1. Mathematics for Computer Graphics

2. Lecture Summary Matrices  Some fundamental operations Vectors  Some fundamental operations Geometric Primitives:  Points, Lines, Curves, Polygons

4. 2D Modeling Transformations Scale Rotate Translate Scale Translate x y World Coordinates Modeling Coordinates

5. 2D Modeling Transformations x y World Coordinates Modeling Coordinates Let’s look at this in detail…

6. 2D Modeling Transformations x y Modeling Coordinates Initial location at (0, 0) with x- and y-axes aligned

7. 2D Modeling Transformations x y Modeling Coordinates Scale.3,.3 Rotate -90 Translate 5, 3

8. 2D Modeling Transformations x y Modeling Coordinates Scale.3,.3 Rotate -90 Translate 5, 3

9. 2D Modeling Transformations x y Modeling Coordinates Scale.3,.3 Rotate -90 Translate 5, 3 World Coordinates

10. Matrices A matrix is a rectangular array of elements (numbers, expression, or function) A matrix with m rows and n columns is said to be an m-by-n matirx ( matrix), e.g In general, we can write an m-by-n matrix as

11. Matrices A matrix with a single row or a single column represent a vector Single row : 1-by-n matrix is a row vector Single column : n-by-1 matrix is a column vector A square matrix is a matrix has the same number of rows as columns In graphics, we frequently work with two-by-two, three-by-three, and four- by-four matrices The zero matrix The identity matrix A diagonal matrix

12. Scalar Multiplication To multiply a martix A by a scalar value s, we multiply each element a mn by the scalar Ex., find 3A = ?

13. Matrix Addition Two matrices A and B may be added together when these two matrices have the same number of rows and column  the same shape The sum is obtained by adding corresponding elements. Ex., find A+B = ?

14. Matrix Multiplication 1x11x33x1 2x2 3x33x1

15. Matrix Multiplication 1x11x33x1 2x2 3x33x1

16. Matrix Multiplication 1x11x33x1 2x2 3x33x1

17. Matrix Multiplication 1x11x33x1 2x2 3x33x1

18. Matrix Multiplication 1x11x33x1 2x2 3x33x1

19. Matrix Multiplication 1x11x33x1 2x2 3x33x1

20. Matrix Multiplication 1x11x33x1 2x2 3x33x1

21. Matrix Multiplication 1x11x33x1 2x2 3x33x1

22. Matrix Multiplication 1x11x33x1 2x2 3x33x1

23. Matrix Multiplication 1x11x33x1 2x2 3x33x1

24. Matrix Multiplication 1x11x33x1 2x2 3x33x1

25. Matrix Multiplication e.g.:

26. Matrix Multiplication e.g.:

27. Matrix Multiplication e.g.:

28. Matrix Multiplication e.g.:

29. Matrix Multiplication e.g.:

30. Matrix Multiplication e.g.:

31. Matrix Multiplication e.g.:

32. Matrix Multiplication e.g.:

33. Matrix Multiplication e.g.:

34. Matrix Multiplication e.g.:

35. Matrix Multiplication e.g.:

36. Matrix Multiplication e.g.:

37. Warning!!! but (AB)C = A(BC) A(B+C) = AB + AC (A+B)C = AC + BC (AB) T = B T A T A(sB) = sAB

38. Determinant of a Matrix

39. Matrix Inverse