6. Exercises

Problem K-1. Consider the bilinear patch $z = f(t,u)$ with corner data $z _ {11} = 1$, $z _ {00} = z _ {01} = z _ {10} = 0$. Describe the horizontal cross-section (level curve) for $z = .5$, and the vertical cross-section for the plane given by $u = .5$.

Problem K-2. For the bilinear patch P$(t,u)$ with corner data $P _ {00}$, $P _ {01}$, $P _ {10}$, $P _ {11}$, find a formula for P$(\frac 1 2, \frac 1 2)$.

Problem K-3. For a linearly blended Coons surface $z = f(t,u)$ consider these corner heights and boundary curves: $z _ {00} = 0$, $z _ {10} = 1$, $z _ {01} = 0$, $z _ {11} = -1$, $z _ {t0} = t ^ 2$ $z _ {t1} = -t$, $z _ {0u} = \sin \pi u$, $z _ {1u} = 2 (\cos {\frac \pi 2} u) - 1$. (a) Is this data consistent? (b) If it is, find an equation for the surface. (The answer may be left as a product of matrices.)

Problem K-4. Explain in detail why the formula for constructing a parametric Lagrange surface does give the desired interpolation.

Problem K-5. (a) Explain why a Lagrange surface that has degree 1 in both $t$ and $u$ and has $t _ 0 = u _ 0 = 0$ and $t _ 1 = u _ 1 = 1$ is an instance of a bilinear patch (Example 10). (b) What about a relaxed cubic spline surface, still with $n=1$ on both coordinates?

Problem K-6. Here are control points for a Bézier surface with tensor-product basis, for $m=n=3$. Describe what the surface should look like, either in words or by an intuitive sketch. The ``middle four'' control points are the corners of a square in the $z = 1$ plane; specifically, $P _ {11} = \left[\begin{array}{r}-1\\ -1\\ 1\end{array}\right]$, $P _ {12} = \left[\begin{array}{r}-1\\ 1\\ 1\end{array}\right]$, $P _ {21} = \left[\begin{array}{r}1\\ -1\\ 1\end{array}\right]$, $P _ {22} = \left[\begin{array}{r}1\\ 1\\ 1\end{array}\right]$. All the other control points are at the origin.

Problem K-7. Consider the parametric Bézier surface $P(t,u)$ with $m = n = 2$ and control points $P _ {ij}$ given by $P _ {ij} = \left[\begin{array}{c}i\\ j\\ 0\end{array}\right]$ except that $P _ {11} = \left[\begin{array}{c}1\\ 1\\ 4\end{array}\right]$. Since $m = n = 2$, you will need the Bernstein polynomials $J _ {2,0} (t) = (1-t) ^ 2$, $J _ {2,1} (t) = 2 (1-t) t$, $J _ {2,2} (t) = t ^ 2$.

(a) Find the coordinate functions of $P(t,u)$. (Use tensor basis; simplify algebraically.)

(b) Sketch $P(t,u)$. (Just sketch the six isoparametric curves for $t$ and $u$ values $0, .5, 1$.)

Problem K-8. Write down an expression for a Lagrange surface with $m = n = 2$ for the data points $P _ {ij} = \left[\begin{array}{c}i\\ j\\ 0\end{array}\right]$, with $t _ i = i$, $u _ j = j$, for $i = 0,1,2$ and $j = 0,1,2$, except that $P _ {11} = \left[\begin{array}{c}1\\ 1\\ 4\end{array}\right]$. Then simplify the expression. (Method: Simplify one coordinate at a time. It can help to use any properties you know about linear combinations of Lagrange basis functions. Of course, if you can guess a function that fits the data, then by the uniqueness of the Lagrange solution that must be correct, so the algebra is avoided.)

Problem K-9. Recall that a partial derivative with respect to one variable is just the derivative when all other variables are held fixed. For a parametric surface P$(t,u)$, observe that the two partial derivatives ${\frac{\displaystyle \partial \mbox{\bf P}}{\displaystyle \partial t}}(t _ 0, u _ 0)$ and ${\frac{\displaystyle \partial \mbox{\bf P}}{\displaystyle \partial u}}(t _ 0, u _ 0)$ give velocity vectors of the two isoparametric curves at P$(t _ 0, u _ 0)$. These two vectors are therefore tangent to the surface, and their cross product is a normal (a perpendicular) to the surface.

(a) For the sphere parameterized as P$(t,u) = R _ u ^ {x\rightarrow y} R _ t ^ {x\rightarrow z} \left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$, or the equivalent written coordinatewise, find a formula for a unit normal at P$(t,u)$.

(b) Does your normal point inwards or outwards? (Either is OK.)

(c) What happens at the poles?

Problem K-10. (a) Using the method of Problem K-, for the Möbius strip described in the handout on parametric surfaces find a normal vector to the surface for $t = 0, u = 0$ and again for $t = 2 \pi, u = 0$, using the cross product of partial derivatives.

(b) Isn't it inconsistent that the two answers are different even though both parameter pairs give the same point on the surface?

Problem K-11. Using the idea of Problem K-, find a formula for a perpendicular vector at the $t = 0, u = 0$ corner of a Bézier surface patch of degree 2 with control points $P _ {ij}$, where $i,j = 0,1,2$.

Problem K-12. Invent the concept of a quadratic blending of three curves and give an example. (Use Lagrange interpolation.)

Problem K-13. Describe surfaces that have the form
P$(t,u) = \left[\begin{array}{c}t\\ u\\ f( \sqrt{t ^ 2 + u ^ 2})\end{array}\right]$.

Problem K-14. Consider a Bézier tensor product surface. Explain how the isoparametric curves are actually Bézier curves in R$^ 3$. With what control points?