3. Other constructions of parametric surfaces

Example 11. A family of curves made into a surface: Take a family of planar curves expressible by a formula that has a constant that determines the individual curve. If you put $z$ for the constant, you get a 3-dimensional surface that contains all the curves in the family as horizontal slices. For a parametric version, the family of curves should be made parametric with a parameter $t$; put $u$ for the constant and set $z = u$.

For example, the parabola $y = x ^ 2$ in $\mathbb{E}^ 2$ can be parameterized by $x = t, y = t ^ 2$. Then $x = t + c, y = t ^ 2 - c ^ 2$ is a parabola with vertex at $\left[\begin{array}{c}c\\ - c ^ 2\end{array}\right]$. As $c$ changes, you get a family of parabolas. Then for a surface you get

P$(t,u) = \left[\begin{array}{c}t + u\\ t ^ 2 - u ^ 2\\ u\end{array}\right]$, say for $-1 \leq t \leq 1, -1 \leq u \leq 1$.

Example 12. A linearly blended Coons patch: Suppose that we have four curves in E$^ 3$, parameterized over unit intervals and connected end-to-end, and we want to fill in a surface between them. We can think of these curves as being given by a function Q$(t,u)$ that is defined only on the boundary of the unit square in parameter space. In other words, Q$(t,u)$ is defined only when one of $t$ and $u$ is 0 or 1 and the other is between 0 and 1; one boundary curve is Q$(t,0)$, and other boundary curves are given similarly. We want to find a nice function P$(t,u)$ defined on the whole unit square and agreeing with Q$(t,u)$ on the boundary.

One idea is to take a linear blending of the two boundary curves Q$(t,0)$ and Q$(t,1)$. However, the resulting surface goes straight across between these two boundary curves and does not follow the other two boundary curves. Similarly, a linear blending of Q$(0,u)$ and Q$(1,u)$ does not work. Combining these two linear blendings by averaging them also does not work.

A solution turns out to be to add the two linear blendings and subtract off the bilinear patch determined by the corner values. Therefore, we let

P$(t,u) = \left ( (1-u)\mbox{\bf Q}(t,0) + u\mbox{\bf Q}(t,1) \right ) + \left ( (1-t)\mbox{\bf Q}(0,u) + t\mbox{\bf Q}(1,u) \right )$
$- \left ( (1-t)(1-u)\mbox{\bf Q}(0,0) + (1-t)u\mbox{\bf Q}(0,1) + t(1-u)\mbox{\bf Q}(1,0) + tu\mbox{\bf Q}(1,1) \right )$.

A compact version written using a matrix of point-valued functions is

P$(t,u) = \left[\begin{array}{ccc}(1-t)&t&1\end{array}\right]\left[\begin{array}... ...t,1)&0\end{array}\right]\left[\begin{array}{c}(1-u)\\ u\\ 1\end{array}\right]$.