Confocal conic sections

Confocal array of ellipses and hyperbolas

In geometry two conics are called confocal if they have the same focal points . Since ellipses and hyperbolas each have two focal points, there are confocal ellipses, confocal hyperbolas, and confocal ellipses and hyperbolas. Confocal ellipses and hyperbolas have the remarkable property: Each ellipse intersects each hyperbola perpendicularly (see below). Parabolas each have only one focal point. Confocal parabolas have the same focus and the same axis of symmetry. Due to this convention, exactly two confocal parabolas go through each point that is not on the symmetry axis, which intersect at right angles (see below).

A formal continuation of the concept of confocal conic sections on surfaces leads to the confocal quadrics.

Confocal ellipses

An ellipse that is not a circle is uniquely defined by its two focal points and a point that does not lie on the line between the focal points (see definition of an ellipse ). The family of confocal ellipses with the focal points can be expressed by the equation ${\displaystyle F_{1},\;F_{2}}$ ${\displaystyle F_{1}=(e,0),\;F_{2}=(-e,0)}$

• ${\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{a^{2}-e^{2}}}=1\ ,\quad a>e>0}$

describe with the coulter parameter . (The linear eccentricity and the center are clearly determined by the focal points.) Since an elliptical point clearly determines the array parameter , the following applies: ${\displaystyle a}$ ${\displaystyle e}$ ${\displaystyle a}$

• Any two ellipses in this family do not intersect.

A geometric figure bordered by two confocal ellipses is called a 2-dimensional focaloid .

Confocal hyperbolas

A hyperbola is uniquely defined by its two focal points and a point that is not on the minor axis and not on the main axis outside the line (see definition of a hyperbola ). The family of confocal hyperbolas with the focal points can be expressed by the equation ${\displaystyle F_{1},\;F_{2}}$ ${\displaystyle {\overline {F_{1}F_{2}}}}$ ${\displaystyle F_{1}=(e,0),\;F_{2}=(-e,0)}$

• ${\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{e^{2}-a^{2}}}=1\ ,\quad 0<a<e}$

describe with the coulter parameter . (The linear eccentricity is clearly determined by the focal points.) Since a hyperbolic point clearly determines the cluster parameter , the following applies: ${\displaystyle a}$ ${\displaystyle e}$ ${\displaystyle a}$

• Any two hyperbolas of this family do not intersect.

Confocal ellipses and hyperbolas

Common description

The above representations of the confocal ellipses and hyperbolas result in a common representation: the equation

• ${\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{a^{2}-e^{2}}}=1}$

describes an ellipse if and a hyperbola if is. ${\displaystyle 0<e<a}$ ${\displaystyle 0<a<e}$

Another representation common in literature is:

• ${\displaystyle {\frac {x^{2}}{a^{2}-\lambda }}+{\frac {y^{2}}{b^{2}-\lambda }}=1\ .}$

Here, the semi-axes of a predetermined ellipse (so are the focal points shown) and the value of coefficient. For one obtains confocal ellipses (it is ) and for confocal hyperbolas with the common focal points . ${\displaystyle a,b}$ ${\displaystyle F_{1},\;F_{2}}$ ${\displaystyle \lambda }$
${\displaystyle \lambda <b^{2}}$ ${\displaystyle a^{2}-\lambda -(b^{2}-\lambda )=e^{2}}$
${\displaystyle b^{2}<\lambda <a^{2}}$ ${\displaystyle F_{1},\;F_{2}}$

Limit curves

Limit curves (blue line, red rays) for resp. ${\displaystyle \lambda {\color {blue}\nearrow }b^{2}\;}$ ${\displaystyle \;\lambda {\color {red}\searrow }b^{2}}$

At this point , the family has the line on the x-axis as the left-hand limit curve (infinitely thin ellipse) and the two rays as the right-hand limit curve (infinitely thin hyperbola) . So: ${\displaystyle \lambda =b^{2}}$ ${\displaystyle [-e,e]}$ ${\displaystyle (-\infty ,-e],[e,\infty )}$

• The two limit curves at this point have the two focal points in common. ${\displaystyle \lambda =b^{2}}$ ${\displaystyle F_{1}=(-e,0),F_{2}=(e,0)}$

This property finds its analog in the 3-dimensional case (see below) and leads to the definition of the (infinitely many) focal points, called focal curves, of confocal quadrics.

