Homework 1 due 10/1/02
1. [5 points] Given a 3D vector [x y z], show that multiplication by any 3x3 matrix is an affine map. Also, show that translation by any vector [A B C] is an affine map.
2. [10 points] Prove that the LaGrange polynomials form a basis for polynomials of degree n.
3. [10 points] Assume you have a 2D parametric curve that passes through the following points at the following times:
Time Point
0 (0,0)
1 (1,3)
2 (5,2)
4 (1,1)
a) [2 points]. Describe the curve parametrically in the LaGrange polynomial basis. State the specific LaGrange polynomials.
b) [3 points]. Describe the curve parametrically in the Newton polynomial basis. Give the specific Newton polynomials.
c) [5 points]. Set up and solve a Vandermonde matrix to determine the parametric form in the power (monomial) basis. You may write or use a simple computer program to solve it, if you wish, but show the matrix setup and your work.
4. [5 points] Find the Bezier control points for the following 2D curve by converting to the Bernstein basis.
X(t) = 6t2+3t+7
Y(t) = 8t3-3t2+9t-1
5. [10 points] Changes in basis can be expressed by a matrix multiplication method. That is, if V = [a b c d]T is a vector in one basis, then it can be converted to a different basis by multiplication by some 4x4 matrix, M. That is, MV would give a vector in the other basis. Determine the following (show any work):
a) [3 points] The basis transformation matrix from the power basis to the Bernstein basis, for degree 5 polynomials.
b) [3 points] The matrix converting from the Bernstein basis to the power basis for degree 5 polynomials.
c) [4 points] The matrix converting from the cubic Hermite basis to the Bernstein basis.
6. [20 points] Assume you want to construct a curve that fits a set of data. The curve should start at point A and end at point C, passing through point B on the way (you can assume that youre at point A at time 0, B at time 2, C at time 3). Assume you also know the velocity (first derivative) at points A and C (i.e. A and C), and the acceleration (second derivative) and change in acceleration at point C (i.e. C and C).
a) [1 point] What degree curve would you need to use to interpolate the data?
b) [3 points] Write the general form such a curve would take, as an expression of coefficients and basis functions.
c) [6 points] Write the constraints that the basis functions should have to meet.
d) [10 points] Determine basis functions that meet the constraints expressed in (c).
Partial credit will be awarded for setting up the problem but not actually solving it. You may (and will probably want to) write or use a computer program to help solve for this information, but should state exactly what you wrote and/or used.