Revise the properties of the hyperbolic paraboloid! (HYPAR - in Croatian)
We observe the pencil of planes [n] and the curvatures of the normal sections in those planes. The normal curvature of a hypar at a point T is a function which has maximum and minimum at the segment
[0,p]. The angles for those extremal values correspond to the principal directions, and the angels where normal curvature is zero correspond to the asymptotic directions.
The asymptotic directions at a point T lying on a hypar are defined with two rulings through T.
The principal directions at a point T lying on a hypar bisect the angle between the asymptotic direction.
The following pictures show the normal sections through the principal and asymptotic directions and a graph of hyper normal curvature at T.
For the following visualisations we used the periodical Mathematica color function Hue (period 1). It is defined in the following way:
A hypar coloured by Hue[G]. | The graph of the Gaussian curvature of a hypar. |
A hypar coloured by Hue[H]. | The graph of the mean curvature of a hypar. |