At this point the flock has an infinitely steep hyperbola as the left-hand limit curve, where both branches fall on the y-axis. There is no right-hand limit curve here, as there are no solutions for the conic section equation . ${\displaystyle a^{2}}$ ${\displaystyle a^{2}<\lambda }$

Double orthogonal system

Confocal ellipses and hyperbolas intersect perpendicularly: proof.

If one looks at the family of ellipses and hyperbolae belonging to two focal points (see first picture), it follows from the respective focal point property (the tangent of an ellipse halves the outer angle of the focal rays, the tangent of a hyperbola halves the angle of the focal rays.):

• Each ellipse of the family intersects each hyperbola of the family vertically (see picture).

This statement can also be proven arithmetically by interpreting the ellipses and hyperbolas as implicit curves and calculating normal vectors at the intersections with the help of partial derivatives (see section Ivory's Theorem ).

The plane can therefore be covered with an orthogonal network of confocal ellipses and hyperbolas.

The orthogonal network of confocal ellipses and hyperbolas is the basis of the elliptical coordinates .

Confocal parabolas

Family of confocal parabolas

Parabolas each have only one focal point. A parabola can be understood as a boundary curve of a family of ellipses (family of hyperbolas), in which one focal point is fixed and the second wanders into infinity. If you carry out this border crossing for a network of ellipses and hyperbolas (see 1st picture), you get a network of two families of confocal parabolas.

If one shifts the parabola with the equation in order in direction, one obtains the equation of a parabola having the origin as a focal point. The following applies: ${\displaystyle y^{2}=2px}$ ${\displaystyle p/2}$ ${\displaystyle x}$ ${\displaystyle y^{2}=2p(x+p/2)=2px+p^{2}}$

• ${\displaystyle y^{2}=2px+p^{2}\ ,\quad p>0\ ,}$are right- opened parabolas and

${\displaystyle y^{2}=-2qx+q^{2}\ ,\quad q>0\ ,}$are parabolas open to the left

with the common focus ${\displaystyle F=(0,0)\ .}$

The guideline definition of a parabola shows:

• The parabolas open to the right (left) do not intersect.

One calculates:

• Every parabola opened to the right intersects every parabola opened to the left at the two points vertically (see picture). ${\displaystyle y^{2}=2px+p^{2}}$ ${\displaystyle y^{2}=-2qx+q^{2}}$ ${\displaystyle ({\tfrac {q-p}{2}},\pm {\sqrt {pq}})\ }$

( are normal vectors at the points of intersection. Their scalar product is .) ${\displaystyle {\vec {n}}_{1}=\left(p,\mp {\sqrt {pq}}\right)^{T},\ {\vec {n}}_{2}=\left(q,\pm {\sqrt {pq}}\right)^{T})}$ ${\displaystyle 0}$

The plane can therefore be covered with an orthogonal network of confocal parabolas (analogous to confocal ellipses and hyperbolas).

Note: The network of confocal parabolas can be understood as the image of an axis-parallel network of the right half-plane under the conformal mapping of the complex plane (see web link). ${\displaystyle w=z^{2}}$

Parabolic coordinates:
Each point in the upper half-plane is the intersection of two confocal parabolas . If you introduce new parameters in such a way that is, you get for the intersection point (see above): ${\displaystyle y^{2}=2px+p^{2},\;y^{2}=-2qx+q^{2},\;p,q>0}$ ${\displaystyle u,\;v\;\geq 0}$ ${\displaystyle p=u^{2},\;v=q^{2}}$

• ${\displaystyle x={\frac {v^{2}-u^{2}}{2}},\quad y=u\;v\ .}$

${\displaystyle u,\;v}$are called parabolic coordinates of the point (see parabolic cylindrical coordinates ). ${\displaystyle (x,y)}$

Graves' theorem: Thread construction of confocal ellipses

Thread construction of a confocal ellipse

In 1850, the Irish Bishop of Limerick Charles Graves ( en ) indicated and proved the following thread construction of an ellipse confocal to a given ellipse:

• If you loop a closed thread around a given ellipse E, which is longer than the circumference of E, and draw a curve as in the Gärtner construction of an ellipse (see picture), this curve is an ellipse that is confocal to E.

