Math Insight

Level set examples

Example 1

Let $f(x,y) = x^2-y^2$. We will study the level curves $c=x^2-y^2$.

First, look at the case $c=0$. The level curve equation $x^2-y^2=0$ factors to $(x-y)(x+y)=0$. This equation is satisfied if either $y=x$ or $y=-x$. Both these are equations for lines, so the level curve for $c=0$ is two lines.

If $c \ne 0$, then we can rewrite the level curve equation $c=x^2-y^2$ as \begin{align*} 1 = \frac{x^2}{c} - \frac{y^2}{c}. \end{align*} If you remember you conic sections, you'll recognize this as the equation for a hyperbola. If $c$ is positive, the hyperbolas open to the left and right. If $c$ is negative, the hyperbolas open up and down.

For example, if $c=1$, the equation is $x^2-y^2=1$. If $c=-1$, the equation is $y^2-x^2=1$. A number of level curves are plotted below.

Level curves of a hyperbolic paraboloid

We can “stack” these level curves on top of one another to form the graph of the function. Below, the level curves are the thick blue lines. Drag the green point from 0 to 1. When the point is at 1, each level curve given by $c=f(x,y)$ will be at the height $z=c$.

Applet: Level curves of a hyperbolic paraboloid

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Applet: Level curves of a hyperbolic paraboloid

Level curves of a hyperbolic paraboloid. When the green point is at 0, as it is in the default view, the figure shows a standard level curve plot of $f(x,y)=x^2-y^2$, though it is floating in a three dimensional space. When you drag the green point to 1, each level curve $f(x,y)=c$ moves to the height $z=c$, so that they are in the same position as in the graph of $z=f(x,y)$. In this way, the figure demonstrates the correspondence between the level curve plot and the graph of the function.

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This is yet another way to visualize the relationship between the level curves and the graph of $z=f(x,y)$ shown below, which is a hyperbolic paraboloid.

Applet: Graph of a hyperbolic paraboloid

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Applet: Graph of a hyperbolic paraboloid

Graph of a hyperbolic paraboloid. The graph of the function $f(x,y)=x^2-y^2$.

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Example 2

Let $f(x,y,z) = x^2+y^2+z^2$. Although we cannot plot the graph of this function, we can graph some of its level surfaces. The equation for a level surface, $x^2+y^2+z^2=c$, is the equation for a sphere of radius $\sqrt{c}$.

Applet: Spherical level surfaces

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Applet: Spherical level surfaces

Spherical level surfaces. The level surfaces $f(x,y,z) = x^2+y^2+z^2=c$ are spheres of radius $\sqrt{c}$. The level surface with $c=1$ is the sphere of radius 1 drawn in dark red. The level surface with $c=4$ is the sphere of radius 2 drawn in light green.

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