1. Week 4 - Monday
CS361

2. Last time
What did we talk about last time?
Vectors

3. Questions?

4. Project 1

5. Geometric Interpretations

6. Cross product
The cross product of two vectors finds a vector that is orthogonal to both
For 3D vectors u and v in an orthonormal basis, the cross product w is:

7. Cross product rules 𝐰 = 𝐮×𝐯 = 𝐮 𝐯 sin θ 𝐮×𝐯=−𝐯×𝐮
𝑎𝐮+𝑏𝐯 ×𝐰=𝑎 𝐮×𝐰 +𝑏(𝐯×𝐰)
In addition
wu and wv
u, v, and w form a right-handed system

8. Things to remember Vectors can represents points or directions
The norm of a vector gives its length
The dot product of two vectors gives a measure of how much they point in the same direction
A scalar!
The cross product of two vectors gives a third vector, orthogonal to both of the original vectors

9. Drawing Primitives with Lighting in MonoGame

10. Student Lecture: Matrices

11. Matrices

12. A matrix
A matrix M is a set of p x q scalars with each element named mij, where 0 ≤ i ≤ p – 1 and 0 ≤ j ≤ q – 1
We display them as p rows and q columns

13. Identity matrix
The identity or unit matrix I is a square matrix whose diagonal is all ones with zeroes elsewhere

14. Operations
We will be interested in a number of operations on matrices, including:
Addition
Scalar multiplication
Transpose
Trace
Matrix-matrix multiplication
Determinant
Inverse

15. Matrix-matrix addition
Similar to vector addition, matrix-matrix addition gives as its result a new matrix made up of element by element additions
The two matrices must be the same size

16. Scalar-matrix multiplication
Similar to scalar-vector multiplication, scalar-matrix addition results in a matrix where each element is multiplied by the scalar
Properties
0M = 0
1M = M
a(bM) = (ab)M
a0 = 0
(a+b)M = aM + bM
a(M + N) = aM + aN

17. Transpose of a matrix
Transposing a matrix means exchanging its rows for columns
It has the effect of mirroring the matrix around its diagonal (or close to it, if not square)
Properties
(aM)T = aMT
(M + N)T = MT + NT
(MT)T = M
(MN)T = NTMT

18. Trace of a matrix
The trace of a square matrix is the sum of its diagonal elements
This is useful in defining quaternion conversions

19. Matrix-matrix multiplication
Multiplication MN is legal only if M is p x q and N is q x r
Each row of M and each column of N are combined with a dot product and put in the corresponding row and column element
𝐌𝐍=𝐓= 𝑡 𝑖𝑗 = 𝑘=0 𝑞−1 𝑚 𝑖𝑘 𝑛 𝑘𝑗

20. Properties of matrix-matrix multiplication
(LM)N = L(MN)
(L + M)N = LN + MN
MI = IM = M
Matrix-matrix multiplication is not commutative
We can treat a vector as an n x 1 matrix and do matrix-vector multiplication similarly

21. Determinant
The determinant is a measure of the "magnitude" of a square matrix
We'll focus on determinants for 2 x 2 and 3 x 3 matrices

22. Subdeterminant
The subdeterminant or cofactor dij of matrix M is the determinant of the (n – 1) x (n – 1) matrix formed when row i and column j are removed
Below is d02 for a 3 x 3 matrix M

23. Adjoint
The adjoint of a matrix is a form useful for transforming surface normals
We can also use the adjoint when finding the inverse of a matrix
We need the subdeterminant dij to define the adjoint
The adjoint A of an arbitrary sized matrix M is:
For a 3 x 3:

24. Multiplicative inverse of a matrix
For a square matrix M where |M| ≠ 0, there is a multiplicative inverse M-1 such that MM-1 = I
For implicit inverse, we only need to find v in the equation u = Mv, done as follows:
For cases up to 4 x 4, we can use the adjoint:

25. Notes about the inverse
For cases larger than 4 x 4, other methods are necessary:
Gaussian elimination
LU decomposition
Fortunately, we never need more than 4 x 4 in graphics
Properties of the inverse:
(M-1)T = (MT)-1
(MN)-1 = N-1M-1

26. Orthogonal matrices
A square matrix is orthogonal if and only if its transpose is its inverse
MMT = MTM = I
Lots of special things are true about an orthogonal matrix M
|M| = ± 1
M-1 = MT
MT is also orthogonal
||Mu|| = ||u||
Mu  Mv iff u  v
If M and N are orthogonal, so is MN
An orthogonal matrix is equivalent to an orthonormal basis of vectors lined up together

27. Homogeneous notation
Why do we often have vectors of 4 things or 4 x 4 matrices in graphics?
We have points (locations) and vectors (directions)
What's really confusing is that we represent them the same way (in what looks like a vector for both)
We need to translate points but translation isn't meaningful for vectors
A 3 x 3 matrix can rotate, scale, or shear, but it can't translate

28. How we do it We add an extra value to our vectors
It's a 0 if it’s a direction
It's a 1 if it's a point
Now we can do a rotation, scale, or shear with a matrix (with an extra row and column):

29. Translations
Then, we multiply by a translation matrix (which doesn't affect a direction vector)

30. Upcoming

31. Next time…
Geometric techniques
Any trigonometry that seems useful

32. Reminders Keep reading Appendix A Read Appendix B
Keep working on Project 1, due Friday