The proof of this theorem uses elliptic integrals and is contained in the book by Felix Klein . Otto Staude generalized this method for the construction of confocal ellipsoids (see book by F. Klein).

If the given ellipse degenerates into the distance between the two focal points , a variant of the gardener's construction of an ellipse is obtained: A closed thread of the length is then looped around the focal points. ${\displaystyle F_{1},F_{2}}$ ${\displaystyle 2a+|F_{1}F_{2}|}$

Confocal Quadrics

Confocal quadrics: (red), (blue), (purple)
${\displaystyle a=1,\;b=0{,}8,\;c=0{,}6,}$
${\displaystyle \lambda _{1}=0{,}1}$ ${\displaystyle \;\lambda _{2}=0{,}5}$ ${\displaystyle \;\lambda _{3}=0{,}8}$

Confocal quadrics as a function of ${\displaystyle \lambda }$

definition

Confocal quadrics are a formal continuation of the concept of confocal conic sections in 3-dimensional space:

Let with selected fixed. Then describe the equation ${\displaystyle a,b,c}$ ${\displaystyle a>b>c>0}$

• ${\displaystyle {\frac {x^{2}}{a^{2}-\lambda }}+{\frac {y^{2}}{b^{2}-\lambda }}+{\frac {z^{2}}{c^{2}-\lambda }}=1}$

for an ellipsoid , ${\displaystyle \lambda <c^{2}}$

for a single-shell hyperboloid (blue in the picture), ${\displaystyle c^{2}<\lambda <b^{2}}$

for a two-shell hyperboloid. ${\displaystyle b^{2}<\lambda <a^{2}}$

For the equation has no solution. ${\displaystyle a^{2}<\lambda }$

Focal curves

Focal conic sections (ellipse, hyperbola, black)

${\displaystyle c^{2}=0.36,\ b^{2}=0.64,\quad }$above: (ellipsoid, red), (1sch. hyperb., blue), (1sch. hyperb., blue), (2sch. hyperb., purple) below: interfaces between the types ${\displaystyle \lambda =}$
${\displaystyle 0.3575}$ ${\displaystyle \ 0.3625}$
${\displaystyle 0.638}$ ${\displaystyle \ 0.642}$

Interfaces for : ${\displaystyle \lambda \to c^{2}}$

If in the family of ellipsoids ( ) the family parameter (from below!) Runs counter to one another, an "infinitely flat" ellipsoid is obtained, more precisely: the area in the xy-plane, that from the ellipse with the equation and its double occupied interior (in the picture below, dark red). If the group parameter (from above!) Is allowed to run counter to the group of single-shell hyperboloids ( ) , an "infinitely flat" hyperboloid is obtained, more precisely: the area in the xy-plane, which consists of the same ellipse and its double-occupied exterior (in Picture below, blue). Ie: ${\displaystyle \lambda <c^{2}}$ ${\displaystyle \lambda }$ ${\displaystyle c^{2}}$ ${\displaystyle E}$ ${\displaystyle {\tfrac {x^{2}}{a^{2}-c^{2}}}+{\tfrac {y^{2}}{b^{2}-c^{2}}}=1}$
${\displaystyle \lambda >c^{2}}$ ${\displaystyle \lambda }$ ${\displaystyle c^{2}}$ ${\displaystyle E}$

The two interfaces at the point have the ellipse ${\displaystyle c^{2}}$

${\displaystyle E:{\frac {x^{2}}{a^{2}-c^{2}}}+{\frac {y^{2}}{b^{2}-c^{2}}}=1}$

together.

