1. Computer Graphics Recitation 1

2. General Office hour: Sunday 16:00 – 17:00 in Schreiber 002 Webpage with the slides: http://www.cs.tau.ac.il/~sorkine/courses/cg/ http://www.cs.tau.ac.il/~sorkine/courses/cg/ E-mail: sorkine@tau.ac.ilsorkine@tau.ac.il Short home assignments will be given from time to time

3. The plan today Basic linear algebra and Analytical geometry

4. Why?? We represent objects using mainly linear primitives:  points  lines, segments  planes, polygons Need to know how to compute distances, transformations…

5. Basic definitions Points specify location in space (or in the plane). Vectors have magnitude and direction (like velocity). Points  Vectors

6. Point + vector = point

7. vector + vector = vector Parallelogram rule

8. point - point = vector A B B – A A B A – B

9. point + point: not defined!!

10. Map points to vectors If we have a coordinate system with origin at point O We can define correspondence between points and vectors:

11. Inner (dot) product Defined for vectors:  L v w

12. Dot product in coordinates v w xvxv yvyv xwxw ywyw x y O

13. Perpendicular vectors v vv

14. Parametric equation of a line P0P0 v t > 0 t < 0 t = 0

15. Parametric equation of a ray P0P0 v

16. P0P0 v t > 0 t = 0

17. Distance between two points B A xBxB yByB xAxA yAyA x y O

18. Distance between point and line Find a point Q’ such that (Q  Q’)  v dist(Q, l) = ||Q  Q ’ || P0P0 v Q Q’ = P 0 +tv l

19. Easy geometric interpretation P0P0 v Q Q’ l L

20. Distance between point and line – also works in 3D! The parametric representation of the line is coordinates-independent v and P 0 and the checked point Q can be in 2D or in 3D or in any dimension…

21. Implicit equation of a line in 2D x y Ax+By+C > 0 Ax+By+C < 0 Ax+By+C = 0

22. Line-segment intersection x y Ax+By+C > 0 Ax+By+C < 0 Q 1 (x 1, y 1 ) Q 2 (x 2, y 2 )

23. Representation of a plane in 3D space The plane  is defined by a normal n and one point in the plane ( P 0 ). A point Q belongs to the plane  = 0 The normal n is perpendicular to all vectors in the plane n P0P0 Q 

24. Distance between point and plane Project the point onto the plane in the direction of the normal: dist(Q,  ) = ||Q’ – Q|| n P0P0 Q’  Q

25. Distance between point and plane n P0P0 Q’  Q

26. Implicit representation of planes in 3D (x, y, z) are coordinates of a point on the plane (A, B, C) are the coordinates of a normal vector to the plane Ax+By+Cz+D > 0 Ax+By+Cz+D < 0 Ax+By+Cz+D = 0

27. Distance between two lines in 3D P1P1 P2P2 u v d The distance is attained between two points Q 1 and Q 2 so that (Q 1 – Q 2 )  u and (Q 1 – Q 2 )  v Q1Q1 Q2Q2 l1l1 l2l2

28. Distance between two lines in 3D P1P1 P2P2 u v d Q1Q1 Q2Q2 l1l1 l2l2

29. P1P1 P2P2 u v d Q1Q1 Q2Q2 l1l1 l2l2

30. P1P1 P2P2 u v d Q1Q1 Q2Q2 l1l1 l2l2

31. Example of usage: warping

32. Barycentric coordinates (2D) Define a point’s position relatively to some fixed points. P =  A +  B +  C, where A, B, C are not on one line, and , ,   R. ( , ,  ) are called Barycentric coordinates of P with respect to A, B, C (unique!) If P is inside the triangle, , ,   [0, 1],  +  +  = 1 A B C P

33. Barycentric coordinates (2D) A B C P

34. Example of usage: warping Tagret A B C We take the barycentric coordinates , ,  of P’ with respect to A’, B’, C’. Color( P ) = Color(  A +  B +  C )

35. See you next time!