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Concept
Confocal conic sections
Résumé
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 (at right angles).
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 (see below).
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.
Any hyperbola or (non-circular) 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 (See and .)
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 (half the distance between and then in this coordinate system
Each ellipse or hyperbola in the pencil is the locus of points satisfying the equation
with semi-major axis as parameter. If the semi-major axis is less than the linear eccentricity the equation defines a hyperbola, while if the semi-major axis is greater than the linear eccentricity it defines an ellipse.
Another common representation specifies a pencil of ellipses and hyperbolas confocal with a given ellipse of semi-major axis and semi-minor axis (so that each conic generated by choice of the parameter
If the conic is an ellipse.
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