1. Vectors Tools for Graphics

2. Vector, Geometry and CG • To review vector arithmetic, and to relate vectors to objects of interest in graphics. • To relate geometric concepts to their algebraic representations. • To describe lines and planes parametrically. • To distinguish points and vectors properly. • To exploit the dot product in graphics topics. • To develop tools for working with objects in 3D space, including the cross product of two vectors.

3. Computer graphics objects • Objects to be drawn • Shape • position • orientation • fundamental mathematical discipline to aid graphics is • vector analysis • transformation

4. Why vector analysis

5. 2-D and 3-D coordinate systems

6. Vector Review The difference between two points is a vector: v = Q - P;

7. Vector and Point • Turning this around, we also say that a point Q is formed by displacing point P by vector v; we say that v offsets P to form Q. Algebraically, Q is then the sum: Q = P + v. • The sum of a point and a vector is a point: P + v = Q.

8. Vector representation • At this point we represent a vector through a list of its components: an n-dimensional vector is given by an n-tuple: • w = (w 1 , w 2 , . . . , w n )

9. Operation with Vectors • Add • Scale

10. Linear Combination of Vectors • a v + b w

11. Affine combination of vectors • A linear combination of vector is affine combination if • ex: 3 a + 2 b - 4 c

12. Convex combination of Vectors • Plus one more requirement • ai >= 0 I = 1…m • .3a+.7b • 1.8a -.8b • The set of coefficients a 1 , a 2 , . . . , a m is sometimes said to form a partition of unity, suggesting that a unit amount of material is partitioned into pieces.

13. The Magnitude of a vector Note that if w is the vector from point A to point B, then |w| will be the distance from A to B

14. Unit vector It is often useful to scale a vector so that the result has a length equal to one. This is called normalizing a vector, and the result is known as a unit vector. For example, we form the normalized version of a, denoted , by scaling it with the value 1/|a|: Ex. a = (3, -4),

15. The dot product

18. The Angle Between Two Vectors.

19. The Sign of b.cand Perpendicularity.

21. The 2D Perp Vector.

22. The perp dot product

23. Orthogonal Projections

24. Calculate K and M

25. The distance from C to The Line the distance from a point C to the line through A in the direction v is:

26. The Cross Product of Two Vectors The cross product (also called the vector product) of two vectors is another vector. It has many useful properties, but the one we use most often is that it is perpendicular to both of the given vectors. The cross product is defined only for three-dimensional vectors.

27. Properties

28. Normal

29. Finding the Normal to a Plane