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COMP 4290

Week 4 - Monday. COMP 4290. Last time. What did we talk about last time? Vectors. Questions?. Project 1. Geometric Interpretations. Interpretations. A vector can either be a point in space or an arrow (direction and distance)

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COMP 4290

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  1. Week 4 - Monday COMP 4290

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

  3. Questions?

  4. Project 1

  5. Geometric Interpretations

  6. Interpretations • A vector can either be a point in space or an arrow (direction and distance) • The norm of a vector is its distance from the origin (or the length of the arrow) • In R2 and R3, the dot product is: where  is the smallest angle between u and v

  7. Projections • We can find the orthogonal projection w of vector u onto vector v • Essentially, this means the part of u that's in v

  8. 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:

  9. Cross product rules • In addition • wu and wv • u, v,and w form a right-handed system

  10. 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

  11. Student Lecture: Matrices

  12. Matrices

  13. 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

  14. Identity matrix • The identity or unit matrixI is a square matrix whose diagonal is all ones with zeroes elsewhere

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

  16. 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

  17. 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

  18. 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

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

  20. 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

  21. Properties of matrix-matrix multiplication • Properties: • (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

  22. 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

  23. 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

  24. 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 subdeterminantdij to define the adjoint • The adjointA of an arbitrary sized matrix M is: • For a 3 x 3:

  25. 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:

  26. 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

  27. 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 Mviffu  v • If M and N are orthogonal, so is MN • An orthogonal matrix is equivalent to an orthonormal basis of vectors lined up together

  28. Upcoming

  29. Next time… • Geometric techniques • Any trigonometry that seems useful

  30. Reminders • CS Club tonight! • 4-6 p.m. in The Point 113 • Keep reading Appendix A • Read Appendix B • Keep working on Project 1, due next Friday

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