confocal conic sections
{{Short description|Conic sections with the same foci}}
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.
Confocal ellipses and hyperbolas
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 {{slink|Ellipse#Definition as locus of points}} and {{slink|Hyperbola#As_locus_of_points}}.)
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 {{nobr|),}} then in this coordinate system
File:Confocal conics parametrized by semi-major axis.png
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 {{nobr|(
Another common representation specifies a pencil of ellipses and hyperbolas confocal with a given ellipse of semi-major axis
:
If
= Limit curves =
As the parameter
This property appears analogously in the 3-dimensional case, leading to the definition of the focal curves of confocal quadrics. See {{slink|#Confocal quadrics}} below.
= Twofold orthogonal system =
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). Any ellipse of the pencil intersects any hyperbola orthogonally (see diagram).
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
File:Parabola as a limit of ellipses and hyperbolas.png
A parabola has only one focus, and can be considered as a limit curve of a set of ellipses (or a set of hyperbolas), 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 {{mvar|x}}-axis as its axis of symmetry is the locus of points satisfying the equation
:
for some value of the parameter
From the definition of a parabola, for any 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
{{clear|left}}
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.{{sfn|Hilbert|Cohn-Vossen|1952|p=[https://archive.org/details/geometryimaginat0000dhil/page/6/ 6]}}
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:Felix Klein: Vorlesungen über Höhere Geometrie, Sringer-Verlag, Berlin, 1926, S.32.
: If one surrounds a given ellipse E by a closed string, which is longer than the given ellipse's circumference, and draws a curve similar to the gardener's construction of an ellipse (see diagram), then one gets an ellipse, that is confocal to E.
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).
If ellipse E collapses to a line segment
Confocal quadrics
[[File:Quadriken-konfok.svg|thumb|upright=1.2|Confocal quadrics:
File:Ello-hypb1-hypb2-lambda.svg
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
:
If
= Focal curves =
[[File:Quadriken-konf-fk-ex-gf.svg|thumb|upright=1.75|
top:
bottom: Limit surfaces between the types]]
Limit surfaces for
As the parameter
:
and its doubly covered interior (in the diagram: below, on the left, red).
As
The two limit surfaces have the points of ellipse
Limit surfaces for
Similarly, as
:
in common.
Focal curves:
The foci of the ellipse
Reverse: Because any quadric of the pencil of confocal quadrics determined by
= Threefold orthogonal system =
Analogous to the case of confocal ellipses/hyperbolas,
: Any point
: The three quadrics through a point
Proof of the existence and uniqueness of three quadrics through a point:
For a point
This function has three vertical asymptotes
Function
Proof of the orthogonality of the surfaces:
Using the pencils of functions
with parameter
:
:
From this equation one gets for the scalar product of the gradients at a common point
:
which proves the orthogonality.
[[File:Ellipsoid-kl.svg|thumb|Ellipsoid with lines of curvature as intersection curves with confocal hyperboloids
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.D. Fuchs, S. Tabachnikov: Ein Schaubild der Mathematik. Springer-Verlag, Berlin/Heidelberg 2011, {{ISBN|978-3-642-12959-9}}, p. 480.
Ivory's theorem
Ivory's theorem (or Ivory's lemma),Ivory used it as a lemma in proving the theorem that equipotential surfaces of the gravitational field external to a homogeneous triaxial ellipsoid are the confocal ellipsoids. named after the Scottish mathematician and astronomer James Ivory (1765–1842), is a statement on the diagonals of a net-rectangle, a quadrangle formed by orthogonal curves:
: For any net-rectangle, which is formed by two confocal ellipses and two confocal hyperbolas with the same foci, the diagonals have equal length (see diagram).
