Slicing Quadric Surfaces

Introduction

After you’ve worked in two dimensions for a while, you’ve probably realized that a lot of the functions you graph can be organized into broad categories. For example, \( x^2-1 \) and \( -4x^2+3 \) are both recognizable as parabolas, even though they don't have the exact same shape. Other shapes you may have learned to recognize in two-dimensional space are circles, ellipsoids, and hyperbolas. Can we identify these same categories in three-dimensional space? The answer is yes, and they are called quadric surfaces.

Quadric Surfaces

When we limit the variables in our functions to having a maximum degree of two in three-dimensional space, the resulting shapes are called quadric surfaces. There are many types of quadric surfaces, including cones, ellipsoids, spheres, hyperboloids of one sheet, and more. For an example, we will examine the hyperbolic paraboloid, which has the following general form: \[ \frac{x^2}{a^2}\ - \frac{y^2}{b^2}\ =z \] Note that the locations of each variable can be switched and will still form a hyperbolic paraboloid, simply rotated about the different axes. Below several functions and graphs of hyperbolic paraboloids are provided:

Slicing Quadric Surfaces

One notable property of Quadric surfaces is that if you cut them in two, their cross section will be a function in two-dimensional space. Let’s see how this works with the first hyperbolic paraboloid in the previous image (\(z = \frac{x^2}{16}\ - \frac{y^2}{9}\ \)). In the process you’ll develop a better idea of what this surface looks like.

In order to cut our hyperbolic paraboloid in two and find its cross section, we will imagine that we are finding the surface’s intersection with a plane. Let’s say our plane can be expressed by the following equation: \[ 0 = -3x + y + 0z \]

Intersecting these two looks like the following, where the blue region is the plane, and the rainbow region is the hyperbolic paraboloid

Now, let’s determine their intersection. By rearranging the equation for our plane, we know that: \[ y=-3x \]

We can substitute this into the function for our quadric surface:

\[ \frac{x^2}{16}\ - \frac{(-3x)^2}{9}\ = z \] \[ \frac{x^2}{16}\ - x^2 = z \] \[ \frac{-15x^2}{16}\ = z \]

Therefore, the intersection of these two is a parabola in the \( xz \) plane. We can see this on the below model, in which the hyperbolic palabra has been cut open along the plane so that we can view the cross section:

Keep in mind this is only one potential plane we could use to slice our quadric surface. What if we used a different one? Take a look at the example below:

In this case, we can see just by looking at the image that the intersection ends up being a hyperbola. Given that slicing this surface gives us a parabola in some places and a hyperbola in others, doesn’t ‘hyperbolic paraboloid’ seem like a fitting name? Try looking at the cross sections of some of the other quadric surfaces, and you should find a similar reasoning behind their names as well.

Why this example?

This example was chosen first of all because the hyperbolic paraboloid is an interesting shape that can be sliced to create both hyperbolas (which is briefly mentioned in this blog post) and parabolas (which is shown mathematically). It was particularly convenient to focus on the parabolic slices because this results in a model with two pieces that stand easily next to each other to display the cut.

Author: Sarah Bombrys