Last update, Dec. 17, 2001

Geometric concepts for Curves in 3D

Focal conics

Focal Conics

Definition :

The vertices h of all real right cones (i.e., which circumscribes a sphere) which contain a given ellipse

E : x²/a² + y²/b² = 1 , z = 0 , a > b

lie on the hyperbola

H : x²/e² - z²/b² = 1 , y = 0 , e² = a² - b²

From [Boehm94], p. 196 .

Properties :

By symmetry, the vertices of all real right cones containing H lie on E.
The circumscribed sphere above is an illustration of Dandelin's sphere. This sphere is tangent to the plane containing E at one of its foci, and tangent to the right cone along a circle.

1st focal property :

H (E) lies in a plane perpendicular to E (H), along the main axis of E (H) and goes through the foci of E (H). That is, the foci of E and H are the vertices of H and E, respectively.

2nd focal property : Confocal quadrics

Every cone circumscribing a quadric of the family

Q : x² / (a² - µ) + y² / (b² - µ) + z²/(-µ) = 1
µ < 0 , then Q is an ellipsoid
0 < µ < b² , then Q is an hyperboloïd of 1 sheet
b² < µ < a² , then Q is an hyperboloïd of 2 sheets

is a right cone if its vertex is on E or H.

E and H are then called the focal conics of Q and this family of quadrics is called confocal.

The focal conics E and H can be viewed as degenerate members of the family Q, where µ = 0 and µ = -b², respectively.
E and H meet each quadric of the confocal family in its umbilical points.

Focal parabolas :

Pair of parabolas which lie in perpendicular planes, and such that the focus of each parabola is the vertex of the other one. One focus of each parabola is considered to be at infinity (w/r to the ellipse-hyperbola case).

Then, the vertices of right cones which are circumscribed about a paraboloid lie on such focal parabolas.

Degenerate cases:

Focal conics are also called "linked conics": A pair of plane conics that lie in perpendicular plane and such that each passes trough the foci of the other. One conic is the linked conic of the other, and vice-versa. Besides the ellipse-hyperbola and parabola-parabola pairs, the following degenerate cases are also possible:

Circle and straight line.
Single straight line
Single point
Empty set

Construction :

Generalized string construction of E with respect to H:

Attach a string on one branch of H at h .
Let the string slide over E at point e .
Attach the other end of the (tight) string on the 2nd branch of H at h0 .
Let the distance between h and e be d, and between e and h0 be d0.
Then, the sum d+d0 is constant for all points part of the conic E, i.e., it is independent of e on E.
N.B.: The angles between the tangent at e and either part of the string from h through e to h0 are equal, i.e., together the focal conics have a reflection property.
A straight line (or tight string) meeting both focal conics is called a focal ray (e.g. the rays between h and e and between e and h0).

From [Boehm94], p. 199 .

Applications :

Ellipse seen as a circle: Xah's Special Plane Curves - Conic Sections .

References :

[Boehm94], pp. 195-199.
[Hirst90], Blending Cones and Planes by Using Cyclides of Dupin .

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Page created & maintained by Frederic Leymarie, 1998-2001.
Comments, suggestions, etc., mail to: leymarie@lems.brown.edu