## Learning Objectives

- 2.6.1 Identify a cylinder as a type of three-dimensional surface.
- 2.6.2 Recognize the main features of ellipsoids, paraboloids, and hyperboloids.
- 2.6.3 Use traces to draw the intersections of quadric surfaces with the coordinate planes.

We have been exploring vectors and vector operations in three-dimensional space, and we have developed equations to describe lines, planes, and spheres. In this section, we use our knowledge of planes and spheres, which are examples of three-dimensional figures called *surfaces*, to explore a variety of other surfaces that can be graphed in a three-dimensional coordinate system.

## Identifying Cylinders

The first surface we’ll examine is the cylinder. Although most people immediately think of a hollow pipe or a soda straw when they hear the word *cylinder*, here we use the broad mathematical meaning of the term. As we have seen, cylindrical surfaces don’t have to be circular. A rectangular heating duct is a cylinder, as is a rolled-up yoga mat, the cross-section of which is a spiral shape.

In the two-dimensional coordinate plane, the equation ${x}^{2}+{y}^{2}=9$ describes a circle centered at the origin with radius $3.$ In three-dimensional space, this same equation represents a surface. Imagine copies of a circle stacked on top of each other centered on the *z*-axis (Figure 2.75), forming a hollow tube. We can then construct a cylinder from the set of lines parallel to the *z*-axis passing through circle ${x}^{2}+{y}^{2}=9$ in the *xy*-plane, as shown in the figure. In this way, any curve in one of the coordinate planes can be extended to become a surface.

## Definition

A set of lines parallel to a given line passing through a given curve is known as a cylindrical surface, or cylinder. The parallel lines are called rulings.

From this definition, we can see that we still have a cylinder in three-dimensional space, even if the curve is not a circle. Any curve can form a cylinder, and the rulings that compose the cylinder may be parallel to any given line (Figure 2.76).

## Example 2.55

### Graphing Cylindrical Surfaces

Sketch the graphs of the following cylindrical surfaces.

- ${x}^{2}+{z}^{2}=25$
- $z=2{x}^{2}-y$
- $y=\text{sin}\phantom{\rule{0.2em}{0ex}}x$

### Solution

- The variable $y$ can take on any value without limit. Therefore, the lines ruling this surface are parallel to the
*y*-axis. The intersection of this surface with the*xz*-plane forms a circle centered at the origin with radius $5$ (see the following figure).

- In this case, the equation contains all three variables $\u2014x,y,$ and $z\u2014$ so none of the variables can vary arbitrarily. The easiest way to visualize this surface is to use a computer graphing utility (see the following figure).

- In this equation, the variable
*z*can take on any value without limit. Therefore, the lines composing this surface are parallel to the*z*-axis. The intersection of this surface with the*xy*-plane outlines curve $y=\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}x$ (see the following figure).

## Checkpoint 2.52

Sketch or use a graphing tool to view the graph of the cylindrical surface defined by equation $z={y}^{2}.$

When sketching surfaces, we have seen that it is useful to sketch the intersection of the surface with a plane parallel to one of the coordinate planes. These curves are called traces. We can see them in the plot of the cylinder in Figure 2.80.

## Definition

The traces of a surface are the cross-sections created when the surface intersects a plane parallel to one of the coordinate planes.

Traces are useful in sketching cylindrical surfaces. For a cylinder in three dimensions, though, only one set of traces is useful. Notice, in Figure 2.80, that the trace of the graph of $z=\text{sin}\phantom{\rule{0.2em}{0ex}}x$ in the *xz*-plane is useful in constructing the graph. The trace in the *xy*-plane, though, is just a series of parallel lines, and the trace in the *yz*-plane is simply one line.

Cylindrical surfaces are formed by a set of parallel lines. Not all surfaces in three dimensions are constructed so simply, however. We now explore more complex surfaces, and traces are an important tool in this investigation.

## Quadric Surfaces

We have learned about surfaces in three dimensions described by first-order equations; these are planes. Some other common types of surfaces can be described by second-order equations. We can view these surfaces as three-dimensional extensions of the conic sections we discussed earlier: the ellipse, the parabola, and the hyperbola. We call these graphs quadric surfaces.

## Definition

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

When a quadric surface intersects a coordinate plane, the trace is a conic section.

