# Derive an atlas of Monge patches for a surface in 3D

Question: If I have an atlas of non-monge charts for a surface, how I can I derive an atlas of monge charts?

In differential geometry, a 'Monge patch' is a chart of the form $$f(u,v) = (u, v, h(u,v))$$. 

In simple terms, it's a parametric equation for a 3D surface patch described by only a height function.

This section from Solid Shape (Koenderink, MIT Press 1990 - p. 233) describes Monge patch representation of surfaces and motivates my interest in learning how to derive them.

The Monge patch representation lends itself particularly well for visualizations and for quick and dirty "back of the envelope" calculations. You should thoroughly familiarize yourself with it. In a local frame you can always use the Monge patch representation for a small neighborhood. Very often you can cover whole surfaces with just a few Monge patches.

An 'atlas' is a set of charts that cover a surface (a differentiable manifold) so what the author is saying is you can often produce an atlas for a surface where every chart is in Monge form.

Typically in the literature on differential geometry, charts for surfaces in $$\mathbb{R}^3$$ are not given in Monge form.

As a specific clarifying example here is a typical non-monge chart on a torus. 

\begin{align} x &= c + a*\cos(v)*\cos(u)\\ y &= c + a*\cos(v)*\sin(u)\\ z &= c + a*\sin(v) \end{align}

$$f(u,v) = (x,y,z)$$

How can I produce an atlas of Monge charts for the torus? That is, a list of parametric equations for surface patches that cover every part of a torus and are all in Monge form.

Specifically, I'd like to understand the process of working the problem in general for simple surfaces in $$\mathbb{R}^3$$, not just the list of equations for the torus.

Here is a visualization of one Monge chart for a torus and its domain. The parametric question for a surface patch like this is what I'm interested in. 

• To get Monge patches you need a cartesian equation for the surface, since you need to be able to write it locally as a graph over the $xy$-, $yz$-, or $xz$-plane. So use your parametric equations to deduce the cartesian equation of the torus. – Ted Shifrin Jan 26 '20 at 7:19
• 'use your parametric equations to deduce the cartesian equation of the torus' is essentially what I'm asking how to do. Do you have any guidance there? "cartesian equation" is a useful hint thank you! – Chandler Jan 26 '20 at 19:42
• You can use obvious trig identities to lead yourself to it. But remember that a surface of revolution has symmetry about (in your case) the $z$-axis, and this means that it's obtained by rotating a curve in the $yz$-plane ($g(y,z)=0$, say) about the $z$-axis. So you get $f(x,y,z)=0$ where $f(x,y,z)=g(?,z)$? – Ted Shifrin Jan 26 '20 at 19:55
• By the way, I think people are getting over-zealous with closing questions here. I voted to re-open. I didn't give a complete answer because you will learn more for thinking it through for yourself, but I realize that annoys some of the folks on MSE. – Ted Shifrin Jan 26 '20 at 20:00

Your torus is a function $$f : (u,v) \rightarrow (x,y,z)$$.

If you choose a xy-point $$(a,b)$$, you can make monge patch using the following: $$(a,b,h(a,b))$$.

To connect $$f$$ with the monge patch, you'll do: $$(x=a, y=b, z=h(a,b))$$.

To calculate $$(u,v)$$ out of chosen $$(a,b)$$, you can do inverse image of $$f$$: $$f^{-1}((x=a,y=b,z=h(a,b))) : (x,y,z) \rightarrow (u,v)$$ $$f^{-1}[B] = \{ x \in X | f(x) \in B \}$$

To calculate this inverse image, we need a function $$g : (x,y,z) \rightarrow 2$$, which can be given by: $$g(x,y,z) = ((x=a) \And (y=b))$$

Once we have $$g$$ function, we can compose it with $$f$$ to get: $$(g \circ f)(u,v) = g(f(u,v)) : (u,v) \rightarrow 2$$. The result we want comes from the equation: $$(g \circ f)(u,v) = {true}$$. This allows us to calculate $$(u,v)$$.

Now that we have $$(u,v)$$, we can use $$f$$ to obtain $$(x,y,z)$$. Then from that, we get $$z=h(a,b)$$ as required.

excersize to the reader: try the pattern with the actual torus formulas.