Interfaces for : ${\displaystyle \lambda \to b^{2}}$

Analogous considerations at this point result in: ${\displaystyle \lambda =b^{2}}$

The two interfaces (in the picture below right, blue and purple) at the point have the hyperbola ${\displaystyle b^{2}}$

${\displaystyle H:\ {\frac {x^{2}}{a^{2}-b^{2}}}-{\frac {z^{2}}{b^{2}-c^{2}}}=1}$

together.

Focal Curves:

It is easy to check that the foci of the ellipse are the vertices of the hyperbola and vice versa. Ie: Ellipse and hyperbola are focal conics . Conversely, since each of the focal conic sections defined by a certain quadric of the confocal family can be constructed with the help of a thread of suitable length, analogous to the thread construction of an ellipse, the focal curves are called the confocal quadrics defined by a certain family. (see thread construction of a 3-axis ellipsoid .) ${\displaystyle E}$ ${\displaystyle H}$ ${\displaystyle E,H}$ ${\displaystyle a,b,c,\lambda }$ ${\displaystyle E,H}$ ${\displaystyle a,b,c}$

Triple orthogonal system

Analogous to the property of confocal ellipses / hyperbolas, the following applies:

• Exactly one area of each of the three types goes through each point with . ${\displaystyle (x_{0},y_{0},z_{0})\in \mathbb {R} ^{3}}$ ${\displaystyle x_{0}\neq 0,\;y_{0}\neq 0,\;z_{0}\neq 0}$

The three quadrics intersect there perpendicularly. ${\displaystyle (x_{0},y_{0},z_{0})}$

Example of a function ${\displaystyle f(\lambda )}$

Proof of the existence of three quadrics through a point:
For a point with let . This function has the 3 poles and is steadily and strictly monotonically increasing in each of the open intervals . From the behavior of the function in the vicinity of the poles and for one recognizes (see picture): The function has exactly 3 zeros with ${\displaystyle (x_{0},y_{0},z_{0})}$ ${\displaystyle x_{0}\neq 0,y_{0}\neq 0,z_{0}\neq 0}$ ${\displaystyle f(\lambda )={\tfrac {x_{0}^{2}}{a^{2}-\lambda }}+{\tfrac {y_{0}^{2}}{b^{2}-\lambda }}+{\tfrac {z_{0}^{2}}{c^{2}-\lambda }}-1}$ ${\displaystyle c^{2}<b^{2}<a^{2}}$ ${\displaystyle (-\infty ,c^{2}),\;(c^{2},b^{2}),\;(b^{2},a^{2}),\;(a^{2},\infty )}$ ${\displaystyle \lambda \to \pm \infty }$
${\displaystyle f}$ ${\displaystyle \lambda _{1},\lambda _{2},\lambda _{3}}$ ${\displaystyle {\color {red}\lambda _{1}}<c^{2}<{\color {red}\lambda _{2}}<b^{2}<{\color {red}\lambda _{3}}<a^{2}\ .}$

Proof of the orthogonality of the surfaces:
With the help of the family of functions with the family parameter , the confocal quadrics can be described by. For two intersecting quadrics with results in a common point ${\displaystyle F_{\lambda }(x,y,z)={\tfrac {x^{2}}{a^{2}-\lambda }}+{\tfrac {y^{2}}{b^{2}-\lambda }}+{\tfrac {z^{2}}{c^{2}-\lambda }}}$ ${\displaystyle \lambda }$ ${\displaystyle F_{\lambda }(x,y,z)=1}$ ${\displaystyle F_{\lambda _{i}}(x,y,z)=1,\;F_{\lambda _{k}}(x,y,z)=1}$ ${\displaystyle (x,y,z)}$

