Are there any examples of non-linear functions whose contour plot is made up of ALL parallel lines?

Question

I know planes are made up of all parallel lines but what about functions in 3-space?

Answer 1

Good question! And great example Jason!

Really, any composition of a nonlinear 1D function with a linear function will result in a nonlinear multivariable function.

What is affected is the spacing between contours in the domain.

If the interval from $a$ to $b$ is inside of the range (set of values hit by the function) of the function $f(x,y)$ and we take an equally spaced partition $c_0=0<c_1<c_2<c_3<...<c_n=b$ of this interval.

For a linear function $f(x,y)=mx+ny$ (or more generally an affine function $f(x,y)=mx+ny+b$), the level curves for each of these $c_k$ values will be equally spaced lines. This gives constant RoC in each and every direction and is why the graph of the function of two variables is a plane

For a nonlinear function composition that I spoke about earlier, you will also get lines, however these lines will not be equally spaced.

For example, take the function $f(x,y)=\sin(mx+ny+b)e^{-(mx+ny+b)^2}$ (I included a Geogebra plot of this below)

This function is definitely not linear, but nevertheless, it has lines for its level curves. The main thing is that the spacing of the level curves is not even and this is where the non-linear behavior comes from. You have regions where the level curves are close together, i.e., the function is changing rapidly compared to other regions where the level curves are more evenly spaced where the function is changing more gradually.

The level curves plotted in this image are for the interval from $-.5$ to $.5$.

I hope this helps!

Answer 2

I may be misunderstanding your question, but any function that can be represented as a function $\mathbb{R}\rightarrow\mathbb{R}$ extruded along a vector in $\mathbb{R}^2$ would have a contour map made up of parallel lines.

The first example that comes to mind is $z(x,y)=x^2$.