What does this ratio of derivatives represent, geometrically?

Question

Suppose I have an arbitrary smooth real-valued function $F(x,y,z)$ with the property that all the first-order partials of $F$ are strictly positive everywhere.

The criterion of St-Robert says that $F$ can be written as a nomogram with three straight scales if and only if

$$\partial_{x}\partial_y \log(\mathcal{M}) = 0$$

where $\mathcal{M}(x,y) \equiv -\frac{\partial_y \widehat{z}(x,y)}{\partial_x \widehat{z}(x,y)}$ and $\widehat{z}$ is the function implicitly defined by solving $F(x,y,z)=0$ for $z$.

I am interested in understanding the geometric significance of $M$. Evidently if $F$ satisfies the criterion, then $M$ can be written as a ratio of single-variable functions.

But regardless of if $F$ satisfies the criterion, the form of $M$ is interesting. It almost has the form of a function defined by the implicit function theorem --- the partial derivative wrt $y$ of the function implicitly defined by solving $F(x,y,z)=0$ for $\widehat{x}(y,z)$—except the arguments to $M$ don't quite match.

I am looking for insight about what $M$ represents. For example: Does it measure some intuitive geometric quantity of $F$? Can you derive it from an application of the implicit function theorem or something similar? Is it related to other familiar quantities?

By analogy, if I were asking about the formula $-\partial_x F(x,y,\widehat{z}) / \partial_z F(x,y,\widehat{z})$, an insightful answer might be "This is a formula for the derivative of $\widehat{z}$, a function you get by applying the IFT to implicitly solve $F(x,y,z)=0$ in the neighborhood of some point. You can visualize it using an example like the sphere $F(x,y,z)=x^2+y^2+z^2-1$, in which case it measures this particular slope of this surface."

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Background on nomography: The function $F(x,y,z)=0$ can be written as a nomogram iff there exist three curves $f,g,h :\mathbb{R}\rightarrow \mathbb{R}^2$ such that $F(x,y,z) = 0$ iff the points $f(x)$, $g(y)$ and $h(z)$ are collinear.

The three curves $f,g,h$ are called scales. A scale is straight if its image is (a subset of) a line.

(A nomogram is a diagram that allows you to graphically solve the equation $F(x,y,z)=0$ for any one of the variables given the values of the other two, by plotting the values of the two variables on their respective curves and drawing a straight line between them to see where it intersects the third curve, reading off the corresponding value.)

Answer 1

Assume an x-y coordinate plane and a nomogram for the function w=w(u,v).

Let $x_u = x_u(u)$ define the x coordinate of the curve for the u scale line with similar definitions for the y coordinate and the other scale lines.

For a given index line, the coordinates lie on a straight line so we can write

$$ x_u y_v + x_v y_w + x_w y_u - x_u y_w - x_v y_u - x_w y_v = 0 $$

For a nomogram with straight vertical lines the x coordinates are constants.

Now take partial derivatives wrt $u$ & $v$

$$ x_v {\partial w \over \partial u} {dy_w \over dw} + x_w {dy_u \over du} - x_u {\partial w \over \partial u} {dy_w \over dw} - x_v {dy_u \over du} = 0 $$

$$ x_u {dy_v \over dv} + x_v {\partial w \over \partial v} {dy_w \over dw} - x_u {\partial w \over \partial v} {dy_w \over dw} - x_w {dy_v \over dv} = 0 $$

Rearrange:

$$ (x_u - x_v) {\partial w \over \partial u} {dy_w \over dw} = (x_w - x_v) {dy_u \over du} $$

$$ (x_u - x_v) {\partial w \over \partial v} {dy_w \over dw} = (x_u - x_w) {dy_v \over dv} $$

and divide: $$ { {\partial w \over \partial v} \over {\partial w \over \partial u} } = { (x_u - x_w) {dy_v \over dv} \over (x_w - x_v) {dy_u \over du} } = - \mathcal{M} $$

From here, there's probably several ways to interpret $\mathcal{M}$ geometrically, but the literal interpretation is that $\mathcal{M}$ is the ratio of 2 boxes, with widths equal to the distance between the scale lines and height equal to the rate of change of the opposite scale line variable.

Since the widths are constant here, it's simpler to just say that $\mathcal{M}$ is proportional the ratio of the rates of change of u & v along their respective scale lines. The constant of proportionality is the ratio of the distance between the scale lines and the w scale line.

A looser interpretation is the index line is like a see-saw pivoting at some point on the w scale line and with ends on the u & v scale lines. As the see-saw goes up and down the the changes in u & v on the nomogram must match the changes in u & v in the function, for constant w.

Note that if we take the log of both sides of the equation above the right hand side separates into individual factors. Taking the partial derivatives wrt $u$ & $v$ clears the right hand side and we are left with St-Roberts criterion.