An ellipsoid is a surface described by an equation of the form $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}+\frac{{z}^{2}}{{c}^{2}}=1.$ Set $x=0$ to see the trace of the ellipsoid in the *yz*-plane. To see the traces in the *xy*- and *xz*-planes, set $z=0$ and $y=0,$ respectively. Notice that, if $a=b,$ the trace in the *xy*-plane is a circle. Similarly, if $a=c,$ the trace in the *xz*-plane is a circle and, if $b=c,$ then the trace in the *yz*-plane is a circle. A sphere, then, is an ellipsoid with $a=b=c.$

## Example 2.56

### Sketching an Ellipsoid

Sketch the ellipsoid $\frac{{x}^{2}}{{2}^{2}}+\frac{{y}^{2}}{{3}^{2}}+\frac{{z}^{2}}{{5}^{2}}=1.$

### Solution

Start by sketching the traces. To find the trace in the *xy*-plane, set $z=0\text{:}$ $\frac{{x}^{2}}{{2}^{2}}+\frac{{y}^{2}}{{3}^{2}}=1$ (see Figure 2.81). To find the other traces, first set $y=0$ and then set $x=0.$

Now that we know what traces of this solid look like, we can sketch the surface in three dimensions (Figure 2.82).

The trace of an ellipsoid is an ellipse in each of the coordinate planes. However, this does not have to be the case for all quadric surfaces. Many quadric surfaces have traces that are different kinds of conic sections, and this is usually indicated by the name of the surface. For example, if a surface can be described by an equation of the form $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=\frac{z}{c},$ then we call that surface an elliptic paraboloid. The trace in the *xy*-plane is an ellipse, but the traces in the *xz*-plane and *yz*-plane are parabolas (Figure 2.83). Other elliptic paraboloids can have other orientations simply by interchanging the variables to give us a different variable in the linear term of the equation $\frac{{x}^{2}}{{a}^{2}}+\frac{{z}^{2}}{{c}^{2}}=\frac{y}{b}$ or $\frac{{y}^{2}}{{b}^{2}}+\frac{{z}^{2}}{{c}^{2}}=\frac{x}{a}.$

## Example 2.57

### Identifying Traces of Quadric Surfaces

Describe the traces of the elliptic paraboloid ${x}^{2}+\frac{{y}^{2}}{{2}^{2}}=\frac{z}{5}.$

### Solution

To find the trace in the *xy*-plane, set $z=0\text{:}$ ${x}^{2}+\frac{{y}^{2}}{{2}^{2}}=0.$ The trace in the plane $z=0$ is simply one point, the origin. Since a single point does not tell us what the shape is, we can move up the *z*-axis to an arbitrary plane to find the shape of other traces of the figure.

The trace in plane $z=5$ is the graph of equation ${x}^{2}+\frac{{y}^{2}}{{2}^{2}}=1,$ which is an ellipse. In the *xz*-plane, the equation becomes $z=5{x}^{2}.$ The trace is a parabola in this plane and in any plane with the equation $y=b.$

In planes parallel to the *yz*-plane, the traces are also parabolas, as we can see in the following figure.

## Checkpoint 2.53

A hyperboloid of one sheet is any surface that can be described with an equation of the form $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}-\frac{{z}^{2}}{{c}^{2}}=1.$ Describe the traces of the hyperboloid of one sheet given by equation $\frac{{x}^{2}}{{3}^{2}}+\frac{{y}^{2}}{{2}^{2}}-\frac{{z}^{2}}{{5}^{2}}=1.$

Hyperboloids of one sheet have some fascinating properties. For example, they can be constructed using straight lines, such as in the sculpture in Figure 2.85(a). In fact, cooling towers for nuclear power plants are often constructed in the shape of a hyperboloid. The builders are able to use straight steel beams in the construction, which makes the towers very strong while using relatively little material (Figure 2.85(b)).

## Example 2.58

### Chapter Opener: Finding the Focus of a Parabolic Reflector

Energy hitting the surface of a parabolic reflector is concentrated at the focal point of the reflector (Figure 2.86). If the surface of a parabolic reflector is described by equation $\frac{{x}^{2}}{100}+\frac{{y}^{2}}{100}=\frac{z}{4},$ where is the focal point of the reflector?

