0 0 −1 0 0 0 1  = ( X Z − Y 1)  0 0 2   0 0  = ( X + Z + − Y + 1) 0  0 1  = ( X + Y + Z + 1)  0  −1 0 0   −1 0  = ( X + Z + − Y − 1) 0  18 Exercises ■ Show that rotation byΠ/2 about X and then Π/2 about Y is equivalent to Π/2 about Y then Π/2 about X ■ Calculate the following matrix Π/2 about X then Π/2 about Y then Π/2 about Z (remember “then” means multiply on the right) What is a simpler form of this matrix? ■ Compose the following matrix translate along X, rotate Π/2 about Y, translate -2 along X Draw a figure with a few points (you will only need 2D) and then its translation under this transformation 19 Summary ■ Rotation, Scale, Translation ■ Composition of transforms ■ The homogenous form 20 [...]... that rotation byΠ/2 about X and then Π/2 about Y is equivalent to Π/2 about Y then Π/2 about X ■ Calculate the following matrix Π/2 about X then Π/2 about Y then Π/2 about Z (remember “then” means multiply on the right) What is a simpler form of this matrix? ■ Compose the following matrix translate 2 along X, rotate Π/2 about Y, translate -2 along X Draw a figure with a few points (you will only need... in matrix form 14 Homogenous Vectors ■ Vectors != Points ■ Remember points can not be added ■ If A and B are points A-B is a vector ■ Vectors have form (x y z 0) ■ Addition makes sense 15 Translation in Homogenous Form ■  1 0 0 0  0 1 0 0 ( x y z 1)  0 0 1 0  = ( x + a y + b z + c 1) Note that the homogenous is preserved (* * * 1), and aside from the translation  a component   b c 1 the matrix. ..Scale ■ ■ Scale uses a diagonal matrix  a 0 0  0 b 0  = ( xa yb zc ) ( ) x y z Scale by 2 along x and -2 along z 0 0 c   2 0 0  ( 3 4 5)  0 1 0  = ( 6 4 − 10)  0 0 − 2 11 Rotation ■ Rotation about z axis Y  cos(θ ) sin(θ
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