${\displaystyle 0=F_{\lambda _{i}}(x,y,z)-F_{\lambda _{k}}(x,y,z)=\dotsb }$

${\displaystyle \quad =(\lambda _{i}-\lambda _{k})\left({\frac {x^{2}}{(a^{2}-\lambda _{i})(a^{2}-\lambda _{k})}}+{\frac {y^{2}}{(b^{2}-\lambda _{i})(b^{2}-\lambda _{k})}}+{\frac {z^{2}}{(c^{2}-\lambda _{i})(c^{2}-\lambda _{k})}}\right)\ .}$

For the scalar product of the gradients in a common point it follows

${\displaystyle \operatorname {grad} F_{\lambda _{i}}\cdot \operatorname {grad} F_{\lambda _{k}}=4\;\left({\frac {x^{2}}{(a^{2}-\lambda _{i})(a^{2}-\lambda _{k})}}+{\frac {y^{2}}{(b^{2}-\lambda _{i})(b^{2}-\lambda _{k})}}+{\frac {z^{2}}{(c^{2}-\lambda _{i})(c^{2}-\lambda _{k})}}\right)=0\ .}$

Ellipsoid with lines of curvature as sections with confocal hyperboloids
${\displaystyle a=1,\;b=0{,}8,\;c=0{,}6}$

Applications:
According to a theorem by Charles Dupin about three-way orthogonal surface systems, the following applies (see web links):

• Two different confocal quadrics each intersect in lines of curvature .
• Analogous to the (plane) elliptical coordinates, there are ellipsoid coordinates in space . The simplest form of ellipsoidal coordinates uses the parameters associated with a point in the confocal quadrics through that point. ${\displaystyle (x_{0},y_{0},z_{0})}$ ${\displaystyle \lambda _{1},\lambda _{2},\lambda _{3}}$

In physics , confocal ellipsoids play a role as equipotential surfaces :

• The equipotential areas of the free charge distribution on an ellipsoid are the confocal ellipsoids.

Ivory's theorem

Confocal Conic Sections: Ivory's Theorem

The sentence of the Scottish mathematician and astronomer James Ivory (1765–1842) makes a statement about diagonals in a "network rectangle" (a square formed by orthogonal curves):

• In each square of a network of confocal ellipses and hyperbolas, which is formed by two ellipses and two hyperbolas, the diagonals are of equal length (see picture).

Points of intersection of an ellipse with a confocal hyperbola: be the ellipse with the focal points and the equation
${\displaystyle E(a)}$ ${\displaystyle F_{1}=(e,0),\;F_{2}=(-e,0)}$

${\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{a^{2}-e^{2}}}=1\ ,\quad a>e>0\ .}$

${\displaystyle H(u)}$ be the confocal hyperbola with the equation

${\displaystyle {\frac {x^{2}}{u^{2}}}+{\frac {y^{2}}{u^{2}-e^{2}}}=1\ ,\quad e>u\ .}$

If you calculate the points of intersection of and , the following four points result: ${\displaystyle E(a)}$ ${\displaystyle H(u)}$

• ${\displaystyle \left(\pm {\frac {au}{e}},\;\pm {\frac {\sqrt {(a^{2}-e^{2})(e^{2}-u^{2})}}{e}}\right)}$

Diagonals in a square grid:
In order to keep the calculation clear, it is assumed here that

1. ${\displaystyle e=1}$ is, which is not a major limitation, since all other confocal networks can be obtained by scaling (stretching at the center).
   
2. Of the alternatives (at the intersections) is only used. At the end of the day, you consider that other sketching combinations lead to the same result. ${\displaystyle \pm }$ ${\displaystyle +}$

Let there be two confocal ellipses and two confocal hyperbolas . The diagonals of the grid square are made up of the 4 intersection points ${\displaystyle E(a_{1}),E(a_{2})}$ ${\displaystyle H(u_{1}),H(u_{2})}$

${\displaystyle P_{11}=\left(a_{1}u_{1},\;{\sqrt {(a_{1}^{2}-1)(1-u_{1}^{2})}}\right)\ ,\quad P_{22}=\left(a_{2}u_{2},\;{\sqrt {(a_{2}^{2}-1)(1-u_{2}^{2})}}\right)\ ,}$

