Hyperbolic paraboloids

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

Hyperbolic paraboloid is a 2nd order surface given in the standard Cartesian coordinates by the eqaution

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

where for the models $a = b$.

Hyperbolic paraboloids are doubly-ruled surfaces, that is they can be constructed by using straight lines. One on the models show the two families of straight line rulings (through any point passes one line from each family).

Another model shows the horizontal cross-sections with planes parallel to $z=0$. In particular, $z=0$ intersect the surface along two rulings.

Yet another model shows behaviour of lines of curvature on the surface.

See more...