So far in this chapter, we have studied the graphs of planes and spheres, but there are other surfaces that we need to talk about. These surfaces are cylinders and quadric surfaces.

FACT:

A cylinder is the surface composed of all the lines that (1) lie parallel to a given line in space and (2) pass through a given plane curve. The plane curve is known as the generating curve for the cylinder.

In the past, you have probably been taught that a cylinder is a right circular cylinder. This is not always the case. I will provide some examples of both right circular cylinders and other types of cylinders.

EXAMPLE 1: z ² + y ² = 9

plot 1

This is a right circular cylinder that has as its center the x-axis. (See plot 1). The generating curve is the circle z ² + y ² = 9 that lies in the yz-plane.

EXAMPLE 2: x = y ²

plot 2

This is a parabolic cylinder. (See plot 2) The generating curve is the parabola x = y ² that lies in the xy-plane.

EXAMPLE 3: z ² - x ² = 1

plot 3

This is a hyperbolic cylinder. (See plot 3) The generating curve is the hyperbola z ² - x ² = 1 that lies in the xz-plane.

FACT:

An equation in any two of three Cartesian coordinates defines a cylinder parallel to the axis of the third coordinate.

A quadric surface is the graph of a second-degree equation in x, y, and z. The most general form is Ax ² + By ² + Cz ² + Dxy + Eyz + Fxz + Gx + Hy + Jz + K = 0.

TYPE 1:

ELLIPSOID

An ellipsoid cuts the coordinate axes at (± a, 0, 0), (0, ± b, 0), and (0, 0, ± c).

When x = 0, you have an ellipse in the yz-plane.

When y = 0, you have an ellipse in the xz-plane.

When z = 0, you have an ellipse in the xy-plane.

If any two of the semi-axes a, b, and c are equal, then the surface is an ellipsoid of revolution.

If a = b = c, then we have a sphere.

EXAMPLE 4: 4x ² + 4y ² + z ² = 16

plot 4

This is an ellipsoid of revolution. (See plot 4) In the xy-plane, we have the circle x ² + y ² = 4. In the xz-plane, we have the ellipse 4x ² + z ² = 16, and in the yz-plane, we have the ellipse 4y ² + z ² = 16.

EXAMPLE 5: x ² + y ² + z ² = 4

plot 5

This is a sphere centered at the origin with radius of 2. (See plot 5) (Remember that a circle is a special case of an ellipse, so a sphere is the special case of an ellipsoid.)

TYPE 2:

ELLIPTIC PARABOLOIDS

If a = b, then we have a circular paraboloid or a paraboloid of revolution.

EXAMPLE 6: z = 4 - 4y ² - x ²

plot 6

This is an elliptic paraboloid. (See plot 6) When z = 0 (xy-plane), we have the ellipse 4y ² + x ² = 4.

EXAMPLE 7: z = 6x ² + 6y ²

plot 7

This is a circular paraboloid (paraboloid of revolution). (See plot 7)

TYPE 3:

ELLIPTIC CONE

If a = b, then the cone is a right circular cone. Remember that a cone has two napes.

EXAMPLE 8: z ² = x ² + y ²

plot 8

This is a right circular cone. (See plot 8)

EXAMPLE 9: 36z ² = 9x ² + 4y ²

plot 9

This is an elliptic cone. (See plot 9)

 TYPE 4:

HYPERBOLOID OF ONE SHEET

 If a = b, then the hyperboloid is a surface of revolution. This shape is commonly used for cooling towers at nuclear power plants.

 EXAMPLE 10: x ² + y ² - z ² = 1

plot 10

 This is a surface of revolution. (See plot 10)

 TYPE 5:

HYPERBOLOID OF TWO SHEETS

 These hyperboloids are asymptotic cones.

 EXAMPLE 11: z ² - x ² - y ² = 1

plot 11

 This is a hyperboloid of two sheets. (See plot 11)

  TYPE 6:

HYPERBOLIC PARABOLOID

 This surface is commonly called a saddle, and it includes a point that is neither a maximum nor a minimum. This point is called the saddle point.

 EXAMPLE 12: z = x ² - y ²

plot 12

 This is a hyperbolic paraboloid. (See plot 12)