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Matrix Math

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) 3 by 6 matrix element 2,3 is (3)

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Matrix Math

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  1. Matrix Math ©Anthony Steed 1999

  2. Overview • To revise • Vectors • Matrices • New stuff • Homogenous co-ordinates • 3D transformations as matrices

  3. Vectors and Matrices • Matrix is an array of numbers with dimensions M (rows) by N (columns) • 3 by 6 matrix • element 2,3 is (3) • Vector can be considered a 1 x M matrix

  4. Identity matrices - I Diagonal Symmetric Diagonal matrices are (of course) symmetric Identity matrices are (of course) diagonal Types of Matrix

  5. Anthony Steed: EQ needs fixing Operation on Matrices • Addition • Done elementwise • Transpose • “Flip” (M by N becomes N by M)

  6. Operations on Matrices • Multiplication • Only possible to multiply of dimensions • x1 by y1 and x2 by y2 iff y1 = x2 • resulting matrix is x1 byy2 • e.g. Matrix A is 2 by 3 and Matrix by 3 by 4 • resulting matrix is 2 by 4 • Just because A x B is possible doesn’t mean B x A is possible!

  7. A is n by k , B is k by m C = A x B defined by BxA not necessarily equal to AxB Matrix Multiplication Order

  8. Example Multiplications

  9. Inverse • If A x B = I and B x A = I then A = B-1 and B = A-1

  10. Anthony Steed: EQ needs fixing 3D Transforms • In 3-space vectors are transformed by 3 by 3 matrices • Example?

  11. Scale • Scale uses a diagonal matrix • Scale by 2 along x and -2 along z

  12. Rotation • Rotation about z axis • Note z values remain the same whilst x and y change Y X

  13. About X About Y Scale (should look familiar) Rotation X, Y and Scale

  14. Homogenous Points • Add 1D, but constrain that to be equal to 1 (x,y,z,1) • Homogeneity means that any point in 3-space can be represented by an infinite variety of homogenous 4D points • (2 3 4 1) = (4 6 8 2) = (3 4.5 6 1.5) • Why? • 4D allows as to include 3D translation in matrix form

  15. 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

  16. Translation in Homogenous Form • Note that the homogenous component is preserved (* * * 1), and aside from the translation the matrix is I

  17. Putting it Together • R is rotation and scale components • T is translation component

  18. Order Matters • Composition order of transforms matters • Remember that basic vectors change so “direction” of translations changed

  19. Exercises • Show that rotation by/2 about X and then /2 about Y is equivalent to /2 about Y then /2 about X • Calculate the following matrix /2 about X then /2 about Y then /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 /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.

  20. Summary • Rotation, Scale, Translation • Composition of transforms • The homogenous form

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