Confocal conic sections

In geometry, two conic sections are called confocal if they have the same foci.
Because ellipses and hyperbolas have 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 have 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.
A circle is an ellipse with both foci coinciding at the center. Circles that share the same focus are called concentric circles, and they orthogonally intersect any line passing through that center.
The formal extension of the concept of confocal conics to surfaces leads to confocal quadrics.

Confocal ellipses and hyperbolas

Any hyperbola or ellipse has two foci, and any pair of distinct points in the Euclidean plane and any third point not on line connecting them uniquely determine an ellipse and hyperbola, with shared foci and intersecting orthogonally at the point
The foci thus determine two pencils of confocal ellipses and hyperbolas.
By the principal axis theorem, the plane admits a Cartesian coordinate system with its origin at the midpoint between foci and its axes aligned with the axes of the confocal ellipses and hyperbolas. If is the linear eccentricity.

Limit curves

As the parameter approaches the value from [|below], the limit of the pencil of confocal ellipses degenerates to the line segment between foci on the -axis. As approaches from above, the limit of the pencil of confocal hyperbolas degenerates to the relative complement of that line segment with respect to the -axis; that is, to the two rays with endpoints at the foci pointed outward along the -axis. These two limit curves have the two foci in common.
This property appears analogously in the 3-dimensional case, leading to the definition of the focal curves of confocal quadrics. See below.

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. Any ellipse of the pencil intersects any hyperbola orthogonally.
This arrangement, in which each curve in a pencil of non-intersecting curves orthogonally intersects each curve in another pencil of non-intersecting curves is sometimes called an orthogonal net. The orthogonal net of ellipses and hyperbolas is the base of an elliptic coordinate system.

Confocal parabolas

A parabola has only one focus, and can be considered as a limit curve of a set of ellipses, where one focus and one vertex are kept fixed, while the second focus is moved to infinity. If this transformation is performed on each conic in an orthogonal net of confocal ellipses and hyperbolas, the limit is an orthogonal net of confocal parabolas facing opposite directions.
Every parabola with focus at the origin and -axis as its axis of symmetry is the locus of points satisfying the equation
for some value of the parameter where is the semi-latus rectum. If then the parabola opens to the right, and if the parabola opens to the left. The point is the vertex of the parabola.
From the definition of a parabola, for any point not on the -axis, there is a unique parabola with focus at the origin opening to the right and a unique parabola with focus at the origin opening to the left, intersecting orthogonally at the point.
Analogous to confocal ellipses and hyperbolas, the plane can be covered by an orthogonal net of parabolas, which can be used for a parabolic coordinate system.
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 .

Concentric circles and intersecting lines

A circle is an ellipse with two coinciding foci. The limit of hyperbolas as the foci are brought together is degenerate: a pair of intersecting lines.
If an orthogonal net of ellipses and hyperbolas is transformed by bringing the two foci together, the result is thus an orthogonal net of concentric circles and lines passing through the circle center. These are the basis for the polar coordinate system.
The limit of a pencil of ellipses sharing the same center and axes and passing through a given point degenerates to a pair of lines parallel with the major axis as the two foci are moved to infinity in opposite directions. Likewise the limit of an analogous pencil of hyperbolas degenerates to a pair of lines perpendicular to the major axis. Thus a rectangular grid consisting of orthogonal pencils of parallel lines is a kind of net of degenerate confocal conics. Such an orthogonal net is the basis for the Cartesian coordinate system.

Graves's theorem

In 1850 the Irish bishop 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

Two quadric surfaces are confocal if they share the same axes and if their intersections with each plane of symmetry are confocal conics. Analogous to conics, nondegenerate pencils of confocal quadrics come in two types: triaxial ellipsoids, hyperboloids of one sheet, and hyperboloids of two sheets; and elliptic paraboloids, hyperbolic paraboloids, and elliptic paraboloids opening in the opposite direction.
A triaxial ellipsoid with semi-axes where determines a pencil of confocal quadrics. Each quadric, generated by a parameter is the locus of points satisfying the equation:
If, the quadric is an ellipsoid; if , it is a hyperboloid of one sheet; if it is a hyperboloid of two sheets. For there are no solutions.

Focal curves

Limit surfaces for :
As the parameter approaches the value from below, the limit ellipsoid is infinitely flat, or more precisely is the area of the --plane consisting of the ellipse
and its doubly covered interior.
As approaches from above, the limit hyperboloid of one sheet is infinitely flat, or more precisely is the area of the --plane consisting of the same ellipse and its doubly covered exterior.
The two limit surfaces have the points of ellipse in common.
Limit surfaces for :
Similarly, as approaches from above and below, the respective limit hyperboloids have the hyperbola
in common.
Focal curves:
The foci of the ellipse are the vertices of the hyperbola and vice versa. So and 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,
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 intersection curve of any two confocal quadrics is a line of curvature. Analogously to the planar elliptic coordinates there exist ellipsoidal coordinates.
In physics confocal ellipsoids appear as equipotential surfaces of a charged ellipsoid.

Ivory's theorem

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:

To simplify the calculation, let without loss of generality and among the four intersections between an ellipse and a hyperbola choose those in the positive quadrant.
Let be two confocal ellipses and two confocal hyperbolas with the same foci. The diagonals of the four points of the net-rectangle consisting of the points
are:
The last expression is invariant under the exchange. Exactly this exchange leads to. Hence
The proof of the statement for confocal parabolas is a simple calculation.
Ivory even proved the 3-dimensional version of his theorem :