1. Math Fundamentals
Maths revisit

2. Overview Vectors Coordinate systems Matrices Homogenous coordinates
Lines
Planes

3. Vector Used to represent In graphics, usually 2D, 3D, 4D quantities
Points in space
E.g. locations of objects, vertices of a triangle mesh
Spatial directions
E.g. orientation of the camera, surface normals
In graphics, usually 2D, 3D, 4D quantities
Quantity with direction and magnitude
E.g. if we connect two points with a directed line segment
Directed line segment has magnitude (its length) and direction (its orientation)
No fixed position

4. Vectors Representation Vector operations Scalar multiplication
Can have their lengths altered by multiplying with real numbers

5. Vectors Vector operations (cont.) Vector addition and subtraction
Can combine using head-to-tail rule
Inverse/negate

6. Vectors
Can use a vector to move from one point to another (point-vector addition): P = Q + v
Any two points define a vector from one point to the second (point-point subtraction): v = P – Q
Unit vector
Vector of magnitude = 1
normalisation

7. Vectors Vectors operations (cont.) Dot product
E.g. [ax ay az ] • [bx by bz ] = axbx + ayby + azbz

8. Vectors Vectors operations (cont.) Vector cross product
3D vectors only
Cross product results in a vector
Length of resulting vector

9. Coordinate Systems
Need a frame of reference to relate points and objects to our physical world
E.g. where is a point?
Can’t answer without a reference system

10. Coordinate Systems Handedness of the system Determined by
Thumb: + x-axis
Index finger: + y-axis
Middle finger: + z-axis
OpenGL: right-handed system
DirectX: left-handed system

11. Matrices Transformations are central to 3D graphics Matrices
Almost every pipeline stage involves a change of coordinate system
Moving from one coordinate system to another requires the use of transformation matrices
Matrices
Rectangular grid of numbers arranged in rows and columns
Dimensions
rows x columns

12. Matrices Square matrices Same number of rows and columns
Diagonal elements
Elements where row and column index the same
Identity matrix
Diagonal elements = 1, all others 0
In some ways, what is 1 is for scalars
M-1 M = I
MI = IM = M

13. Matrices Transposition Flip the matrix diagonally
Multiplying a matrix with a scalar

14. Matrices
Multiplying two matrices
AB ≠ BA
AI = IA = A

15. Matrices
Multiplying two matrices (cont.)

16. Matrices
Multiplying two matrices (cont.)

17. Matrices
Multiplying two matrices (cont.)

18. Homogenous Coordinates
Homogenous coordinates are key to all computer graphics systems
Points are typically represented by (x,y) or (x,y,z)
Add a 3rd or 4th coordinate, w
All standard transformations (rotation, translation, scaling) can be implemented with matrix multiplication using 3 x 3 or 4 x 4 matrices
Hardware pipeline works with 4-dimensional representations

19. Lines and Rays A line extends infinitely in two directions
A line segment is a finite portion of a line that has two endpoints
A ray is half of a line that has an origin and extends infinitely in one direction
Sometimes defined as a directed line segment (finite length)

20. Lines and Rays
Parametric definition of a line segment with endpoints (x0, y0) and (x1, y1)
x(t) = x0 + t ∆x
y(t) = y0 + t ∆y
In vector notation, p(t) = p0 + td
Straightforward extension of r3D, add z(t)
The ray starts at point p0 : (x0, y0, z0). p0 contains information about the position of the ray
0 ≤ t ≤ 1
∆x = (x1 – x0)
∆y = (y1 – y0)

21. Plane
A plane is a flat surface (3D), has no thickness, extends infinitely
Equation of a plane ax + by + cz + d = 0
p = (x, y, z)
A 3D point on a plane
n = (a, b, c)
Normal vector, perpendicular to the plane
n.p + d = 0