Die einfachsten Kurven und Flachen
Hyperboloid
About hyperloid, Hilbert and Cohn-Vossen wrote
Uber die beiden Scahren von Geraden, die auf dem einschaligen Hyperboloid und auf dem hyperbolischen Paraboloid
verlaufen, lasst sich ein uberrachender Satz ableiten.
What is the theorem that they stated "uberrachend" (surprising) ?
As the following figure shows, there are two classes,
say L1 and L2, of straight lines
that entirely lie on a hyperboloid.
Since all straight lines in these classes are not parallel to the horizontal plane,
they traverse the circle C that is the intersection of the hyperloid with the horizontal plane.
We say that two straight lines in each class are adjacent
if the intersection points that are made by these lines with the circle C are adjacent.
Let us take two adjacent straight lines arbitrarily from each class.
Then these four straight lines can form a skew quadrilateral.
In this way, on the hyperboloid, there is a network that consists of skew quadrilaterals.
Now we ask if it is possible to deform shape of the network
while lengths of all edges of skew quadrilaterals are preserved.
The "surprising theorem" claims that it is possible.
Therefore,
if the hyperboloid is made of fibres, we can deform its shape
without stretching or relaxing fibres.
The above figure is made by my Open-GL program.
In fact the program can draw not only static figures, but also show deformation
that the "surprising theorem" states.
To understand content of the program,
we need some mathematical background.
Hyperbolic paraboloid
Among quadratic surfaces, hyperbolic parabolids enjoy the same marvellous property as hyperboloids.
That is,
(1) a hyperbolic parabolid is made of two classes of fibre (straight lines),
so that the surface is covered with a large number (to be exact, infiniltely many number) of 'skew' quadrilaterals,
(2) and the surface can be deformed while preserving all lengths of edges of these skew quadrilaterals.
Construct ellipsoid using a string