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 a,b,c where a>b>c>0, determines a pencil of confocal quadrics. Each quadric, generated by a parameter \lambda, is the locus of points satisfying the equation: :\frac{x^2}{a^2-\lambda}+\frac{y^2}{b^2-\lambda}+\frac{z^2}{c^2-\lambda} = 1. If \lambda, the quadric is an
ellipsoid; if c^2 (in the diagram: blue), it is a
hyperboloid of one sheet; if b^2 it is a
hyperboloid of two sheets. For a^2 there are no solutions.
Focal curves Limit surfaces for \lambda\to c^2: As the parameter \lambda approaches the value c^2 from
below, the limit ellipsoid is infinitely flat, or more precisely is the area of the --plane consisting of the ellipse :E : \frac{x^2}{a^2-c^2}+\frac{y^2}{b^2-c^2}=1 and its doubly covered
interior (in the diagram: below, on the left, red). As \lambda approaches c^2 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 E and its doubly covered
exterior (in the diagram: bottom, on the left, blue). The two limit surfaces have the points of ellipse E in common.
Limit surfaces for \lambda\to b^2: Similarly, as \lambda approaches b^2 from above and below, the respective limit hyperboloids (in diagram: bottom, right, blue and purple) have the hyperbola :H:\ \frac{x^2}{a^2-b^2}+\frac{z^2}{c^2-b^2}=1 in common.
Focal curves: The foci of the ellipse E are the vertices of the hyperbola H and vice versa. So E and H are a pair of
focal conics. Reverse: Because any quadric of the pencil of confocal quadrics determined by a,b,c can be constructed by a pins-and-string method (see
ellipsoid) the focal conics E,H 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, : Any point (x_0, y_0, z_0)\in \R^3 with x_0 \ne 0,\; y_0 \ne 0,\; z_0 \ne 0 lies on
exactly one surface of any of the three types of confocal quadrics. : The three quadrics through a point (x_0, y_0, z_0) intersect there
orthogonally (see external link).
Proof of the
existence and uniqueness of three quadrics through a point: For a point (x_0,y_0,z_0) with x_0\ne 0, y_0\ne 0,z_0\ne 0 let be f(\lambda)=\frac{x_0^2}{a^2-\lambda}+\frac{y_0^2}{b^2-\lambda}+\frac{z_0^2}{c^2-\lambda}-1. This function has three vertical
asymptotes c^2 and is in any of the open intervals (-\infty,c^2),\;(c^2,b^2),\;(b^2,a^2),\;(a^2,\infty) a
continuous and
monotone increasing function. From the behaviour of the function near its vertical asymptotes and from \lambda \to \pm \infty one finds (see diagram): Function f has exactly 3 zeros \lambda_1, \lambda_2, \lambda_3 with {\color{red}\lambda_1}
Proof of the
orthogonality of the surfaces: Using the pencils of functions F_\lambda(x,y,z)=\frac{x^2}{a^2-\lambda}+\frac{y^2}{b^2-\lambda}+\frac{z^2}{c^2-\lambda} with parameter \lambda the confocal quadrics can be described by F_\lambda(x,y,z)=1. For any two intersecting quadrics with F_{\lambda_i}(x,y,z)=1,\; F_{\lambda_k}(x,y,z)=1 one gets at a common point (x,y,z) :0=F_{\lambda_i}(x,y,z) - F_{\lambda_k}(x,y,z)= \dotsb :\ =(\lambda_i-\lambda_k)\left(\frac{x^2}{(a^2-\lambda_i)(a^2-\lambda_k)}+\frac{y^2}{(b^2-\lambda_i)(b^2-\lambda_k)}+\frac{z^2}{(c^2-\lambda_i)(c^2-\lambda_k)}\right)\ . From this equation one gets for the scalar product of the gradients at a common point : \operatorname{grad} F_{\lambda_i}\cdot \operatorname{grad} F_{\lambda_k}=4\;\left(\frac{x^2}{(a^2-\lambda_i)(a^2-\lambda_k)}+\frac{y^2}{(b^2-\lambda_i)(b^2-\lambda_k)}+\frac{z^2}{(c^2-\lambda_i)(c^2-\lambda_k)}\right)=0\ , 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 ==