6. Exercises

Problem FF-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 FF-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 FF-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 FF-4. Explain in detail why the formula for constructing a parametric interpolating spline surface does give the desired interpolation.

Problem FF-5. (a) If $h(t,u)$ is the nonparametric relaxed bicubic spline surface with $n=m=2$ and with value 0 at $(0,0)$, $(0,1)$, $(0,2)$, $(1,0)$, $(1,1)$, $(1,2)$, $(2,0)$, $(2,1)$ and $h(2,2) = 1$, find $h(.5, .5)$. (You may use any earlier problems that deal with the case $n = 2$, $t = {\frac 1 2}$.)

(b) If $P(t,u)$ is the parametric relaxed bicubic spline surace with $n=m=2$ and $P(t,u) = (0, h(t,u), u)$ for $t,u$ among $0,1,2$, find $P(.5,.5)$. (Here $h$ is from part (a). If you handle each coordinate separately it should not be necessary to consider many basis functions.)

Problem FF-6. A piecewise bicubic spline surface (as made with a tensor basis of uniform relaxed interpolating splines) can be thought of as one patch over a rectangle, or it can be regarded as made up of many smaller patches glued together, with each one having a unit square as a domain. What kind of function is being used on each unit square?

Problem FF-7. (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 FF-8. 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, $P _ {11} = (-1,-1,1)$, $P _ {12} = (-1,1,1)$, $P _ {21} = (1,-1,1)$, $P _ {22} = (1,1,1)$. All the other control points are at the origin.

Problem FF-9. Consider the parametric Bézier surface $P(t,u)$ with $m = n = 2$ and control points $P _ {ij}$ given by $P _ {ij} = (i,j,0)$ except that $P _ {11} = (1,1,4)$. 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 FF-10. Write down an expression for a Lagrange surface with $m = n = 2$ for the data points $P _ {ij} = (i,j,0)$, with $t _ i = i$, $u _ j = j$, for $i = 0,1,2$ and $j = 0,1,2$, except that $P _ {11} = (1,1,4)$. 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 FF-11. 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$(t,u) = (1,0,0) R _ t ^ {x\rightarrow z} R _ u ^ {x\rightarrow y}$, 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 FF-12. (a) Using the method of Problem FF-, 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 FF-13. Using the idea of Problem FF-, 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 FF-14. Invent the concept of a quadratic blending of three curves and give an example. (Use Lagrange interpolation.)

Problem FF-15. Describe surfaces that have the form
P$(t,u) = (t,u,f( \sqrt{t ^ 2 + u ^ 2}))$.

Problem FF-16. Consider a Bézier tensor product surface. Explain how the isoparametric curves are actually Bézier curves in R$^ 3$. With what control points?

Problem FF-17. Suppose you want to interpolate a nonparametric relaxed uniform cubic spline surface $z = g(x,y)$ with $m = n = 2$ for data values $z _ {11} = 4$ and $z _ {ij} = 0$ otherwise, where $i = 0,1,2$, $j = 0,1,2$. Find $g(0,{\frac 1 2})$. (Use a tensor basis. The answer is simpler than you might anticipate.)