# Parametric Equation of a Surface

If $\left\{P_{k,l}^{0}\right\}_{k,l=0}^{n}$ is a set of $\left(n+1\right)^2$ three-dimensional points, and

$$P_{k,l}^{r+1}\left(t,s\right)=\left(P_{k+1,l+1}^{r}+P_{k+1,l}^{r}+P_{k,l+1}^{r}+P_{k,l}^{r}\right)ts,\tag{*}$$

then, is $P_{0,0}^{n}(t,s)$ the parametric equation of a surface?

I tried plotting this with a set of test points and obtained a spatial, straight line, when I was instead aiming to obtain a surface (since I am making use of the two parameters $t$ and $s$).

What is the problem with $(*)$? I am just interested in knowing whether this is a surface or not. Thanks in advance!

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The notation is very confusing (I assume this is some special case of something more general, or I am completely misunderstanding). I assume "three-dimensional point" means vector in $\mathbb{R}^3$. In which case, I don't understand the summation. Effectively, you are asking, "Given $v\in\mathbb{R}^3$, does $vts$ define a surface as $t$ and $s$ vary?" Well, you are only getting the linear span of $v$ no matter what $t$ and $s$ are, so you actually always get a straight line! (Unless that sum is $0$, in which case you only get a point). – Matt Nov 13 '12 at 22:42
@Matt, the way I read it, the sum itself depends on $t$ and $s$. The superscript $r$ is the key: the 0th generation is constant, the 1st linear in them, and the $r$th generation has degree $r$ of both. So in the end, you will probably have something curved. – MvG Nov 14 '12 at 0:07
Thank you, @Matt. Yes! You're completely right; this is a very special case of something way more general on which I'm currently working on. Indeed, the points are best thought of as vectors in $\mathbb{R}^3$, and now that you broke it down like that for me, I see how this is just either a point or a straight line regardless of the values of $t$ and $s$ (I could even merge them into one variable). I will have a talk with my adviser about this and post any clarifications promptly. – EsX_Raptor Nov 14 '12 at 0:11

As far as I read your notation, the formulka does not describe a surface. You work with two parameters, but only ever use the product of these. So the set of points $P_{0,0}^n(t,s)$ can be described using a single parameter, e.g. $P_{0,0}^n(u,1)$. You can see that by taking $u=ts$, there is a one-to-one correspondence between your two-parameter form any my one-parameter version. Therefore, the best you can hope for is a curved line, not a surface.