I answered any pending questions about the previous lecture or the assignment.
I talked a bit about my background, my teaching philosophy and the number of
times I've taught this course. I talked about the limitations of a large class,
and how the only resource limitation we can't really resolve by throwing more
manpower at is the number of workstations available.
Vectors/Scalars
What is a scalar? (simple number)
What is a vector? (multi-dimensional scalar)
How many dimensions can vectors have? (no limit)
What are more dimensions we might associate with a vector, other than x,y,z? (time, temperature, density, permeability, etc)
How are vectors written? (Bold, with a little arrow on top, or a little upside-down 'v')
How does vector addition/subtration work? (Put the tail of one at the head of another)
How does vector and scalar multiplication work? (aP = (ax,ay,az,...))
What is the length of a vector? (|P| = sqrt(x*x + y*y + ...))
What is a normal vector? (unit length)
How do you normalize a vector? (divide P by |P|)
Can you normalize all vectors? (not when |P| = 0)
Vector multiplication : Dot Product
Is the dot product a scalar or vector? (scalar)
What are the two ways of computing the dot product? (P . Q = xp * xq + yp * yq + ...) and (|P| |Q| cos(theta))
How can we use them to calculate the angle between two vectors? (set them equal to each other and solve for theta)
Can that always be calculated? (nope, not when |P| or |Q| = 0)
Can that be used for two vectors in any dimension, or is it limited to 2- (and maybe 3-) dimensions? (it works in any dimension because P.Q can be calculated as can |P| and |Q|)
What does the dot product mean geometrically? (length of the projection of one vector, A, onto the line in the direction of the other (B) times the length of B)
What must the geometric relationship between two vectors be to make the dot product the greatest value it can have? The least? (parallel, anti-parallel)
What is another term for Dot Product? (Inner Product)
Vector multiplication : Cross Product
How does one calculate the cross product? (A x B = the determinant of the matrix:
i_hat j_hat k_hat
xa ya za
xb yb zb
where [ijk]_hat are unit vectors defined in the direction of x, y and z.)
Is the cross product a scalar or vector? (vector)
How does one easily calculate the magnitude of the cross product? (|A| |B| sin(theta))
How does one easily calculate the direction of the cross product? (Right hand rule)
How can one use this information to figure out the angle between two vectors (if the dot product weren't available)? (same way as in dot product above, solve for theta)
What does the cross product mean geometrically? (area of parallelogram)
What must the geometric relationship between two vectors be to make the cross product the greatest value it can have? The least? (right angles = max, parallel or anti-parallel = min (considering lengths of vector as the metric))
Matrices
What is a row/column vector? (a vector written out in matrix notation, i.e. [x y z])
What is a matrix? (a stacked set of row or column vectors, possibly representing coefficients in an equation)
What is a determinant? (The volume of a parallelepiped P in n-dimensional space, provided the edges of P come from the rows of A)
What is the transpose of a matrix? (The ith row of A becomes the ith column of AT - it flips across its NW-SE axis)
What is the inverse? (A-1 is the matrix that when multiplied with A, gives I)
Can all matrices be inverted? (Not if they are singular)
What is an orthogonal matrix? (QTQ = QQT = I, and QT = Q-1)
What does it mean for a matrix's rows or columns to be linearly independent? (none can be expressed as linear combinations of the others) Linearly dependent? (there exists one that can be expressed as a linear combination of the others)
The line/plane equation
What is the line equation the old high school way? (y = mx + b)
What's wrong with that represention? (vertical lines require another representation)
What is the non-parametric line equation that solves the problem? (ax + by = c))
What's the parametric line equation? (P(t) = P0 * (1-t) + P1 * t)
In how many dimensions is it? (A line is a one-dimensional quantity but it is embedded in whatever space the points that define it are in)
What does spanning the space mean w.r.t. choosing points for parametric line/plane equations? (For a line, the points must be unique, and for a plane, the points must not be collinear)
References
Computer Graphics Principles and Practice by Foley, vanDam, Feiner & Hughes in the Appendix: Mathematics for Computer Graphics
Computer Graphics by Hearn & Baker in the Appendix: Mathematics for Computer Graphics
