1. Under the Hood – How the Graphics Pipeline Works
Part II
Under the Hood – How the Graphics Pipeline Works

2. Points and Vectors
Lecture 18
Fri, Oct 5, 2007

3. 3D Coordinate Space
There are three mutually orthogonal axes: the x-axis, the y-axis, and the z-axis.
In the standard viewing position, the x- and y-axes look the same as in the usual 2D coordinate system.
The positive z-axis points towards the viewer; the negative z-axis points away.

4. 3D Coordinate Space y x z

5. 3D Coordinate Space y
Point x z

6. 3D Coordinate Space y
Point y x z z x

7. Points and Vectors A point specifies a position in space.
A vector specifies a direction and a magnitude.
Often, our vectors will be normalized to have a length of 1.
In those cases, we will use vectors for direction only.

8. 3D Coordinate Space y
Point y x z z x

9. 3D Coordinate Space y
Point
Vector y x z z x

10. Homogeneous Coordinates
Points and vectors are in homogeneous coordinates (x, y, z, w).
For points, w  0.
For vectors, w = 0.
For points, the true 3-dimensional coordinates are (x/w, y/w, z/w).
For this reason, we usually want w = 1.

11. Points and Vectors
Two points determine a vector. A B

12. Points and Vectors
Two points determine a vector. A B
v = B – A

13. Points and Vectors
The difference between two points A = (a1, a2, a3, 1) and B = (b1, b2, b3, 1) is the vector
B – A = (b1 – a1, b2 – a2, b3 – a3, 0).
It is not permissible to add points or to multiply points by a scalar.
Why not?

14. Vector Addition
The sum of two vectors is a vector, as defined by the parallogram rule. u v

15. Vector Addition
The sum of two vectors is a vector, as defined by the parallogram rule. u v

16. Vector Addition
The sum of two vectors is a vector, as defined by the parallogram rule. u
u + v v

17. Vector Addition
The sum of the vectors u = (u1, u2, v3, 0) and v = (v1, v2, v3, 0) is the vector
u + v = (u1 + v1, u2 + v2, u3 + v3, 0).

18. Vector Subtraction
The difference between two vectors is a vector. u v

19. Vector Subtraction The difference between two vectors is a vector.
u – v v

20. Vector Subtraction
The difference between the vectors u = (u1, u2, v3, 0) and v = (v1, v2, v3, 0) is the vector
u – v = (u1 – v1, u2 – v2, u3 – v3, 0).

21. Scalar Multiplication
The product of a vector and a scalar is a vector. v

22. Scalar Multiplication
The product of a vector and a scalar is a vector. v 4v

23. Scalar Multiplication
The product of a vector and a scalar is a vector. v -v

24. Scalar Multiplication
The product of the vector v = (v1, v2, v3, 0) and the scalar a is the vector
av = (av1, av2, av3, 0).

25. Point-Vector Addition
The sum of a point and vector is a point. A v

26. Point-Vector Addition
The sum of a point and vector is a point. A
B = A + v v

27. Point-Vector Addition
The sum of the point A = (a1, a2, a3, 1) and the vector v = (v1, v2, v3, 0) is the point
A + v = (a1 + v1, a2 + v2, a3 + v3, 1).

28. Point-Vector Subtraction
A point minus a vector is a point. A v

29. Point-Vector Subtraction
The sum of a point and vector is a vector. A v -v
B = A – v

30. Point-Vector Subtraction
The difference between the point A = (a1, a2, a3, 1) and the vector v = (v1, v2, v3, 0) is the point
A – v = (a1 – v1, a2 – v2, a3 – v3, 1).
It is the same as A + (-v).

31. Point-Vector Arithmetic
The following operations should be avoided.
Point + point: A + B
Scalar * point: aA
Vector – point: v – A
Suppose we want to find the midpoint of the line segment from A to B. How could we do it?

32. Affine Combinations of Points
An affine combination of points P1, P2, …, Pn, is a linear combination
a1P1 + a2P2 + … + anPn,
where a1, a2, …, an are real numbers and
a1 + a2 + … + an = 1.
This is a permissible calculation.
What will it produce?

33. Vector Magnitude
The magnitude of a vector is given by the distance formula.
Let v = (v1, v2, v3).
The magnitude of v, denoted |v|, is given by

34. Normalized Vectors To normalize a vector, we divide it by its length.
That is, for any vector v  0, the unit vector n with the same direction as v is

35. The Point3D and Vector3D Classes
point3d.h, point3d.cpp
vector3d.h, vector3d.cpp