.
: For there are no solutions.
(In this context the parameter is ''not'' the linear eccentricity of an ellipse !)
Focal curves
Limit surfaces for :
Varying the ellipsoids by ''increasing'' parameter such that it approaches the value from below one gets an infinite flat ellipsoid. More precise: the area of the x-y-plane, that consists of the ellipse with equation and its doubly covered ''interior'' (in the diagram: below, on the left, red).
Varying the 1-sheeted hyperboloids by ''decreasing'' parameter such that it approaches the value from above one gets an infinite flat hyperboloid. More precise: the area of the x-y-plane, that consists of the same ellipse and its doubly covered ''exterior'' (in the diagram: bottom, on the left, blue).
That means: The two limit surfaces have the points of ellipse
:
in common.
Limit surfaces for :
Analogous considerations at the position yields:
The two limit surfaces (in diagram: bottom, right, blue and purple) at position have the hyperbola
:
in common.
Focal curves:
One easily checks, that the foci of the ellipse are the vertices of the hyperbola and vice versa. That means: Ellipse and hyperbola are a pair of focal conics
In geometry, focal conics are a pair of curves consisting of
either
*an ellipse and a hyperbola, where the hyperbola is contained in a plane, which is orthogonal to the plane containing the ellipse. The vertices of the hyperbola are the foci of ...
.
Reverse: Because any quadric of the pencil of confocal quadrics determined by can be constructed by a pins-and-string method (see ellipsoid
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
An ellipsoid is a quadric surface; that is, a surface that may be defined as the z ...
) the focal conics play the role of infinite many foci and are called focal curves of the pencil of confocal quadrics.
Threefold orthogonal system
Analogous to the case of confocal ellipses/hyperbolas one has:
* Any point with lies on ''exactly one surface'' of any of the three types of confocal quadrics.
: The three quadrics through a point intersect there ''orthogonally'' (see external link).
Proof of the ''existence and uniqueness'' of three quadrics through a point:
For a point with let be
.
This function has three vertical asymptote
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
s