Kuzee
Vector Algebra and Analytic Geometry of Space

Paraboloids

The Elliptic Paraboloid

Elliptic paraboloid

An elliptic paraboloid is the set of points whose zz-coordinate equals the square of the distance from the origin to the projection of the point onto the xyxy-plane.

The Standard Equation

The standard equation of the elliptic paraboloid

x2a2+y2b2=2z\frac{x^2}{a^2} + \frac{y^2}{b^2} = 2z

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

The Geometric Properties

  • The direction it opens: upwards along the positive zz-axis
  • The vertex: at the origin (0,0,0)(0, 0, 0)
  • Symmetry: symmetric about the xzxz-plane and the yzyz-plane

The Nature of the Sections

  • the sections by planes parallel to the xzxz-plane are parabolas
  • the sections by planes parallel to the yzyz-plane are parabolas
  • the sections by planes parallel to the xyxy-plane are ellipses

The Hyperbolic Paraboloid

Hyperbolic paraboloid

A hyperbolic paraboloid is the set of points whose zz-coordinate equals a linear combination of the coordinates of the projection of the point onto the xyxy-plane.

The Standard Equation

The standard equation of the hyperbolic paraboloid

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

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

The Geometric Properties

  • The direction it opens: upwards along the xx-axis and downwards along the yy-axis
  • The saddle point: it takes a saddle shape at the origin (0,0,0)(0, 0, 0)
  • Symmetry: symmetric about the xzxz-plane and the yzyz-plane

The Nature of the Sections

  • the sections by planes parallel to the xzxz-plane are parabolas
  • the sections by planes parallel to the yzyz-plane are parabolas
  • the sections by planes parallel to the xyxy-plane are hyperbolas

The Properties of Paraboloids

The Behaviour at Infinity

  • the elliptic paraboloid: as z→+∞z \to +\infty, the cross-sectional ellipse grows without bound
  • the hyperbolic paraboloid: it has an asymptotic cone, opening out towards infinity

Curvature

  • the elliptic paraboloid: it has only one principal direction of curvature
  • the hyperbolic paraboloid: it has two principal directions of curvature of opposite sign, forming a saddle surface

Applications

  • the elliptic paraboloid: satellite antennas, the reflecting surfaces of car headlamps
  • the hyperbolic paraboloid: saddle roofs in architecture, cooling towers

Exercises

Exercise 1

Determine the type of the surface x2+y2=2zx^2 + y^2 = 2z.

paraboloidrecognising the equationthe elliptic paraboloid

Approach:

Put the equation into standard form.

Steps:

  1. The original equation: x2+y2=2zx^2 + y^2 = 2z
  2. The standard form: x212+y212=2z\frac{x^2}{1^2} + \frac{y^2}{1^2} = 2z
  3. This is the standard equation of the elliptic paraboloid

Answer: this is an elliptic paraboloid, opening upwards.

Exercise 2

Determine the type of the surface x2−y2=2zx^2 - y^2 = 2z.

paraboloidrecognising the equationthe hyperbolic paraboloid

Approach:

Recognise the standard form of the hyperbolic paraboloid.

Steps:

  1. The original equation: x2−y2=2zx^2 - y^2 = 2z
  2. The standard form: x212−y212=2z\frac{x^2}{1^2} - \frac{y^2}{1^2} = 2z
  3. This is the standard equation of the hyperbolic paraboloid

Answer: this is a hyperbolic paraboloid, forming a saddle surface.

Summary

Symbols Used in This Article

SymbolTypePronunciation/ExplanationMeaning in This Article
x2a2+y2b2=2z\frac{x^2}{a^2} + \frac{y^2}{b^2} = 2zElliptic paraboloidelliptic paraboloidThe standard equation of the elliptic paraboloid
x2a2−y2b2=2z\frac{x^2}{a^2} - \frac{y^2}{b^2} = 2zHyperbolic paraboloidhyperbolic paraboloidThe standard equation of the hyperbolic paraboloid
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Hyperboloids

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