1. 3D Geometry for Computer Graphics Class 1

2. General Office hour: Sunday 11:00 – 12:00 in Schreiber 002 (contact in advance) Webpage with the slides: http://www.cs.tau.ac.il/~sorkine/courses/cg/cg2005/ http://www.cs.tau.ac.il/~sorkine/courses/cg/cg2005/ E-mail: sorkine@tau.ac.ilsorkine@tau.ac.il

3. The plan today Basic linear algebra and Analytical geometry

4. Why??

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

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

7. Point + vector = point

8. vector + vector = vector Parallelogram rule

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

10. point + point: not defined!!

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

12. Inner (dot) product Defined for vectors:  L v w Projection of w onto v

13. Dot product in coordinates (2D) v w xvxv yvyv xwxw ywyw x y O

14. Perpendicular vectors (2D) v vv

15. Parametric equation of a line p0p0 v t > 0 t < 0 t = 0

16. Parametric equation of a ray 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 A 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. 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, then  +  +  =1, , ,  > 0 A B C P

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

33. Example of usage: warping

34. 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!