Kuzee
Vector Algebra and Analytic Geometry of Space

Hyperboloids

The Hyperboloid of One Sheet

Hyperboloid of one sheet

A hyperboloid of one sheet is the set of points whose coordinates satisfy that the square of one variable minus the squares of the other two equals a constant.

The Standard Equation

The standard equation of the hyperboloid of one sheet

x2a2+y2b2−z2c2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1

where a,b,c>0a, b, c > 0.

The Geometric Properties

  • Topology: a connected surface with only one sheet
  • Symmetry: symmetric about the three coordinate planes
  • The direction it opens: it opens in both the positive and the negative zz-directions

The Hyperboloid of Two Sheets

Hyperboloid of two sheets

A hyperboloid of two sheets is the set of points whose coordinates satisfy that the square of one variable minus the squares of the other two equals a negative constant.

The Standard Equation

The standard equation of the hyperboloid of two sheets

x2a2+y2b2−z2c2=−1\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = -1

where a,b,c>0a, b, c > 0.

The Geometric Properties

  • Topology: a disconnected surface made up of two separate sheets
  • Symmetry: symmetric about the three coordinate planes
  • Position: the two sheets lie on the positive and the negative zz-directions respectively

The Properties of Hyperboloids

The Nature of the Sections

The sections of a hyperboloid

The sections of a hyperboloid by the coordinate planes are hyperbolas or ellipses:

  • the section by the xyxy-plane: x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (an ellipse)
  • the section by the xzxz-plane: x2a2−z2c2=1\frac{x^2}{a^2} - \frac{z^2}{c^2} = 1 (a hyperbola)
  • the section by the yzyz-plane: y2b2−z2c2=1\frac{y^2}{b^2} - \frac{z^2}{c^2} = 1 (a hyperbola)

The Behaviour at Infinity

A hyperboloid has an asymptotic cone:

The asymptotic cone

x2a2+y2b2−z2c2=0\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 0

The points of the hyperboloid approach this cone as their distance from the origin goes to infinity.

The Ruling Lines

A hyperboloid of one sheet can be generated by rotating two intersecting straight lines.

Special Cases

The Hyperboloid of Revolution

When a=ba = b, a hyperboloid of revolution is formed:

  • the hyperboloid of one sheet of revolution: x2a2+y2a2−z2c2=1\frac{x^2}{a^2} + \frac{y^2}{a^2} - \frac{z^2}{c^2} = 1
  • the hyperboloid of two sheets of revolution: x2a2+y2a2−z2c2=−1\frac{x^2}{a^2} + \frac{y^2}{a^2} - \frac{z^2}{c^2} = -1

The Equilateral Hyperboloid

When a=b=ca = b = c:

  • the equilateral hyperboloid of one sheet: x2+y2−z2=a2x^2 + y^2 - z^2 = a^2
  • the equilateral hyperboloid of two sheets: x2+y2−z2=−a2x^2 + y^2 - z^2 = -a^2

Exercises

Exercise 1

Determine the type of the surface x29+y24−z216=1\frac{x^2}{9} + \frac{y^2}{4} - \frac{z^2}{16} = 1.

hyperboloidrecognising the equationthe hyperboloid of one sheet

Approach:

Recognise the standard equation form of the hyperboloid.

Steps:

  1. The form of the equation: x2a2+y2b2−z2c2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1
  2. The parameters: a=3a = 3, b=2b = 2, c=4c = 4
  3. This is the standard equation of the hyperboloid of one sheet

Answer: this is a hyperboloid of one sheet

Exercise 2

Determine the type of the surface x29+y24−z216=−1\frac{x^2}{9} + \frac{y^2}{4} - \frac{z^2}{16} = -1.

hyperboloidrecognising the equationthe hyperboloid of two sheets

Approach:

Recognise the standard equation form of the hyperboloid of two sheets.

Steps:

  1. The form of the equation: x2a2+y2b2−z2c2=−1\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = -1
  2. The parameters: a=3a = 3, b=2b = 2, c=4c = 4
  3. This is the standard equation of the hyperboloid of two sheets

Answer: this is a hyperboloid of two sheets

Summary

Symbols Used in This Article

SymbolTypePronunciation/ExplanationMeaning in This Article
x2a2+y2b2−z2c2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1One-sheet hyperboloidone-sheet hyperboloidThe standard equation of the hyperboloid of one sheet
x2a2+y2b2−z2c2=−1\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = -1Two-sheet hyperboloidtwo-sheet hyperboloidThe standard equation of the hyperboloid of two sheets
x2a2+y2b2−z2c2=0\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 0Asymptotic coneasymptotic coneThe asymptotic cone of the hyperboloid
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Cylinders

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