${\displaystyle P_{12}=\left(a_{1}u_{2},\;{\sqrt {(a_{1}^{2}-1)(1-u_{2}^{2})}}\right)\ ,\quad P_{21}=\left(a_{2}u_{1},\;{\sqrt {(a_{2}^{2}-1)(1-u_{1}^{2})}}\right)}$

calculated:

${\displaystyle {\begin{aligned}|P_{11}P_{22}|^{2}&=(a_{2}u_{2}-a_{1}u_{1})^{2}+\left({\sqrt {(a_{2}^{2}-1)(1-u_{2}^{2})}}-{\sqrt {(a_{1}^{2}-1)(1-u_{1}^{2})}}\right)^{2}=\dotsb \\&=a_{1}^{2}+a_{2}^{2}+u_{1}^{2}+u_{2}^{2}-2\,\left(1+a_{1}a_{2}u_{1}u_{2}+{\sqrt {(a_{1}^{2}-1)(a_{2}^{2}-1)(1-u_{1}^{2})(1-u_{2}^{2})}}\right)\end{aligned}}}$

The last term is obviously invariant to the interchange . Exactly this exchange leads to . So: ${\displaystyle u_{1}\leftrightarrow u_{2}}$ ${\displaystyle |P_{1\color {red}2}P_{2\color {red}1}|^{2}}$

• ${\displaystyle |P_{11}P_{22}|=|P_{12}P_{21}|}$

The validity of Ivory's theorem for confocal parabolas can also be easily proven mathematically.

Ivory has also proven the 3-dimensional form of the sentence (see Blaschke, p. 111):

• In a "curved parallelepiped" delimited by confocal quadrics, the 4 diagonals of opposite points are of equal length. This also applies to each pair of diagonals on the side surfaces.

See also

• Focaloid

Individual evidence

1. ↑ AM Schoenflies , Max Dehn : Introduction to the Analytical Geometry of Plane and Space. Basic teachings of math. Knowledge in Einzeleld., Volume XXI, Springer, Berlin 1931, p. 135.
2. ^ Felix Klein : Lectures on Higher Geometry. Springer-Verlag, Berlin 1926, p. 32.
3. ^ Felix Klein: Lectures on Higher Geometry , Springer-Verlag, 2013, ISBN 3642886744 , p. 20.
4. ^ Staude, O .: About thread constructions of the ellipsoid . Math. Ann. 20, 147-184 (1882)
5. ↑ Staude, O .: About new focal properties of surfaces of the 2nd degree. Math. Ann. 27: 253-271 (1886).
6. ↑ Staude, O .: The algebraic fundamentals of the focal properties of the 2nd order surfaces Math. Ann. 50, 398-428 (1898)
7. ↑ D. Fuchs , S. Tabachnikov : A diagram of mathematics. Springer-Verlag, Berlin / Heidelberg 2011, ISBN 978-3-642-12959-9 , p. 480.

literature

• W. Blaschke : Analytical geometry. Springer, Basel 1954, ISBN 978-3-0348-6813-6 , p. 111.
• D. Hilbert , S. Cohn-Vossen : Illustrative geometry. Springer, 1996, ISBN 978-3-642-19947-9 .
• Horst Knörrer : Geometry. Vieweg-Verlag, Wiesbaden 2006, ISBN 978-3-8348-0210-1 , p. 198.
• E. Pascal : Repertory of higher mathematics. Teubner, Leipzig / Berlin 1910, p. 257.
• R. Resel : Journey to the Center of Mathematics. Logos-Verlag, Berlin, ISBN 978-3-8325-3672-5 , p. 259.
• K. Baer : Parabolic coordinates in the plane and in space. Royal Hofbuchdruckerei Trowizsch u. Son, Frankfurt a. O. 1888.

Web links

• T. Hoffmann : Miniskript Differentialgeometrie I. S. 48 ff.
• B. Springborn : Curves and Surfaces, 12th Lecture: Confocal Quadrics. P. 22 f.
• H. Walser : Conformal illustrations. P. 8.