Confocal conic sections

Consequently, any point not on the axis of symmetry lies on two confocal parabolas which intersect orthogonally (see below).

Any hyperbola or (non-circular) ellipse has two foci, and any pair of distinct points

not on line connecting them uniquely determine an ellipse and hyperbola, with shared foci

Each ellipse or hyperbola in the pencil is the locus of points satisfying the equation with semi-major axis

), the equation defines a hyperbola, while if the semi-major axis is greater than the linear eccentricity (

This representation generalizes naturally to higher dimensions (see § Confocal quadrics).

from above, the limit of the pencil of confocal hyperbolas degenerates to the relative complement of that line segment with respect to the x-axis; that is, to the two rays with endpoints at the foci pointed outward along the x-axis (an infinitely flat hyperbola).

This property appears analogously in the 3-dimensional case, leading to the definition of the focal curves of confocal quadrics.

Considering the pencils of confocal ellipses and hyperbolas (see lead diagram) one gets from the geometrical properties of the normal and tangent at a point (the normal of an ellipse and the tangent of a hyperbola bisect the angle between the lines to the foci).

The orthogonal net of ellipses and hyperbolas is the base of an elliptic coordinate system.

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 x-axis as its axis of symmetry is the locus of points satisfying the equation for some value of the parameter

not on the x-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 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.

[1] 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.

In 1850 the Irish bishop Charles Graves proved and published the following method for the construction of confocal ellipses with help of a string:[2] 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 (see Klein's book).

, one gets a slight variation of the gardener's method drawing an ellipse with foci

from below, the limit ellipsoid is infinitely flat, or more precisely is the area of the x-y-plane consisting of the ellipse and its doubly covered interior (in the diagram: below, on the left, red).

from above, the limit hyperboloid of one sheet is infinitely flat, or more precisely is the area of the x-y-plane consisting of the same ellipse

and its doubly covered exterior (in the diagram: bottom, on the left, blue).

from above and below, the respective limit hyperboloids (in diagram: bottom, right, blue and purple) have the hyperbola in common.

play the role of infinite many foci and are called focal curves of the pencil of confocal quadrics.

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.

without loss of generality (any other confocal net can be obtained by uniform scaling) and among the four intersections between an ellipse and a hyperbola choose those in the positive quadrant (other sign combinations yield the same result after an analogous calculation).