How to read contour plot?

Question

I have a problem on reading a contour plot. I have tried to learn about it on some sites but i still can't understand it.

Please any help to fill in the blank. Thanks before :) !

Answer 1

(Throughout the following, one could claim that my neighboring contours are not from level sets whose level differs by $1$. It should be clear that a monotonic re-scaling of the output of the functions used can put the levels on the shown contours to land exactly on a contiguous set of integers, even meeting the requirement "values on neighboring level curves differ by $1$" excepting the annular example which I mention is excluded by the "differ by $1$" condition.)

Let's start with the graph of a function of two variables. (This function is $\sin x \cos y + \frac{1}{5}\sqrt{|x|}$, graphed on $[-5,5]\times[-5,5]$, but the details of the function are not important.)

By various means, we can find that the minimum value of the function shown in the graph is $-0.752\dots$ (at $(x,y) = (-1.488\dots,0)$) and the maximum such value is $1.435\dots$ (at $(-4.758\dots, 0)$). So the total range of values of the function shown in the graph is $2.187\dots$. So let's make a contour graph. We need to pick a set of values from the range of values of the function. Let's pick $$ \{ -0.7, -0.3, 0.1, 0.5, 0.9, 1.3 \} \text{.} $$ (There is much freedom in this choice in general -- we have chosen equally spaced values, but that is not required. The number of values chosen depends on how much precision you wish to show in your contour plot -- more values is more precise, but the result is more cluttered and fewer values give less precision, but are much easier to interpret quickly by eye.)

We plot the above graph with a plane at one of the selected heights (in order from $-0.7$ to $1.3$), the same graph viewed from directly above, and the intersection of the plane with the function's graph.

The right-most column is a collection of graphs of level sets. If we plot all the level sets together, we get a contour plot.

There are some ambiguities in this graph. (Topographical maps tend to avoid these ambiguities, either by indicating the height of each level set using a small numerical label on each connected component of the curve(s), or by indicating the direction of the gradient (or minus the gradient) on each level set.) First, we lose all the data about which curve goes with which level set. For instance, if we were to plot the negative of our function, we could get exactly this contour plot by picking the negatives of the heights that we actually picked. Consequently, we have no way of knowing which (homeomorphic to a) disk regions -- regions bounded by one curve which region contains no other curve -- are local minima or local maxima.

On the above contour plot, there are almost self-intersections along the $x$ axis. (A very easy way to get this is to contour plot $x^2 - y^2$ with the levels $\{-1,0,1\}$. The $0$ level set self intersects at the origin.

) If a curve self-intersects transversely (that is, not self-tangentially), there is an ambiguous stationary point at the intersection (there are two directions along which the directional derivative is zero -- along both intersecting segments of the level curve). If there are regions above this level set and regions below this level set meeting at the intersection point, the intersection is a saddle. If all the regions meeting at that point are above the level set for that contour, the level set is a local minimum at each of its points. (Draw an "8" on the $xy$-plane, set the height of every point to be the square of the minimal distance from that point to the points on the "8". The "8" is a level set for this function and it is a local minimum at each of its points.) If all the regions meeting at that point are below the level set for that contour, the level set is a local maximum at each of its points. (Take the previous example of squared distances from "8" and negate every height.)

Notice that we do not know which regions are above and which are below in our contour plot. (In fact, our contour plot could be showing only the $0$ level set, so the only thing we know about the regions between the curves is all the heights of the points on the interior of each region have the same sign.) So determining what kind of stationary point requires additional information. The following plot is $(x^2 - y^2)^2$ with level sets $\{-1,0,1\}$. I challenge you to distinguish it from the unsquared function's graph, above.

(I cheated and actually used the same graph. It's a cheat because the second graph has an empty $-1$ level set, but the first does not. Of course, without isocline labels or gradient labels, there is no way to know that the two plots are different.) Here's the 3D plot of $(x^2 - y^2)^2$.

Notice that the $0$ level set is transversely self-intersecting at $(0,0,0)$, but this is not a saddle point of the function, it's a local minimum. Consequence: the point labelled $F$ in the given graph does not have an unambiguous interpretation. It could be a local minimum, a local maximum, or a saddle point and there is not enough information to resolve which it is.

Here are a pair of function graphs with the same contour plot.

The level set for the shown contours of the left function is $\{5\}$ and the level set for the right function is $\sqrt{\frac{1}{2}(6 \pm \sqrt{6})}$. A thing we know is true is the disk-like region contains at least one local maximum and/or at least one local minimum. It could in fact contain infinitely many if there is a region of positive area all at the same maximal or minimal height. But what I want to focus on is the annular region between the two contours. For the left function, that region contains local minima on a curve running between (and parallel) to the two contous shown. For the right function, there are no local extrema on the annulus between the contours. The requirement that the neighboring curves differ in height by $1$ prevents this from happening in the given problem.

Finally, even if a region happens to enclose a local extremum, you have no way of knowing where in the region that local extrmum is. So, point $A$ might be at a local extremum, but there is no way to know this from a contour plot.

I also imagine that your instructor believes that the phrase "values at neighboring level curves differ by $1$" removes these ambiguities. This is not so : the entire set of levels could be $\{0,1\}$. (For instance, here's $\sin^2(r)$ :

Notice that the $0$ and $1$ level sets alternate (as you proceed from one contour to the next) and all the minima and maxima are on the level sets. ) From the information given, every point on every shown contour could be a local maximum or local minimum, alternating as you move from one contour to the next. As given, the problem is hopeless without inventing a great deal of information that is not shown in the image you have put in your Question.