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A x 2 + B y 2 + C z 2 + D x y + E x z + F y z + G x + H y + J z + K = 0 .

The following figures summarizes the most important ones.

This figure is of a table with two columns and three rows. The three rows represent the first 6 quadric surfaces: ellipsoid, hyperboloid of one sheet, and hyperboloid of two sheets. The equations and traces are in the first column. The second column has the graphs of the surfaces. The ellipsoid graph is a vertical oblong round shape. The hyperboloid of one sheet is circular on the top and the bottom and narrow in the middle. The hyperboloid in two sheets has two parabolic domes opposite of each other.
Characteristics of Common Quadratic Surfaces: Ellipsoid, Hyperboloid of One Sheet , Hyperboloid of Two Sheets .
This figure is of a table with two columns and three rows. The three rows represent the second 6 quadric surfaces: elliptic cone, elliptic paraboloid, and hyperbolic paraboloid. The equations and traces are in the first column. The second column has the graphs of the surfaces. The elliptic cone has two cones touching at the points. The elliptic paraboloid is similar to a cone but oblong. The hyperbolic paraboloid has a bend in the middle similar to a saddle.
Characteristics of Common Quadratic Surfaces: Elliptic Cone , Elliptic Paraboloid , Hyperbolic Paraboloid.

Identifying equations of quadric surfaces

Identify the surfaces represented by the given equations.

  1. 16 x 2 + 9 y 2 + 16 z 2 = 144
  2. 9 x 2 18 x + 4 y 2 + 16 y 36 z + 25 = 0
  1. 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
    x 2 9 + y 2 16 + z 2 9 = 1 .

    So, this is, in fact, an ellipsoid, centered at the origin.
  2. 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
    9 x 2 18 x + 4 y 2 + 16 y 36 z + 25 = 0 9 x 2 18 x + 4 y 2 + 16 y + 25 = 36 z 9 ( x 2 2 x ) + 4 ( y 2 + 4 y ) + 25 = 36 z 9 ( x 2 2 x + 1 1 ) + 4 ( y 2 + 4 y + 4 4 ) + 25 = 36 z 9 ( x 1 ) 2 9 + 4 ( y + 2 ) 2 16 + 25 = 36 z 9 ( x 1 ) 2 + 4 ( y + 2 ) 2 = 36 z ( x 1 ) 2 4 + ( y 2 ) 2 9 = z .

    This is an elliptic paraboloid centered at ( 1 , 2 , 0 ) .
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Identify the surface represented by equation 9 x 2 + y 2 z 2 + 2 z 10 = 0 .

Hyperboloid of one sheet, centered at ( 0 , 0 , 1 )

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Key concepts

  • A set of lines parallel to a given line passing through a given curve is called a cylinder , or a cylindrical surface . The parallel lines are called rulings .
  • The intersection of a three-dimensional surface and a plane is called a trace . To find the trace in the xy -, yz -, or xz -planes, set z = 0 , x = 0 , or y = 0 , respectively.
  • Quadric surfaces are three-dimensional surfaces with traces composed of conic sections. Every quadric surface can be expressed with an equation of the form A x 2 + B y 2 + C z 2 + D x y + E x z + F y z + G x + H y + J z + K = 0 .
  • To sketch the graph of a quadric surface, start by sketching the traces to understand the framework of the surface.
  • Important quadric surfaces are summarized in [link] and [link] .

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.

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[T] z = cos ( 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.

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[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.

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

  1. Specify the name of the quadric surface.
  2. 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.

a. Cylinder; b. The x -axis

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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.

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

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

  1. x 2 4 + y 2 9 z 2 12 = 1
  2. x 2 4 y 2 9 z 2 12 = 1
  3. x 2 4 + y 2 9 + z 2 12 = 1
  4. z 2 = 4 x 2 + 3 y 2
  5. z = 4 x 2 y 2
  6. 4 x 2 + y 2 z 2 = 0

Hyperboloid of two sheets

b.

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Hyperboloid of one sheet

a.

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

Practice Key Terms 9

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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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