Intersection points of an ellipse and a confocal hyperbola:
Let
:
and
:
Computing the intersection points of
:
Diagonals of a net-rectangle:
To simplify the calculation, let
Let be
:
P_{11}&=\left(a_1u_1,\; \sqrt{(a_1^2-1)(1-u_1^2)}\right), &
P_{22}&=\left(a_2u_2,\; \sqrt{(a_2^2-1)(1-u_2^2)}\right), \\[5mu]
P_{12}&=\left(a_1u_2,\; \sqrt{(a_1^2-1)(1-u_2^2)}\right), &
P_{21}&=\left(a_2u_1,\; \sqrt{(a_2^2-1)(1-u_1^2)}\right)
\end{align}
are:
:
|P_{11}P_{22}|^2 &= (a_2u_2-a_1u_1)^2+\left(\sqrt{(a_2^2-1)(1-u_2^2)}-\sqrt{(a_1^2-1)(1-u_1^2)}\right)^2 \\[5mu]
&= a_1^2+a_2^2+u_1^2+u_2^2 - 2\left(1+a_1a_2u_1u_2+\sqrt{(a_1^2-1)(a_2^2-1)(1-u_1^2)(1-u_2^2)}\right)
\end{align}
The last expression is invariant under the exchange
The proof of the statement for confocal parabolas is a simple calculation.
Ivory even proved the 3-dimensional version of his theorem (s. Blaschke, p. 111):
:For a 3-dimensional rectangular cuboid formed by confocal quadrics the diagonals connecting opposite points have equal length.
See also
References
- {{cite book
| last = Blaschke
| first = Wilhelm
| author-link = Wilhelm Blaschke
| year = 1954
| title = Analytische Geometrie
| lang = de
| trans-title = Analytic Geometry
| publisher = Springer
| place = Basel
| chapter = VI. Konfokale Quadriken
| trans-chapter = Confocal Quadrics
| pages = 108–132
| chapter-url = https://archive.org/details/analytischegeome0000wilh/page/108
| chapter-url-access = limited
}}
- {{cite book
| last1 = Glaeser
| first1 = Georg | author1-link = Georg Glaeser
| last2 = Stachel
| first2 = Hellmuth
| last3 = Odehnal
| first3 = Boris
| year = 2016
| title = The Universe of Conics
| chapter = 2. Euclidean Plane
| pages = 11–60
| publisher = Springer
| doi = 10.1007/978-3-662-45450-3_2
| isbn = 978-3-662-45449-7
}} See also "10. Other Geometries", {{doi|10.1007/978-3-662-45450-3_10}}.
- {{citation
| last1 = Hilbert
| first1 = David
| author1-link = David Hilbert
| last2 = Cohn-Vossen
| first2 = Stephan
| author2-link = Stephan Cohn-Vossen
| year = 1952
| title = Geometry and the Imagination
| chapter = §1.4 The Thread Construction of the Ellipsoid, and Confocal Quadrics
| pages = 19–25
| chapter-url = https://archive.org/details/geometryimaginat0000dhil/page/19
| chapter-url-access = limited
| publisher = Chelsea
}}
- {{cite book
| last1 = Odehnal
| first1 = Boris
| last2 = Stachel
| first2 = Hellmuth
| last3 = Glaeser
| first3 = Georg | author3-link = Georg Glaeser
| year = 2020
| title = The Universe of Quadrics
| chapter = 7. Confocal Quadrics
| pages = 279–325
| publisher = Springer
| doi = 10.1007/978-3-662-61053-4_7
| isbn = 978-3-662-61052-7
| s2cid = 242527367
}}
- Ernesto Pascal: Repertorium der höheren Mathematik. Teubner, Leipzig/Berlin 1910, p. 257.
- A. Robson: An Introduction to Analytical Geometry Vo. I, Cambridge, University Press, 1940, p. 157.
- {{cite book
| last = Sommerville
| first = Duncan MacLaren Young
| author-link = Duncan Sommerville
| year = 1934
| title = Analytical Geometry of Three Dimensions
| publisher = Cambridge University Press
| chapter = XII. Foci and Focal Properties
| chapter-url = https://archive.org/details/analyticalgeomet0000somm/page/224/
| chapter-url-access = limited
| pages = 224–250
}}
External links
- T. Hofmann: [https://www-m10.ma.tum.de/foswiki/pub/Lehre/DifferentialGeometrieWS0607/VL01-19_dg1_06.pdfMiniskript Differentialgeometrie I, p. 48]
- B. Springborn: [https://www3.math.tu-berlin.de/geometrie/Lehre/SS06/KundF/Blaetter_Skript/miniskript-kuf-SS06.pdf Kurven und Flächen, 12. Vorlesung: Konfokale Quadriken] (S. 22 f.).
- H. Walser: [http://e-collection.library.ethz.ch/eserv/eth:25629/eth-25629-08.pdf Konforme Abbildungen.] p. 8.