Quadric Surfaces
Identified and Explained w/ Examples!

Did you know all we have to do to understand a three-dimensional object is to first look at its cross section?

Why?

Because a quadric surface is nothing more than a three-dimensional extension of a conic section.

This means that if we can identify the two-dimensional curve — line, ellipse, parabola, or hyperbola — then identifying and sketching the resulting surface in the three-dimensional space will be a snap.

Cross Section

But what do we mean by cross-section?

A cross section of an object or surface is what would be exposed or revealed if we slice the object with a plane: \(x=a\) , \(y=b\) , or \(z=c\).

The 2D sketch (i.e., cross-section) of the surface is called the trace.

For example, suppose we slice a cone with the xy-plane as seen in the image below. The resulting cross section is an ellipse, and the subsequent sketch of the slice in the 2D plane is the trace.

So, if the traces of a surface are nothing more than a cross-section created when the surface intersects a plane parallel to one of the coordinate planes, what’s a quadric surface?

Identifying Quadric Surfaces

Quadric surfaces are the graphs of equations that can be expressed in the form:

\begin{equation}
A x^{2}+B y^{2}+C z^{2}+D x y+E x z+F y z+G x+H y+J z+k=0
\end{equation}

Please note that when a quadric surface intersects the coordinate plane (xy-plane, xz-plane, or yz-plane), the trace is a conic section:

• Line
• Parabola
• Circle
• Ellipse
• Hyperbola

Types of Quadric Surfaces

Let’s take a look by comparing the 2D curve with its 3D counterpart.

Circle Vs Cylinder

The absence of the z-variable allows the three-dimensional solid to vary along the z-axis; thus, giving it height, and transforming it from a circle to a cylinder.

Ellipse vs Ellipsoid

Each slice (trace) of the ellipsoid is an ellipse and all the signs are positive. Moreover, it’s important to note that if \(a=b=c\) then the surface is a sphere.

Ellipse vs Cone

Notice how two of the variables are on one side of the equation and the third is on the other. If we move all three variables to the same side the resulting equation is now equal to zero, not one, which tells us the resulting shape will be a cone.

Parabola vs Paraboloid

Notice how one of the variables is not squared. This tells use it is a paraboloid and that it will open along the non-squared axis.

Hyperbola vs Hyperboloid of One-Sheet

Notice how only “one” of the variables is negative. This tells use it is a hyperboloid of one-sheet. Additionally, the variable that is negative also indicates to which axis the hyperboloid will open.

Hyperbola vs Hyperboloid of Two-Sheet

Notice how “two” of the variables are negative. This tells use it is a hyperboloid of two-sheets. And this time, the variable that is positive will indicate which axis the hyperboloid will open.

Elliptic Paraboloid & Hyperbolic Paraboloid

And here’s the cool part! We can even combine our quadric surfaces to yield such surfaces as elliptic paraboloids or hyperbolic paraboloids.

Traces Of Quadric Surfaces

So, how do we determine the resulting surface?

We look for traces.

Example

For instance, let’s name the shapes by identifying the traces:
\begin{equation}
x-5 y^{2}+2 z^{2}=0
\end{equation}

First, we notice that one of the variables is not squared. This instantly tells us that we are dealing with a paraboloid. Now, all we have to do is determine what kind of paraboloid (i.e., elliptic, or hyperbolic)

How?

Finding traces means we let each variable become a constant (number), and identify it’s resulting curve.

• If \(x=k\) where \(k\) is any constant, then \(k-5 y^{2}+2 z^{2}=0\) which is a hyperbola.
• If \(y=k\) where \(k\) is any constant, then \(x-5 k^{2}+2 z^{2}=0\) which is a parabola.
• If \(z=k\) where \(k\) is any constant, then \(x-5 y^{2}+2 k^{2}=0\) which is a parabola.

This means that \(x-5 y^{2}+2 z^{2}=0\) in the Cartesian coordinate system is a hyperbolic paraboloid that will open along the x-axis because it is the non-squared term.

Example

Let’s tackle this problem.

Name the shapes by identifying the traces
\begin{equation}
x^{2}-5 y^{2}+2 z^{2}=8
\end{equation}

First, we notice that all of the variables are squared, but one of the variables is negative. This instantly tells us that we are dealing with a hyperboloid of one-sheet.

Let’s confirm our suspicions by finding the traces.

• If \(x=k\) where \(k\) is any constant, then \(k^{2}-5 y^{2}+2 z^{2}=8\) which is a hyperbola.
• If \(y=k\) where \(k\) is any constant, then \(x^{2}-5 k^{2}+2 z^{2}=8\) which is a circle.
• If \(z=k\) where \(k\) is any constant, then \(x^{2}-5 y^{2}+2 k^{2}=8\) which is a hyperbola.

This confirms that our equation represents a circular hyperboloid of one sheet that will open along the y-axis because it is the negative term.

Easy, right?

Together in our video we will review our two-dimensional conic sections so we can quickly and easily identify the three-dimensional quadric surfaces.

We will learn how to find traces and sketch level curves, which are just a collection of traces or cross-sections of a graph, so we can easily identify and graph surfaces in 3-space.

Let’s jump on in!

Video Tutorial w/ Full Lesson & Detailed Examples (Video)

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