Hyperbolic paraboloid  

From The Art and Popular Culture Encyclopedia

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The hyperbolic paraboloid is a doubly ruled surface: it contains two families of mutually skew lines. The lines in each family are parallel to a common plane, but not to each other. Hence the hyperbolic paraboloid is a conoid.

These properties characterize hyperbolic paraboloids and are used in one of the oldest definitions of hyperbolic paraboloids: a hyperbolic paraboloid is a surface that may be generated by a moving line that is parallel to a fixed plane and crosses two fixed skew lines. This property makes it simple to manufacture a hyperbolic paraboloid from a variety of materials and for a variety of purposes, from concrete roofs to snack foods.

Pringles potato chips resemble a truncated hyperbolic paraboloid. Their uniform shape allows them to be stacked in sturdy tubular containers, and the strength of the hyperbolic paraboloid shape helps prevent them from breaking while stacked.

Examples in architecture

• St. Mary's Cathedral, Tokyo
• Cathedral of Saint Mary of the Assumption, San Francisco, California
• Saddledome in Calgary, Alberta, Canada
• London Velopark
• Dogra Hall, Indian Institute of Technology Delhi in New Delhi, India

Plane sections

As plane sections of a hyperbolic paraboloid with equation

<math>z = \frac{x^2}{a^2} - \frac{y^2}{b^2}</math>

one gets the following cases:

• a parabola, if the plane is parallel to the z-axis with equation <math> ux+vy+w=0,\ au\ne \pm bv</math>,
• a line, if the plane is parallel to the z-axis with equation <math> ux+vy+w=0,\ au= \pm bv</math>,
• a pair of intersecting lines, if the plane is a tangent plane,
• a hyperbola, if the plane is not parallel to the z-axis and not a tangent plane.

Remarks:

1. A hyperbolic paraboloid is a ruled surface (contains lines), but not developable (in this case it is unlike a cylinder or cone).
2. The Gauss curvature at any point is negative. Hence it is a saddle surface.
3. The unit hyperbolic paraboloid with equation <math>z=x^2-y^2 </math> can be represented by <math>z=2xy</math> after a rotation around the z-axis with an angle of 45° degrees.
4. A hyperbolic paraboloid is projectively equivalent to a hyperboloid of one sheet.