1. COS 397 Computer Graphics Svetla Boytcheva AUBG, Spring 2013

2. Lecture 2Ray Tracing & Transformations

3. Outline • Points and Vectors • Matrices • Vector Operations • Ray Tracing • Homogenous coordinates • Affine Transformations • Translation • Rotation • Scaling • Projection • View Point • Coordinates of objects on the surface of other objects

4. Points and Vectors

5. Points and Vectors • Drawing systems of particles • Determining coordinates of other objects • Drawing objects

6. Points and Vectors • Points • (x,y,z) • Representation • Absolute coordinates (point) • Relative coordinate (radius-vector) • Distance (vector) • Direction (vector)

7. Points in 3D • 3D absolute coordinates X P(x,y,z) x y Y z Z

8. Radius-vector • Relative coordinate (radius-vector) (0,0,0) X x P(x,y,z) y Y z Z

9. Length and Magnitude • Calculating the length/magnitude • Requires reference point R |p| X R Y z  x  y Z

10. Direction • Determines the direction in 3D • This is not sufficient for complete orientation X |p| Y Z

11. Points and Vectors operations

12. Vectors representation • Vectors (points) • Vector defined by points X1and X2

13. Examples • Vector p(5,2,3) • Vectorq from point (3,0,8) to (0,2,7)

14. P2 y2 V y1 P1 x1 x2 Points and Vectors • Point • Vector: Pointposition and direction • Magnitude- Direction

15. Length • Vector p length • Unit vector • Vector with length=1

16. Examples • Vectorp(5,2,3) • Vectorp from (3,0,8) to (0,2,7)

17. Addition and Subtraction • Vector can be added or subtracted only to other vector

18. Examples • Addition of p(5,2,3) andq(-1,-1,1) • Subtraction of q(-1,-1,1) and r(4,1,4)

19. Product • Various product operations • Product with scalar (i.e. vector and number) • Scalar product of vectors • Vector product of vectors • Three of them are used in Computer Graphics for important operations

20. Product with scalar • Product with number • Geometrical meaning • Scaling (extending, shortening) vector • Preserves the direction for k>0 • Reverses direction when k<0 (the opposite direction)

21. Examples • Doubled vector p(5,2,3) • Reversed vector q(-1,-1,1)

22. Unit Vector • Unit vector • Geometrical meaning • Vector with length = 1 • Preserves the direction • Can be applied only over non-zero vectors

23. Examples • Unit vector of p(5,2,3) • Check

24. Scalar Product • Product of two vectors, the result is scalar (number) • Angle between two vectors • Geometrical/Graphical meaning • Checking for perpendicularity • Lightening of surface • Calculating the area

25. p.q calculating • Using unit axis vectors X p Y ey ex ez Z

26. Multiply Remember that Find

27. Examples (2,2,0).(0,1,0) • Lucky method • Lying in the same plane • Angle between = 45о • First class method • "First class" in terms of quality • but also because it may be applied by first grader student

28. Example • Check whether the vectors are perpendicular • p(4,0,1)and q(-2,3,8) • Solution

29. V2  V1 |V2|cos Scalar Product • Definition • For Cartesian Reference Frame • Properties • Commutative • Distributive Dot Product, Inner Product

30. Vector product • Vector product and the result is vector • Anglebetween two vectors • Geometrical meaning • Finding normal vectors • Parallelogram Area

31. V1 V2 V2 u  V1 Where vector points on? • The Result vector of Vector product is: • Perpendicular to the plane defined by to vector in this product • Lies in the plane where the direction of rotation from the first to the second vector is positive (i. e. counterclockwise)

32. How to remember? • Use right hand • Coordinate system PQRis right-hand Снимка: FreeDigitalPhotos.net

33. Unit Vectors X X + - + - Z Z Y Y ey ey ex ex ez ez + -

34. Calculation of pхq • Using unit axis vectors

35. After vector product of unit vectors Regroup

36. Final Result

37. V1 V2 V2 u  V1 Vector Product Cross Product, Outer Product • Definition • For Cartesian Reference Frame • Properties • AntiCommutative • Not Associative • Distributive

38. Scalar Product Vector Product Examples (x2,y2) V2  (x1,y1) V1 (x0,y0) Angle between Two Edges Normal Vector of the Plane

39. Determinant of Matrix • Definition • For a square matrix, combining the matrix elements to product a single number • 2  2 matrix • Determinant of nn Matrix A (n 2)

40. Inverse Matrix • Definition • Non-singular matrix • If and only if the determinant of the matrix is non-zero • 2  2 matrix • Properties

41. 2D Geometric Transformations

42. Geometric Transformation • Definition • Translation, Rotation, Scaling • Motivation – Why do we need geometric transformations in CG? • As a viewing aid • As a modeling tool • As an image manipulation tool

43. Example: 2D Geometric Transformation Modeling Coordinates World Coordinates

44. Example: 2D Scaling Modeling Coordinates Scale(0.3, 0.3) World Coordinates

45. Example: 2D Rotation Modeling Coordinates Scale(0.3, 0.3) Rotate(-90) World Coordinates

46. Example: 2D Translation Modeling Coordinates Scale(0.3, 0.3) Rotate(-90) Translate(5, 3) World Coordinates

47. Example: 2D Geometric Transformation Modeling Coordinates Again? World Coordinates

48. Example: 2D Geometric Transformation Modeling Coordinates Scale Translate Scale Rotate Translate World Coordinates

49. Basic 2D Transformations • Translation • Scale • Rotation • Shear

50. Basic 2D Transformations • Translation • Scale • Rotation • Shear Transformations can be combined (with simple algebra)