How do I accurately visualize and describe level curves?

answered 2014-07-02 21:54:08 -0600

Kyouko
231 ●4 ●12

This doesn't really answer your question directly, but level curves can be thought of as a value-bound portion of a 2D or 3D graph/figure where anything between two of the level curves describes the value between a certain range.

To help you visualize this, think of a hiking map. When you see these, there are various enclosed figures (for simplicity sake lets call them circles) that represent a span of altitude (say 1000-1100 ft above sea level). As you look over the map, you'll see many of these circles denoting various elevations as one would climb up or down the mountain.

The concept of level curves is important in computing a "steepest descent" sort of problem where you want to find the local or absolute minimum/maximum of a function. Let's assume we're looking at the bowl from yesterday's class where we have

$$ f(x,y) = x^2 + y^2 $$

(i'm not sure how i am able to graph this and put it in here, and I don't have mathmatica either)

We could then apply level curves along each "z" component (or in this case f(x,y)), and what you would see are rising circles that go up the sides of the bowl.

I hope this helps you get an idea, but i'm not sure if this explanation is very good

answered 2014-07-02 23:27:06 -0600

Gear Junky
278 ●2 ●7 ●13

Running off of Kyouko and from what I understand, the level curves take shape of their functions for that axis in the $xy$ plane. For example, $f(x,y)=x^2-y^2$ can be broke down as $y=x^2$ (easy visual) and $y=\sqrt x$ for $0 \leq x \leq \infty $ as $0 \geq x \geq -\infty$ can be represented with $-\sqrt {\mid x \mid}$ (also easily visualized). For the proximity of the lines from one to another, $f(x,y)$ is also recognized as $z$. The larger the magnitude of "$z$" from the set interval change should result in the lines appearing closer together on the contour plot. So like Kyouko stated, it's very similar to a topographic map with $z$ representing the elevation change.