### Solution

Since *z* is the first-power variable, the axis of the reflector corresponds to the *z*-axis. The coefficients of ${x}^{2}$ and ${y}^{2}$ are equal, so the cross-section of the paraboloid perpendicular to the *z*-axis is a circle. We can consider a trace in the *xz*-plane or the *yz*-plane; the result is the same. Setting $y=0,$ the trace is a parabola opening up along the *z*-axis, with standard equation ${x}^{2}=4pz,$ where $p$ is the focal length of the parabola. In this case, this equation becomes ${x}^{2}=100\xb7\frac{z}{4}=4pz$ or $25=4p.$ So *p* is $6.25$ m, which tells us that the focus of the paraboloid is $6.25$ m up the axis from the vertex. Because the vertex of this surface is the origin, the focal point is $\left(0,0,6.25\right).$

Seventeen standard quadric surfaces can be derived from the general equation

The following figures summarizes the most important ones.

In the following two figures, the “axis” of a quadric surface may or may not be an axis of symmetry. However, all traces on the surface formed by any plane perpendicular to an “axis” will be of the same conic section type.

## Example 2.59

### Identifying Equations of Quadric Surfaces

Identify the surfaces represented by the given equations.

- $16{x}^{2}+9{y}^{2}+16{z}^{2}=144$
- $9{x}^{2}-18x+4{y}^{2}+16y-36z+25=0$

### Solution

- The $x,y,$ and $z$ terms are all squared, and are all positive, so this is probably an ellipsoid. However, let’s put the equation into the standard form for an ellipsoid just to be sure. We have

$$16{x}^{2}+9{y}^{2}+16{z}^{2}=144.$$

Dividing through by 144 gives

$$\frac{{x}^{2}}{9}+\frac{{y}^{2}}{16}+\frac{{z}^{2}}{9}=1.$$

So, this is, in fact, an ellipsoid, centered at the origin. - We first notice that the $z$ term is raised only to the first power, so this is either an elliptic paraboloid or a hyperbolic paraboloid. We also note there are $x$ terms and $y$ terms that are not squared, so this quadric surface is not centered at the origin. We need to complete the square to put this equation in one of the standard forms. We have

$$\begin{array}{}\\ \hfill 9{x}^{2}-18x+4{y}^{2}+16y-36z+25& =\hfill & 0\hfill \\ \hfill 9{x}^{2}-18x+4{y}^{2}+16y+25& =\hfill & 36z\hfill \\ \hfill 9\left({x}^{2}-2x\right)+4\left({y}^{2}+4y\right)+25& =\hfill & 36z\hfill \\ \hfill 9\left({x}^{2}-2x+1-1\right)+4\left({y}^{2}+4y+4-4\right)+25& =\hfill & 36z\hfill \\ \hfill 9{\left(x-1\right)}^{2}-9+4{\left(y+2\right)}^{2}-16+25& =\hfill & 36z\hfill \\ \hfill 9{\left(x-1\right)}^{2}+4{\left(y+2\right)}^{2}& =\hfill & 36z\hfill \\ \hfill \frac{{\left(x-1\right)}^{2}}{4}+\frac{{\left(y-2\right)}^{2}}{9}& =\hfill & z.\hfill \end{array}$$

This is an elliptic paraboloid centered at $\left(1,2,0\right).$

## Checkpoint 2.54

Identify the surface represented by equation $9{x}^{2}+{y}^{2}-{z}^{2}+2z-10=0.$

## Section 2.6 Exercises

For the following exercises, sketch and describe the cylindrical surface of the given equation.

**[T]** ${x}^{2}+{y}^{2}=9$

**[T]** $z={e}^{x}$

**[T]** $z=\text{ln}\left(x\right)$

For the following exercises, the graph of a quadric surface is given.

- Specify the name of the quadric surface.
- Determine the axis of the quadric surface.

For the following exercises, match the given quadric surface with its corresponding equation in standard form.

- $\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}-\frac{{z}^{2}}{12}=1$
- $\frac{{x}^{2}}{4}-\frac{{y}^{2}}{9}-\frac{{z}^{2}}{12}=1$
- $\frac{{x}^{2}}{4}+\frac{{y}^{2}}{9}+\frac{{z}^{2}}{12}=1$
- $z=4{x}^{2}+3{y}^{2}$
- $z=4{x}^{2}-{y}^{2}$
- $4{x}^{2}+{y}^{2}-{z}^{2}=0$

Ellipsoid

Hyperbolic paraboloid

Elliptic cone

For the following exercises, rewrite the given equation of the quadric surface in standard form. Identify the surface.

