Quadric Surfaces

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 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 ([link]), 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.

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 ([link]).

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 [link].

Traces are useful in sketching cylindrical surfaces. For a cylinder in three dimensions, though, only one set of traces is useful. Notice, in [link], that the trace of the graph of

z = sin 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.

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 .

to see the trace of the ellipsoid in the yz-plane. To see the traces in the y- and xz-planes, set

z = 0

then the trace in the yz-plane is a circle. A sphere, then, is an ellipsoid with

a = b = c .

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 ([link]). 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}

Hyperboloids of one sheet have some fascinating properties. For example, they can be constructed using straight lines, such as in the sculpture in [link](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 ([link](b)).

Key Concepts

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

[T] $x^{2} + z^{2} = 1$

The surface is a cylinder with the rulings parallel to the y-axis.* * *

This figure is a circular cylinder inside of a box. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.

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

[T] $z = cos (\frac{π}{2} + x)$

The surface is a cylinder with rulings parallel to the y-axis.* * *

This figure is a surface inside of a box. Its cross section parallel to the x z plane would be a cosine curve. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.

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

[T] $z = 9 - y^{2}$

The surface is a cylinder with rulings parallel to the x-axis.* * *

This figure is a surface inside of a box. Its cross section parallel to the y z plane would be an upside down parabola. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.

[T] $z = ln (x)$

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

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

![This figure is a surface inside of a box. Its cross section parallel to the y z plane would be an upside down parabola. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.](/calculus-book/resources/CNX_Calc_Figure_12_06_215.jpg)

a. Cylinder; b. The x-axis

![This figure is a surface inside of a box. It is an elliptical cone. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.](/calculus-book/resources/CNX_Calc_Figure_12_06_214.jpg)

![This figure is a surface in the 3-dimensional coordinate system. There are two conical shapes facing away from each other. They have the x axis through the center.](/calculus-book/resources/CNX_Calc_Figure_12_06_221.jpg)

a. Hyperboloid of two sheets; b. The x-axis

![This figure is a surface in the 3-dimensional coordinate system. It is a parabolic surface with the x axis through the center.](/calculus-book/resources/CNX_Calc_Figure_12_06_222.jpg)

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

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

Hyperboloid of two sheets

Ellipsoid

Elliptic paraboloid

Hyperbolic paraboloid

Hyperboloid of one sheet

Elliptic cone

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

− x^{2} + 36 y^{2} + 36 z^{2} = 9

- \frac{x^{2}}{9} + \frac{y^{2}}{\frac{1}{4}} + \frac{z^{2}}{\frac{1}{4}} = 1,

hyperboloid of one sheet with the x-axis as its axis of symmetry

−4 x^{2} + 25 y^{2} + z^{2} = 100

−3 x^{2} + 5 y^{2} - z^{2} = 10

- \frac{x^{2}}{\frac{10}{3}} + \frac{y^{2}}{2} - \frac{z^{2}}{10} = 1,

hyperboloid of two sheets with the y-axis as its axis of symmetry

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

5 y = x^{2} - z^{2}

y = - \frac{z^{2}}{5} + \frac{x^{2}}{5},

hyperbolic paraboloid with the y-axis as its axis of symmetry

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

x^{2} + 5 y^{2} + 3 z^{2} - 15 = 0

\frac{x^{2}}{15} + \frac{y^{2}}{3} + \frac{z^{2}}{5} = 1,

ellipsoid

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

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

\frac{x^{2}}{40} + \frac{y^{2}}{8} - \frac{z^{2}}{5} = 0,

elliptic cone with the z-axis as its axis of symmetry

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

6 x = 3 y^{2} + 2 z^{2}

x = \frac{y^{2}}{2} + \frac{z^{2}}{3},

elliptic paraboloid with the x-axis as its axis of symmetry

49 y = 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] $x^{2} + z^{2} + 4 y = 0, z = 0$

Parabola $y = - \frac{x^{2}}{4},$

This figure is the graph of an upside down parabola with its highest point at the origin of a rectangular coordinate system.

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

[T] $−4 x^{2} + 25 y^{2} + z^{2} = 100, x = 0$

Ellipse $\frac{y^{2}}{4} + \frac{z^{2}}{100} = 1,$

This figure is the graph of an ellipse centered at the origin of a rectangular coordinate system.

[T] $−4 x^{2} + 25 y^{2} + z^{2} = 100, y = 0$

[T] $x^{2} + \frac{y^{2}}{4} + \frac{z^{2}}{100} = 1, x = 0$

Ellipse $\frac{y^{2}}{4} + \frac{z^{2}}{100} = 1,$

This figure is the graph of an ellipse centered at the origin of a rectangular coordinate system.

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

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

This figure is a surface inside of a box. It is an oval solid on its side. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.

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.

a. Ellipsoid; b. The third equation; c. $\frac{x^{2}}{100} + \frac{y^{2}}{400} + \frac{z^{2}}{225} = 1$

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

This figure is a surface inside of a box. It is a parabolic solid opening up vertically. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.

