12.6

12.6 Cylinders and Quadric Surfaces

Cylinders and Quadric Surfaces

We have already looked at two special types of surfaces:

planes and spheres.

Here we investigate two other types of surfaces: cylinders and quadric surfaces.

In order to sketch the graph of a surface, it is useful to determine the curves of intersection of the surface with planes parallel to the coordinate planes.

These curves are called traces (or cross-sections) of the surface.

Cylinders

Cylinders

A cylinder is a surface that consists of all lines

(called rulings) that are parallel to a given line and pass through a given plane curve.

Example 1

Sketch the graph of the surface z = x². Solution:

Notice that the equation of the graph, z = x², doesn’t involve y.

This means that any vertical plane with equation

y = k(parallel to the xz-plane) intersects the graph in a curve with equation z = x².

So these vertical traces are parabolas.

Example 1 – Solution

Figure 1 shows how the graph is formed by taking the parabola z = x² in the xz-plane and moving it in the direction of the y-axis.

The graph is a surface, called a parabolic cylinder, made up of infinitely many shifted copies of the same parabola.

Here the rulings of the cylinder are parallel to the y-axis.

cont’d

The surface z = x² is a parabolic cylinder.

Figure 1

Cylinders

We noticed that the variable y is missing from the equation of the cylinder in Example 1. This is typical of a surface

whose rulings are parallel to one of the coordinate axes.

If one of the variables x, y or z is missing from the equation of a surface, then the surface is a cylinder.

Note

When you are dealing with surfaces, it is important to recognize that an equation like x² + y² = 1 represents a cylinder and not a circle. The trace of the cylinder

x² + y² = 1 in the xy-plane is the circle with equations x² + y² = 1, z = 0.

Quadric Surfaces

Quadric Surfaces

A quadric surface is the graph of a second-degree

equation in three variables x, y, and z. The most general such equation is

where A,B,C,…, J are constants, but by translation and rotation it can be brought into one of the two standard forms

Ax² + By² + Cz² + J = 0 or Ax² + By² + Iz = 0

Quadric surfaces are the counterparts in three dimensions of the conic sections in the plane.

Example 3

Use traces to sketch the quadric surface with equation

Solution:

By substituting z = 0, we find that the trace in the xy-plane is x² + y²/9 = 1, which we recognize as an equation of an ellipse. In general, the horizontal trace in the plane z = k is

which is an ellipse, provided that k² < 4, that is, –2 < k < 2.

Example 3 – Solution

Similarly, vertical traces parallel to the yz- and xz-planes are also ellipses:

(if –1 < k < 1)

(if –3 < k < 3)

cont’d

Example 3 – Solution

Figure 4 shows how drawing some traces indicates the shape of the surface.

It’s called an ellipsoid because all of its traces are ellipses.

Notice that it is symmetric with respect to each coordinate plane;

this is a reflection of the fact that its equation involves only even powers of x, y, and z.

cont’d

The ellipsoid

Example 4

Use traces to sketch the surface z = 4x² + y². Solution:

If we put x = 0, we get z = y², so the yz-plane intersects the surface in a parabola. If we put x = k (a constant), we get z = y² + 4k².

This means that if we slice the graph with any plane

parallel to the yz-plane, we obtain a parabola that opens upward.

Similarly, if y = k, the trace is z = 4x² + k², which is again a parabola that opens upward.

Example 4 – Solution

If we put z = k, we get the horizontal traces 4x² + y² = k, which we recognize as a family of ellipses. Knowing the shapes of the traces, we can sketch the graph in Figure 5.

Because of the elliptical and parabolic traces, the quadric surface z = 4x² + y² is called an elliptic paraboloid.

cont’d

Figure 5

The surface z = 4x²+ y² is an elliptic paraboloid. Horizontal traces are ellipses; vertical traces are parabolas.

Example 5

Sketch the surface z = y² – x². Solution:

The traces in the vertical planes x = k are the parabolas z = y² – k², which open upward. The traces in y = k are the parabolas z = –x² + k², which open downward.

The horizontal traces are y² – x² = k, a family of hyperbolas.

Example 5 – Solution

We draw the families of traces in Figure 6, and we show how the traces appear when placed in their correct planes in Figure 7.

cont’d

Figure 6

Vertical traces are parabolas; horizontal traces are hyperbolas. All traces are labeled with the value of k.

Traces in x = k

are z = y²– k² Traces in z = k

are y² – x² = k Traces in y = k

are z = –x²+ k²

Example 5 – Solution

Traces in x = k Traces in y = k Traces in z = k

Traces moved to their correct planes

Figure 7

Example 5 – Solution

In Figure 8 we fit together the traces from Figure 7 to form the surface z = y² – x², a hyperbolic paraboloid.

Notice that the shape of the surface near the origin resembles that of a saddle.

cont’d

Two views of the surface z = y²– x², a hyperbolic paraboloid.

Figure 8

Example 6

Sketch the surface Solution:

The trace in any horizontal plane z = k is the ellipse z = k

but the traces in the xz- and yz-planes are the hyperbolas y = 0 and x = 0

Example 6 – Solution

This surface is called a hyperboloid of one sheet and is sketched in Figure 9.

cont’d

Figure 9

Quadric Surfaces

The idea of using traces to draw a surface is employed in three-dimensional graphing software.

In most such software, traces in the vertical planes x = k and y = k are drawn for equally spaced values of k, and parts of the graph are eliminated using hidden line removal.

Quadric Surfaces

Table 1 shows computer-drawn graphs of the six

basic types of quadric surfaces in standard form.

All surfaces are

symmetric with respect to the z-axis.

If a quadric surface is

symmetric about a different axis, its equation changes

Table 1

Graphs of Quadric Surfaces

Applications of Quadric

Surfaces

Applications of Quadric Surfaces

Examples of quadric surfaces can be found in the world around us. In fact, the world itself is a good example.

Although the earth is commonly modeled as a sphere, a more accurate model is an ellipsoid because the earth’s rotation has caused a flattening at the poles.

Circular paraboloids, obtained by rotating a parabola about its axis, are used to collect and reflect light, sound, and

radio and television signals.

Applications of Quadric Surfaces

In a radio telescope, for instance, signals from distant stars that strike the bowl are all reflected to the receiver at the focus and are therefore amplified.

The same principle applies to microphones and satellite dishes in the shape of paraboloids.

Cooling towers for nuclear reactors are usually designed in the shape of hyperboloids of one sheet for reasons of

structural stability.

Pairs of hyperboloids are used to transmit rotational motion between skew axes.