One-sheeted hyperboloids with asymptotic cones

Categories

• Curves
  • Types of curves on surfaces
    • Lines of curvature

• Surfaces
  • Quadrics (surfaces of order two)
    • Classification of surfaces of 2nd order

• Surfaces
  • Quadrics (surfaces of order two)
    • Rulings on quadrics

Description

One-sheeted hyperboloid in the standard Cartesian coordinates is given by the equation

$\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$,

where in the model $a \neq b$ (thus the horizontal cross-sections are ellipses).

Any one-sheeted hyperboloid is a doubly-ruled surface, that can be constructed from straight lines.

The same surface is modeled in plaster in three different versions: plain, with two families of straight rulings, with lines of curvature. Each model has a hollow inside in the shape of the asymptotic cone.

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