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  • Identify a cylinder as a type of three-dimensional surface.
  • Recognize the main features of ellipsoids, paraboloids, and hyperboloids.
  • 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 ( [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.

This figure a 3-dimensional coordinate system. It has a right circular center with the z-axis through the center. The cylinder also has points labeled on the x and y axis at (3, 0, 0) and (0, 3, 0).
In three-dimensional space, the graph of equation x 2 + y 2 = 9 is a cylinder with radius 3 centered on the z -axis. It continues indefinitely in the positive and negative directions.

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 ( [link] ).

This figure has a 3-dimensional surface that begins on the y-axis and curves upward. There is also the x and z axes labeled.
In three-dimensional space, the graph of equation z = x 3 is a cylinder, or a cylindrical surface with rulings parallel to the y -axis.

Graphing cylindrical surfaces

Sketch the graphs of the following cylindrical surfaces.

  1. x 2 + z 2 = 25
  2. z = 2 x 2 y
  3. y = sin x
  1. 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).
    This figure is the 3-dimensional coordinate system. It has a right circular cylinder on its side with the y-axis in the center. The cylinder intersects the x-axis at (5, 0, 0). It also has two points of intersection labeled on the z-axis at (0, 0, 5) and (0, 0, -5).
    The graph of equation x 2 + z 2 = 25 is a cylinder with radius 5 centered on the y -axis.
  2. In this case, the equation contains all three variables x , y , and z 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).
    This figure has a surface in the first octant. The cross section of the solid is a parabola.
  3. 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 yz -plane outlines curve y = sin x (see the following figure).
    This figure is a three dimensional surface. A cross section of the surface parallel to the x y plane would be the sine curve.
    The graph of equation y = sin x is formed by a set of lines parallel to the z -axis passing through curve y = sin x in the xy -plane.
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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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