Sample midterm

(In an actual version, problems would be on separate pages with space for answers. Each problem counts 10 points.)

1. Give a map that is a rotation of $\mathbf{E}^ 2$ by $90 ^ \circ$ clockwise with center $\left[\begin{array}{c}2\\ 1\end{array}\right]$. Your answer may include one or more matrices, but with explicit numeric entries.

2. Sketch a cubic parametric curve $Q(t)$ such that $Q(0) = \left[\begin{array}{c}0\\ 0\end{array}\right]$, $Q'(0) = \left[\begin{array}{c}6\\ 0\end{array}\right]$, $Q(1) = \left[\begin{array}{c}0\\ 0\end{array}\right]$, $Q'(1) = \left[\begin{array}{c}0\\ 6\end{array}\right]$. Indicate axes and label some points on the axes to indicate the scale. Take care to plot $Q(\frac 12)$ exactly.

3. Suppose $f(t,u)$ is a biquadratic polynomial function that is 0 at all the points $(i,j)$ for $i = 0,1,2$ and $j = 0,1,2$, except that $f(1,1) = 1$. Find $f(\frac 12, \frac 12)$. (There is only one function that fits this description, so if you can think of one, that's it. A biquadratic polynomial is quadratic in each variable when the other is held fixed.)

4. For the parametric bilinear patch with data points $P _ {ij}$, $i=0,1$, $j = 0,1$, find a normal vector at the point where $t=0$, $u=0$.

5. Short-answer questions:

(a) Sketch the points that are convex combinations of the vertices of the standard triangle. (A convex combination is a barycentric linear combination with nonnegative coefficients.)

(b) Draw a Bézier control polygon with four control points at random and indicate graphically how to use the de Casteljau method to calculate $P(\frac 14)$.

(c) For the cubic Bézier curve $P(t)$ in R$^ 2$ with control points $\left[\begin{array}{c}0\\ 0\end{array}\right]$, $\left[\begin{array}{c}1\\ 10\end{array}\right]$, $\left[\begin{array}{c}2\\ 20\end{array}\right]$, $\left[\begin{array}{c}0\\ 0\end{array}\right]$, find $P''(0)$ explicitly.

(d) Invent an example of a ruled parametric surface (a surface that can be swept out by a moving straight line). Your answer should be some kind of explicit formula for a function $P(t,u)$.