Conjugate diameters

Definitions of conjugate diameters of an ellipse:
top: KD1, middle: KD2, bottom KD3

In geometry, conjugate diameters are two diameters of an ellipse that have a special relationship to one another.

Here, diameter means a chord through the center point. If the ellipse is a circle, then two diameters are conjugate if and only if they are orthogonal.

The following equivalent definitions can be found in the literature:

• KD1: The tangents at the end points of a diameter of an ellipse are parallel. If the diameter is parallel to these tangents, then the reverse also applies: The tangents at the end points of are too parallel. ${\displaystyle d_{1}}$ ${\displaystyle d_{2}}$ ${\displaystyle d_{2}}$ ${\displaystyle d_{1}}$

Two diameters of an ellipse are called conjugate if the tangents at the end points of one diameter are parallel to the other diameter. ${\displaystyle d_{1},d_{2}}$

• KD2: The centers of the chords of an ellipse parallel to a diameter lie on a diameter . And vice versa: The midpoints of the tendons that are to be parallel lie on . ${\displaystyle d_{1}}$ ${\displaystyle d_{2}}$ ${\displaystyle d_{2}}$ ${\displaystyle d_{1}}$

Two diameters are called conjugate if the midpoints of the parallel chords lie on. ${\displaystyle d_{1},d_{2}}$ ${\displaystyle d_{1}}$ ${\displaystyle d_{2}}$

• KD3: If one understands an ellipse as an affine image of the unit circle, then the images of orthogonal circle diameters are called conjugated .

The proof of the properties in KD1 and KD2 results from the fact that any ellipse is an affine image of the unit circle (see Ellipse (Descriptive Geometry) ). Because the two properties are obviously correct for a circle, and an affine mapping maps center points onto center points and maintains parallelism.

The main axes of an ellipse are always conjugate.

Two conjugate radiuses of an ellipse are two half-diameters lying on different conjugate diameters.
Two conjugate points of an ellipse are two ellipse points lying on different conjugate diameters.
Two directions (vectors) are called conjugate if there is a parallel pair of conjugate diameters of the ellipse.

Conjugate diameters play an important role in the descriptive geometry of the Rytz axis construction (see ellipse in descriptive geometry ). The main axes of an ellipse are reconstructed from the knowledge of two conjugate radiuses.

• 

  Hyperbola: centers of parallel tendons

• 

  Parabola: centers of parallel tendons

Comment:

1. The centers of parallel chords of a hyperbola also lie on a straight line through the center. However, this straight line does not have to be a tendon, namely when the parallel tendons intersect both branches of the hyperbola. Therefore we only speak of conjugate directions here. When conjugate diameters are mentioned for a hyperbola, the diameter means a diameter of the given hyperbola or of the hyperbola conjugated to it. (The hyperbola conjugated to the hyperbola has the equation .) ${\displaystyle {\tfrac {x^{2}}{a^{2}}}-{\tfrac {y^{2}}{b^{2}}}=1}$ ${\displaystyle -{\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1}$
2. The centers of parallel chords of a parabola also lie on a straight line. This straight line is always parallel to the parabolic axis (see picture). Since a parabola does not have a center point, one does not speak of conjugate diameters here. Sometimes a parallel to the parabolic axis is called a diameter.

Calculation of conjugate points of an ellipse

The tangent to the ellipse at the point has the equation (see ellipse ). A point to be conjugated must lie on the straight line parallel to the tangent through the zero point (center point). So it applies ${\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1}$ ${\displaystyle (x_{1},y_{1})}$ ${\displaystyle {\frac {x_{1}x}{a^{2}}}+{\frac {y_{1}y}{b^{2}}}=1}$ ${\displaystyle (x_{1},y_{1})}$ ${\displaystyle (x_{2},y_{2})}$

• Two points of the ellipse are conjugated if and only if the equation ${\displaystyle (x_{1},y_{1}),(x_{2},y_{2})}$ ${\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1}$

${\displaystyle {\frac {x_{1}x_{2}}{a^{2}}}+{\frac {y_{1}y_{2}}{b^{2}}}=0}$

is satisfied.

