Confocal conic sections


In geometry, two conic sections are called confocal, if they have the same foci. Because ellipses and hyperbolas possess two foci, there are confocal ellipses, confocal hyperbolas and confocal mixtures of ellipses and hyperbolas. In the mixture of confocal ellipses and hyperbolas, any ellipse intersects any hyperbola orthogonally. Parabolas possess only one focus, so, by convention, confocal parabolas have the same focus and the same axis of symmetry. Consequently, any point not on the axis of symmetry lies on two confocal parabolas which intersect orthogonally.
The formal extension of the concept of confocal conics to surfaces leads to confocal quadrics.

Confocal ellipses

An ellipse which is not a circle is uniquely determined by its foci and a point not on the major axis. The pencil of confocal ellipses with the foci can be described by the equation
with semi-major axis as parameter.
Because a point of an ellipse uniquely determines the parameter,
A hyperbola is uniquely determined by its foci and a point not on the axes of symmetry. The pencil of confocal hyperbolas with the foci can be described by the equation
with the semi-axis as parameter.
Because a point of the hyperbola determines the parameter uniquely,

Common representation

From the previous representations of confocal ellipses and hyperbolas one gets a common representation: The equation
describes an ellipse, if, and a hyperbola, if .
In the literature one finds another common representation:
with the semi-axes of a given ellipse and is the parameter of the pencil.

For one gets confocal ellipses and

for confocal hyperbolas with the foci in common.

Limit curves

At position the pencil of confocal curves have as left sided limit curve the line section on the x-axis and the right sided limit curve the two intervalls. Hence:
This property appears in the 3-dimensional case in an analogous one and leads to the definition of the focal curves of confocal quadrics.

Twofold orthogonal system

Considering the pencils of confocal ellipses and hyperbolas one gets from the geometrical properties of the normal and tangent at a point :
Hence, the plane can be covered by an orthogonal net of confocal ellipses and hyperbolas.
This orthogonal net can be used as the base of an elliptic coordinate system.

Confocal parabolas

Parabolas possess only one focus. A parabola can be considered as a limit curve of a pencil of confocal ellipses, where one focus is kept fixed, while the second one is moved to infinity. If one performs this transformation for a net of confocal ellipses and hyperbolas, one gets a net of two pencils of confocal parabolas.
The equation describes a parabola with the origin as focus and the x-axis as axis of symmetry. One considers the two pencils of parabolas:
From the definition of a parabola one gets
It follows by calculation that,
Analogous to confocal ellipses and hyperbolas, the plane can be covered by an orthogonal net of parabolas.
The net of confocal parabolas can be considered as the image of a net of lines parallel to the coordinate axes and contained in the right half of the complex plane by the conformal map .

Graves's theorem: the construction of confocal ellipses by a string

In 1850 the Irish bishop of Limerick Charles Graves proved and published the following method for the construction of confocal ellipses with help of a string:
The proof of this theorem uses elliptical integrals and is contained in Klein's book.
Otto Staude extended this method to the construction of confocal ellipsoids.
If ellipse E collapses to a line segment, one gets a slight variation of the gardener's method drawing an ellipse with foci.

Confocal quadrics

Definition

The idea of confocal quadrics is a formal extension of the concept of confocal conic sections to quadrics in 3-dimensional space:
Fix three real numbers with.
The equation
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.
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.
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 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.
Reverse: Because any quadric of the pencil of confocal quadrics determined by can be constructed by a pins-and-string method 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:
Proof of the existence and uniqueness of three quadrics through a point:

For a point with let be
This function has three vertical asymptotes and is in any of the open intervals a continuous and monotone increasing function. From the behaviour of the function near its vertical asymptotes and from one finds :

Function has exactly 3 zeros with
Proof of the orthogonality of the surfaces:

Using the pencils of functions
with parameter the confocal quadrics can be described by. For any two intersecting quadrics with one gets at a common point
From this equation one gets for the scalar product of the gradients at a common point
which proves the orthogonality.
Applications:

Due to Dupin's theorem on threefold orthogonal systems of surfaces the following statement is true:
In physics confocal ellipsoids appear as equipotential surfaces:
Ivory's theorem, named after the Scottish mathematician and astronomer James Ivory, is a statement on the diagonals of a net-rectangle, a quadrangle formed by orthogonal curves:
Intersection points of an ellipse and a confocal hyperbola:

Let be the ellipse with the foci and the equation
and the confocal hyperbola with equation
Computing the intersection points of and one gets the four points:
Diagonals of a net-rectangle:

In order to keep the calculation simple, it is supposed that
  1. , which is no essential restriction, because any other confocal net can be obtained by a uniform scaling.
  2. From the possible alternatives ) only is used. At the end, one considers easily, that any other combination of signs yields the same result.
Let be two confocal ellipses and two confocal hypervbolas with the same foci. The diagonals of the four points of the net-rectangle consisting of the points
are:
Obviously the last expression is invariant, if one performs the exchange
. Exactly this exchange leads to. Hence one gets :
The proof of the statement for confocal parabolas is a simple calculation.
Ivory even proved the 3-dimensional version of his theorem :