# Understanding tensor product surfaces

Recently I bought a book about curves and surfaces (Curves and Surfaces for Computer Graphics by David Salomon), and I'm having trouble understanding the tensor product surfaces.

I think I understand the idea behind the basic curves:

• Linear interpolation
• Lagrange polynomials
• Hermite curves
• Bézier curves
• B-splines

And the basic surfaces:

• Ruled surfaces (two curves on opposite sides from u=0 to u=1, and in the w-direction each point is a linear combination of these two curves)
• Bilinear surfaces (like above, but now the two mentioned curves are linear as well)
• Translational surfaces (one curve in the u-direction and another in w, having the same point $\mathbf{P}_{00}$. Just translate them to get the surface)
• Coons patches (the four boundary curves are given, and the middle points are obtained by adding the linear interpolation in both u- and w-direction, and then subtracting the bilinear surface made from the corners)

Can I use some of the surfaces mentioned above to get a grasp of the tensor product surfaces? In the book it says, you start with two parametric curves:

\begin{align} \mathbf{Q}(u) = \sum_{i=1}^n f_i(u) \mathbf{Q}_i\\ \mathbf{R}(w) = \sum_{i=1}^n g_i(w) \mathbf{R}_i \end{align}

And then you can write the function

\begin{align} \mathbf{P}(u,w) = \sum_{i=1}^n \sum_{j=1}^m f_i(u) g_j(w) \mathbf{P}_{ij} \end{align}

I find it difficult to interpret this double summation (luckily the writer included matrix notation), but more importantly, where do these points $\mathbf{P}_{ij}$ come from? Like, how does a biquadratic surface work? And a bicubic surface? By the way, I'm talking about the simplest case, the interpolating polynomial surfaces -- as soon as I understand these, the others (e.g. approximating Bézier patches) won't be that difficult I guess.

I hope some of you can provide a basic explanation, or perhaps referral to other readable material (mind you, I'm a mechanical engineer, so the more explanation and examples, the better). Thanks!

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