Theorem of Apollonios

This article deals with Apollonios' theorem in the geometry of ellipses. For further sentences of Apollonios see Theorem of Apollonios .

To a theorem by Apollonios about conjugate diameters / radii of an ellipse

The set of Apollonius (or set of Apollonius ) is a classic theorem of analytical geometry , one of the part areas of mathematics . It goes back to the ancient Greek mathematician Apollonios von Perge and deals with the metric properties of the conjugate diameters and radiuses of the ellipses in the Euclidean plane .

Formulation of the sentence

The sentence consists of two sub-clauses, which are also called the first and second movements of Apollonios and which are to be stated as follows:

Given is an ellipse of the Euclidean plane with major and minor axes of the lengths . ${\displaystyle E}$ ${\displaystyle 2a,2b>0}$

Then:

First theorem of Apollonios : For each pair of conjugate diameters and radiuses of the ellipse , the sum of squares of the respective lengths is always the same. The length always applies to a pair of conjugate radiuses  . ${\displaystyle E}$ ${\displaystyle c_{1},c_{2}>0}$ ${\displaystyle {c_{1}}^{2}+{c_{2}}^{2}=a^{2}+b^{2}}$

Second set of Apollonius : For each pair of conjugated radii owns the these within the ellipse spanned triangle always the same area , namely  . ${\displaystyle \Delta }$ ${\displaystyle F_{\Delta }}$ ${\displaystyle F_{\Delta }={\frac {a\cdot b}{2}}}$

Alternative formulations

In the Bronstein , the sentence of Apollonios is given in a different way. Instead of the identity equation of the above second sentence of Apollonios, the following is formulated:

If in the ellipse there are a pair of conjugate radii   and the acute angles of these two with the main axis, then it always holds  . ${\displaystyle E}$ ${\displaystyle \alpha }$ ${\displaystyle \beta }$ ${\displaystyle c_{1}\cdot c_{2}\cdot \sin(\alpha +\beta )=a\cdot b}$

In a third version, the second sentence of Apollonios appears in Volume IV of the Encyclopedia of Elementary Mathematics . This can be represented as follows:

If the ellipse is circumscribed as a pair of conjugate diameters, the associated parallelogram , the sides of which are parallel to one of the two conjugate diameters, then always has the same area , namely  . ${\displaystyle E}$ ${\displaystyle \Pi }$ ${\displaystyle \Pi }$ ${\displaystyle F_{\Pi }}$ ${\displaystyle F_{\Pi }=4\cdot a\cdot b}$

Proof of the statements

The proof of the statements results from the description of conjugate points of an ellipse (see conjugate diameter ): 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: ${\displaystyle {\big (}\cos t,\sin t{\big )},\ 0\leq t<2\pi }$ ${\displaystyle {\big (}x(t),y(t){\big )},\ {\big (}x(t\pm {\tfrac {\pi }{2}}),y(t\pm {\tfrac {\pi }{2}}{\big )}}$

• The vector (radius) is conjugated to the vector . ${\displaystyle {\vec {c}}_{2}=(-a\sin t,b\cos t)^{T}}$ ${\displaystyle {\vec {c}}_{1}=(a\cos t,b\sin t)^{T}}$

It is

${\displaystyle |{\vec {c}}_{1}|^{2}+|{\vec {c}}_{2}|^{2}=a^{2}\cos ^{2}t+b^{2}\sin ^{2}t+a^{2}\sin ^{2}t+b^{2}\cos ^{2}t=a^{2}+b^{2}\ .}$

The area of the triangle spanned by the vectors is: ${\displaystyle {\vec {c}}_{1},{\vec {c}}_{2}}$

${\displaystyle F_{\Delta }={\tfrac {1}{2}}\det({\vec {c}}_{1},{\vec {c}}_{2})=\cdots ={\tfrac {1}{2}}ab={\tfrac {1}{2}}c_{1}c_{2}\sin(\alpha +\beta )}$(See picture and triangular area .). So it applies

${\displaystyle c_{1}c_{2}\sin(\alpha +\beta )=ab}$.

Note: A proof that also uses the determinant but does not require trigonometric functions can be found in the proof archive, aa0 under (6.1) and (6.2).

The parallelogram of conjugate diameters circumscribed by the ellipse is composed of 8 triangles of equal area. The last of the statements follows from this.

