1. Matrices, Transformations and the 3D Pipeline Matthew Rusch Paul Keet

2. Matrices What is a MATRIX  “A concise and useful way of uniquely representing and working with linear transformations.” Notation Column or row major?  Above is row major

3. Matrix Operations - Scalar Scalar - Matrix  Addition  Subtraction  Multiplication  Division

4. Matrix Operations - Matrix Matrix – Matrix  Transpose  Determinant  Matrix Addition  Matrix Multiplication  Matrix Inverse

5. Transpose The object obtained by replacing all elements a ij with a ji Optimization note: Matrix elements down the diagonal can be left alone Properties of the transpose:

6. Determinant The determinant of a matrix A is usually denoted det(A) or |A| For a 2x2 matrix, the determinant is defined as: An NxN matrix can be expanded by minors to obtain:  Sign is given by (-1) i+j Matrices with a determinant of zero are called singular matrices

7. Matrix Multiplication Einstein summation Associative Not generally commutative  [A][B] != [B][A] Identity matrices and matrices that are diagonal and of the same dimension are exceptions

8. Matrix Inverse A matrix times its inverse yields an identity matrix Same as the transpose for orthogonal matrices Gauss-Jordan elimination Gaussian elimination LU Decomposition

9. Matrix Inverse 2x2 matrix 3x3 matrix

10. Matrix Transforms Scale Rotation  Around isolated axis  Around arbitrary axis Translation  Can be a vector addition  Homogenous coordinates can be used Shearing Affine  Preserves colinearity and relations of distances  Projection matrices are NOT affine

11. Transformations in the 3D Pipeline Model to World, or World Transform World to View, or View Transform  OpenGL combines this with the world transform and calls it the modelview matrix Lighting is done in world space in this case View to Projection, or Projection Transform How do we get from projection space to screen space?  Screen Transform Homogenezation, or divide by w Concatination, or combining transforms