Specify the name of the quadric surface.
Which of the equations— $36 z = 9 x^{2} + y^{2}, 9 x^{2} + 4 y^{2} = 36 z, or - 36 z = −81 x^{2} + 4 y^{2}$
—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.

x^{2} + 2 z^{2} + 6 x - 8 z + 1 = 0

a. $\frac{{(x + 3)}^{2}}{16} + \frac{{(z - 2)}^{2}}{8} = 1;$

b. Cylinder centered at $(−3, 2)$

with rulings parallel to the y-axis

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

x^{2} + 4 y^{2} - 4 z^{2} - 6 x - 16 y - 16 z + 5 = 0

a. $\frac{{(x - 3)}^{2}}{4} + {(y - 2)}^{2} - {(z + 2)}^{2} = 1;$

b. Hyperboloid of one sheet centered at $(3, 2, −2),$

with the z-axis as its axis of symmetry

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

x^{2} + \frac{y^{2}}{4} - \frac{z^{2}}{3} + 6 x + 9 = 0

a. ${(x + 3)}^{2} + \frac{y^{2}}{4} - \frac{z^{2}}{3} = 0;$

b. Elliptic cone centered at $(−3, 0, 0),$

with the z-axis as its axis of symmetry

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

Write the standard form of the equation of the ellipsoid centered at the origin that passes through points $A (2, 0, 0), B (0, 0, 1),$

and $C (\frac{1}{2}, \sqrt{11}, \frac{1}{2}) .$

\frac{x^{2}}{4} + \frac{y^{2}}{16} + z^{2} = 1

Write the standard form of the equation of the ellipsoid centered at point $P (1, 1, 0)$

that passes through points $A (6, 1, 0), B (4, 2, 0)$

and $C (1, 2, 1) .$

Determine the intersection points of elliptic cone $x^{2} - y^{2} - z^{2} = 0$

with the line of symmetric equations $\frac{x - 1}{2} = \frac{y + 1}{3} = z .$

(1, −1, 0)

and $(\frac{13}{3}, 4, \frac{5}{3})$

Determine the intersection points of parabolic hyperboloid $z = 3 x^{2} - 2 y^{2}$

with the line of parametric equations $x = 3 t, y = 2 t, z = 19 t,$

where $t \in ℝ .$

Find the equation of the quadric surface with points $P (x, y, z)$

that are equidistant from point $Q (0, −1, 0)$

and plane of equation $y = 1 .$

Identify the surface.

x^{2} + z^{2} + 4 y = 0,

elliptic paraboloid

Find the equation of the quadric surface with points $P (x, y, z)$

that are equidistant from point $Q (0, 2, 0)$

and plane of equation $y = −2 .$

Identify the surface.

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

find the focal point of the reflector.

(0, 0, 100)

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

Find its focal point.

Show that quadric surface $x^{2} + y^{2} + z^{2} + 2 x y + 2 x z + 2 y z + x + y + z = 0$

reduces to two parallel planes.

Show that quadric surface $x^{2} + y^{2} + z^{2} - 2 x y - 2 x z + 2 y z - 1 = 0$

reduces to two parallel planes passing.

[T] The intersection between cylinder ${(x - 1)}^{2} + y^{2} = 1$

and sphere $x^{2} + y^{2} + z^{2} = 4$

is called a Viviani curve.

This figure is a surface inside of a box. It is a sphere with a right circular cylinder through the sphere vertically. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.

Solve the system consisting of the equations of the surfaces to find the equation 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 $x^{2} + y^{2} + z^{2} = 4 .$

a. $x = 2 - \frac{z^{2}}{2}, y = \pm \frac{z}{2} \sqrt{4 - z^{2}},$

where $z \in [−2, 2];$

b.* * *

This figure is a surface inside of a box. It is a sphere with a figure eight curve on the side of the sphere. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.

Hyperboloid of one sheet $25 x^{2} + 25 y^{2} - z^{2} = 25$

and elliptic cone $−25 x^{2} + 75 y^{2} + z^{2} = 0$

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.)

This figure is a surface inside of a box. It is a hyperbolic paraboloid with a hyperbola of two sheets intersecting. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.

[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.

This figure is a surface inside of a box. It is a solid oval with an elliptical cylinder vertically intersecting. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.

two ellipses of equations $\frac{x^{2}}{2} + \frac{y^{2}}{\frac{9}{2}} = 1$

in planes $z = ± 2 \sqrt{2}$

[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 $a = 8.7 mm and c = 9.6 mm .$
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.

a. $\frac{x^{2}}{3963^{2}} + \frac{y^{2}}{3963^{2}} + \frac{z^{2}}{3950^{2}} = 1;$

b.* * *

This figure is a surface inside of a box. It is a sphere. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system. ;* * *

c. The intersection curve is the ellipse of equation $\frac{x^{2}}{3963^{2}} + \frac{y^{2}}{3963^{2}} = \frac{(2950) (4950)}{3950^{2}},$

and the intersection is an ellipse.; d. The intersection curve is the ellipse of equation $\frac{2 y^{2}}{3963^{2}} + \frac{z^{2}}{3950^{2}} = 1 .$

[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 ${(x^{2} + \frac{9}{4} y^{2} + z^{2} - 1)}^{3} - x^{2} z^{3} - \frac{9}{80} y^{2} z^{3} = 0 .$

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.

a.* * *

This figure is a surface inside of a box. It is a heart. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.

b. The intersection curve is ${(x^{2} + z^{2} - 1)}^{3} - x^{2} z^{3} = 0 .$

This figure is a curve on a rectangular coordinate system. It is the shape of a heart centered about the y-axis.

[T] The ring torus symmetric about the z-axis is a special type of surface in topology and its equation is given by ${(x^{2} + y^{2} + z^{2} + R^{2} - r^{2})}^{2} = 4 R^{2} (x^{2} + y^{2}),$

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 $R = 2 and r = 1 .$

This figure is a surface inside of a box. It is a torus, a doughnut shape. The outside edges of the 3-dimensional box are scaled to represent the 3-dimensional coordinate system.

Write the equation of the ring torus with $R = 2 and r = 1,$
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.

Identifying Cylinders

Quadric Surfaces

Key Concepts

Glossary