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 shown floating in a three-dimensional plot. Drag the green point to the right. When the green point is all the way to the right, 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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Level curves of a hyperbolic paraboloid. When the green point on the slider is to the left, 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 the right, 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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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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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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