Background of the area calculation

Both the first as well as the second sentence by Apollonios can essentially be derived from school mathematics .

For the background of the second Apollonian theorem, it is significant that here - as suggested by the elliptical axis construction according to Rytz von Brugg - the ellipse can also be understood as a compact area of the real coordinate plane , which is a vertical axis-affine image of the closed circular disk given around the origin of the radius arises. ${\displaystyle E}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle {\overline {B}}_{a}^{2}{\begin{pmatrix}0\\0\end{pmatrix}}}$ ${\displaystyle a}$

The linear transformation used for this  

${\displaystyle T\colon \,\mathbb {R} ^{2}\to \mathbb {R} ^{2}}$

${\displaystyle {\begin{pmatrix}x\\y\end{pmatrix}}\mapsto {\begin{pmatrix}x\\{\frac {b}{a}}y\end{pmatrix}}={\begin{pmatrix}1&0\\0&{\frac {b}{a}}\\\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}}$

is a homeomorphism of the coordinate plane on itself.

Hence, using the transformation theorem, one obtains the area of each compact patch ${\displaystyle A\subseteq \mathbb {R} ^{2}}$

${\displaystyle F_{T(A)}=\det {\begin{pmatrix}1&0\\0&{\frac {b}{a}}\\\end{pmatrix}}\cdot F_{A}={\frac {b}{a}}\cdot F_{A}}$

and therefore in particular

${\displaystyle F_{\Delta }={\frac {b}{a}}\cdot {\bigl (}{\frac {a^{2}}{2}}{\bigr )}={\frac {a\cdot b}{2}}}$

such as

${\displaystyle F_{\Pi }={\frac {b}{a}}\cdot {\bigl (}4\cdot a^{2}{\bigr )}=4\cdot a\cdot b}$  .

In the same way one proves that the area of ​​the entire ellipse

${\displaystyle F_{E}={\frac {b}{a}}\cdot {\bigl (}\pi \cdot a^{2}{\bigr )}=\pi \cdot a\cdot b}$

amounts.

literature

• PS Alexandroff , AI Markuschewitsch , AJ Chintschin : Encyclopedia of Elementary Mathematics (=  university books for mathematics . Volume 10 ). Volume IV. Geometry . VEB Deutscher Verlag der Wissenschaften, Berlin 1969.
• IN Bronstein, KA Semendjajev, G. Musiol, H. Mühlig (Hrsg.): Taschenbuch der Mathematik . 7th, completely revised and expanded edition. Verlag Harri Deutsch, Frankfurt am Main 2008, ISBN 978-3-8171-2007-9 .
• György Hajós : Introduction to Geometry . BG Teubner Verlag, Leipzig 1970 (Hungarian: Bevezetés A Geometriába . Translated by G. Eisenreich [Leipzig, also editorial office]).
• Hans Honsberg: Analytical Geometry . With appendix “Introduction to vector calculation” (=  mathematics for high schools ). 3. Edition. Bayerischer Schulbuch-Verlag, Munich 1971, ISBN 3-7627-0677-8 .
• Apollonius von Perga's seven books on conic sections

References and comments

1. ^ György Hajós : Introduction to Geometry . BG Teubner Verlag, Leipzig 1970, p. 510–511 (Hungarian: Bevezetés A Geometriába . Translated by G. Eisenreich [Leipzig, also editorial]).
2. ↑ Within that is the main axis is the longest and the minor axis is the shortest route . As usual, the length of the major and the length of the minor semiaxis is here. ${\displaystyle E}$ ${\displaystyle a}$ ${\displaystyle b}$
3. ^ IN Bronstein, KA Semendjajev et al .: Taschenbuch der Mathematik. 2008, p. 205
4. ↑ ^{a b} P. S. Alexandroff et al .: Encyclopedia of Elementary Mathematics. Volume IV 1969, p. 598
5. ↑ A parallelogram circumscribed by the ellipse is characterized by the fact that each of its four sides lies on a tangent of , i.e. only touches it at a single point . ${\displaystyle E}$ ${\displaystyle E}$ ${\displaystyle E}$
6. ↑
   

   Wikibooks: Evidence Archive: Geometry: Conjugated Diameters  - Learning and Teaching Materials
7. ↑ Hans Honsberg: Analytical Geometry. 1971, pp. 88-90, 95-96
8. ↑ If you omit the edge curve , the area naturally remains unchanged.