$\mathrm{-4}{x}^{2}+25{y}^{2}+{z}^{2}=100$

$3{x}^{2}-{y}^{2}-6{z}^{2}=18$

$8{x}^{2}-5{y}^{2}-10z=0$

$63{x}^{2}+7{y}^{2}+9{z}^{2}-63=0$

$5{x}^{2}-4{y}^{2}+20{z}^{2}=0$

$49y={x}^{2}+7{z}^{2}$

For the following exercises, find the trace of the given quadric surface in the specified plane of coordinates and sketch it.

**[T]** ${x}^{2}+{z}^{2}+4y=0,x=0$

**[T]** $\mathrm{-4}{x}^{2}+25{y}^{2}+{z}^{2}=100,y=0$

**[T]** ${x}^{2}-y-{z}^{2}=1,y=0$

Use the graph of the given quadric surface to answer the questions.

- Specify the name of the quadric surface.
- Which of the equations—$16{x}^{2}+9{y}^{2}+36{z}^{2}=3600,9{x}^{2}+36{y}^{2}+16{z}^{2}=3600,$ or $36{x}^{2}+9{y}^{2}+16{z}^{2}=3600$—corresponds to the graph?
- Use b. to write the equation of the quadric surface in standard form.

Use the graph of the given quadric surface to answer the questions.

- Specify the name of the quadric surface.
- Which of the equations—$36z=9{x}^{2}+{y}^{2},9{x}^{2}+4{y}^{2}=36z,\phantom{\rule{0.2em}{0ex}}\text{or}\phantom{\rule{0.2em}{0ex}}-36z=\mathrm{-81}{x}^{2}+4{y}^{2}$—corresponds to the graph above?
- Use b. to write the equation of the quadric surface in standard form.

For the following exercises, the equation of a quadric surface is given.

- Use the method of completing the square to write the equation in standard form.
- Identify the surface.

$4{x}^{2}-{y}^{2}+{z}^{2}-8x+2y+2z+3=0$

${x}^{2}+{z}^{2}-4y+4=0$

${x}^{2}-{y}^{2}+{z}^{2}-12z+2x+37=0$

Write the standard form of the equation of the ellipsoid centered at the origin that passes through points $A\left(2,0,0\right),B\left(0,0,1\right),$ and $C\left(\frac{1}{2},\sqrt{11},\frac{1}{2}\right).$

Write the standard form of the equation of the ellipsoid centered at point $P\left(1,1,0\right)$ that passes through points $A\left(6,1,0\right),B\left(4,2,0\right)$ and $C\left(1,2,1\right).$

Determine the intersection points of elliptic cone ${x}^{2}-{y}^{2}-{z}^{2}=0$ with the line of symmetric equations $\frac{x-1}{2}=\frac{y+1}{3}=z.$

Determine the intersection points of parabolic hyperboloid $z=3{x}^{2}-2{y}^{2}$ with the line of parametric equations $x=3t,y=2t,z=19t,$ where $t\in \mathbb{R}.$

Find the equation of the quadric surface with points $P\left(x,y,z\right)$ that are equidistant from point $Q\left(0,\mathrm{-1},0\right)$ and plane of equation $y=1.$ Identify the surface.

Find the equation of the quadric surface with points $P\left(x,y,z\right)$ that are equidistant from point $Q\left(0,2,0\right)$ and plane of equation $y=\mathrm{-2}.$ Identify the surface.

If the surface of a parabolic reflector is described by equation $400z={x}^{2}+{y}^{2},$ find the focal point of the reflector.

Consider the parabolic reflector described by equation $z=20{x}^{2}+20{y}^{2}.$ Find its focal point.

Show that quadric surface ${x}^{2}+{y}^{2}+{z}^{2}+2xy+2xz+2yz+x+y+z=0$ reduces to two parallel planes.

Show that quadric surface ${x}^{2}+{y}^{2}+{z}^{2}-2xy-2xz+2yz-1=0$ reduces to two parallel planes passing.