If , d. H. the ellipse is a circle, two conjugate points belong to two orthogonal radiuses and the last equation has the familiar form (scalar product = 0). ${\displaystyle a=b}$ ${\displaystyle x_{1}x_{2}+y_{1}y_{2}=0}$

Is the ellipse through the parametric representation

${\displaystyle x(t)=a\cos t,\quad y(t)=b\sin t}$

given d. H. as an affine image of the unit circle , the points, as images of orthogonal radii of the unit circle, belong to conjugate points of the ellipse. With the help of the addition theorems it follows: ${\displaystyle (\cos t,\sin t),\ 0\leq t<2\pi }$ ${\displaystyle (x(t),y(t),\ (x(t\pm {\tfrac {\pi }{2}}),y(t\pm {\tfrac {\pi }{2}})}$

• The two points are conjugated to the point . ${\displaystyle (-a\sin t,b\cos t),\ (a\sin t,-b\cos t)}$ ${\displaystyle (a\cos t,b\sin t)}$

Connection with orthogonality relations

The previous section has shown that the ellipse is directly related to the symmetrical bilinear shape ${\displaystyle {\tfrac {x^{2}}{a^{2}}}+{\tfrac {y^{2}}{b^{2}}}=1}$

• ${\displaystyle \ f((x_{1},y_{1}),(x_{2},y_{2}))={\frac {x_{1}x_{2}}{a^{2}}}+{\frac {y_{1}y_{2}}{b^{2}}}}$

related. This bilinear form defines

• on an orthogonality relation : ${\displaystyle \mathbb {R} ^{2}}$

${\displaystyle (x_{1},y_{1}),(x_{2},y_{2})\neq (0,0)}$are orthogonal if and only if is, and ${\displaystyle f((x_{1},y_{1}),(x_{2},y_{2}))={\frac {x_{1}x_{2}}{a^{2}}}+{\frac {y_{1}y_{2}}{b^{2}}}=0}$

• on a metric : ${\displaystyle \mathbb {R} ^{2}}$

${\displaystyle f((x,y),(x,y))={\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}}$ is the length of the vector and ${\displaystyle (x,y)}$

• an elliptical polarity on the straight line . ( Elliptical means here: the polarity has no fixed points. This is equivalent to no vector being orthogonal to itself )

Two conjugate directions are thus orthogonal in the sense defined here and the given ellipse is the "unit circle" with regard to the metric defined here.

Remark 1: The hyperbola leads to the symmetrical bilinear form with analogous considerations ${\displaystyle {\tfrac {x^{2}}{a^{2}}}-{\tfrac {y^{2}}{b^{2}}}=1}$

• ${\displaystyle \ f((x_{1},y_{1}),(x_{2},y_{2}))={\frac {x_{1}x_{2}}{a^{2}}}-{\frac {y_{1}y_{2}}{b^{2}}}}$.

Here, too, you can define an orthogonality relation and a metric. The special thing about this case is: There are directions that are orthogonal to themselves, namely the asymptote directions, and there are vectors of length 0 other than (0,0)! This metric is also called the Minkowski metric and the associated "circles" (= hyperbolas) Minkowski circles or pseudo-Euclidean circles . This case plays an essential role in the theory of relativity . On the distance line, the bilinear shape induces a hyperbolic polarity . (Here, hyperbolic means: the polarity has two fixed points.)

Remark 2: If one tries analogous considerations for a parabola, this leads to an "unusable" orthogonality relation. In this case all directions would be orthogonal to the direction of the parabolic axis.

See also

• Theorem of Apollonios

Individual evidence

1. ^ Ulrich Graf, Martin Barner: Descriptive Geometry. Quelle & Meyer, Heidelberg 1961, ISBN 3-494-00488-9 , p. 106.
2. ↑ C. Leopold: Geometric Basics of Architectural Representation. Verlag W. Kohlhammer, Stuttgart 2005, ISBN 3-17-018489-X , p. 61.
3. ↑ Rudolf Fucke, Konrad Kirch, Heinz Nickel: Descriptive Geometry. Fachbuch-Verlag, Leipzig 1998, ISBN 3-446-00778-4 , p. 183.