**[T]** The intersection between cylinder ${\left(x-1\right)}^{2}+{y}^{2}=1$ and sphere ${x}^{2}+{y}^{2}+{z}^{2}=4$ is called a *Viviani curve*.

- Solve the system consisting of the equations of the surfaces to find the equations of the intersection curve. (
*Hint:*Find $x$ and $y$ in terms of $z.)$ - Use a computer algebra system (CAS) to visualize the intersection curve on sphere ${x}^{2}+{y}^{2}+{z}^{2}=4.$

Hyperboloid of one sheet $25{x}^{2}+25{y}^{2}-{z}^{2}=25$ and elliptic cone $\mathrm{-25}{x}^{2}+75{y}^{2}+{z}^{2}=0$ are represented in the following figure along with their intersection curves. Identify the intersection curves and find their equations (*Hint:* Find *y* from the system consisting of the equations of the surfaces.)

**[T]** Use a CAS to create the intersection between cylinder $9{x}^{2}+4{y}^{2}=18$ and ellipsoid $36{x}^{2}+16{y}^{2}+9{z}^{2}=144,$ and find the equations of the intersection curves.

**[T]** A spheroid is an ellipsoid with two equal semiaxes. For instance, the equation of a spheroid with the *z*-axis as its axis of symmetry is given by $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{a}^{2}}+\frac{{z}^{2}}{{c}^{2}}=1,$ where $a$ and $c$ are positive real numbers. The spheroid is called *oblate* if $c<a,$ and *prolate* for $c>a.$

- The eye cornea is approximated as a prolate spheroid with an axis that is the eye, where $a=8.7\phantom{\rule{0.2em}{0ex}}\text{mm and}\phantom{\rule{0.2em}{0ex}}c=9.6\phantom{\rule{0.2em}{0ex}}\text{mm}.$ Write the equation of the spheroid that models the cornea and sketch the surface.
- Give two examples of objects with prolate spheroid shapes.

**[T]** In cartography, Earth is approximated by an oblate spheroid rather than a sphere. The radii at the equator and poles are approximately $3963$ mi and $3950$ mi, respectively.

- Write the equation in standard form of the ellipsoid that represents the shape of Earth. Assume the center of Earth is at the origin and that the trace formed by plane $z=0$ corresponds to the equator.
- Sketch the graph.
- Find the equation of the intersection curve of the surface with plane $z=1000$ that is parallel to the
*xy*-plane. The intersection curve is called a*parallel*. - Find the equation of the intersection curve of the surface with plane $x+y=0$ that passes through the
*z*-axis. The intersection curve is called a*meridian*.

**[T]** A set of buzzing stunt magnets (or “rattlesnake eggs”) includes two sparkling, polished, superstrong spheroid-shaped magnets well-known for children’s entertainment. Each magnet is $1.625$ in. long and $0.5$ in. wide at the middle. While tossing them into the air, they create a buzzing sound as they attract each other.

- Write the equation of the prolate spheroid centered at the origin that describes the shape of one of the magnets.
- Write the equations of the prolate spheroids that model the shape of the buzzing stunt magnets. Use a CAS to create the graphs.

**[T]** A heart-shaped surface is given by equation ${\left({x}^{2}+\frac{9}{4}{y}^{2}+{z}^{2}-1\right)}^{3}-{x}^{2}{z}^{3}-\frac{9}{80}{y}^{2}{z}^{3}=0.$

- Use a CAS to graph the surface that models this shape.
- Determine and sketch the trace of the heart-shaped surface on the
*xz*-plane.

**[T]** The ring torus symmetric about the *z*-axis is a special type of surface in topology and its equation is given by ${\left({x}^{2}+{y}^{2}+{z}^{2}+{R}^{2}-{r}^{2}\right)}^{2}=4{R}^{2}\left({x}^{2}+{y}^{2}\right),$ where $R>r>0.$ The numbers $R$ and $r$ are called are the major and minor radii, respectively, of the surface. The following figure shows a ring torus for which $R=2\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}r=1.$

- Write the equation of the ring torus with $R=2\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}r=1,$ and use a CAS to graph the surface. Compare the graph with the figure given.
- Determine the equation and sketch the trace of the ring torus from a. on the
*xy*-plane. - Give two examples of objects with ring torus shapes.