Cycloids from Ceva

Cycloids from Ceva

The cycloid of Ceva or Trisektrix of Ceva is a flat curve named after Tommaso Ceva (1648–1736) , which can be used to divide angles into three (hence Trisektrix ). Ceva himself described the curve as cycloidum anomalarum .

Geometric definition

Animation of the construction of Ceva's cycloids

Angular property of the cycloid by Ceva: ( basic set of angles ) (set of external angles , set of basic angles ) ( secondary angle , sum of angles )
${\displaystyle \varphi =\angle P_{1}OP_{2}=\angle P_{1}P_{2}O}$
${\displaystyle 2\varphi =\angle P_{3}P_{1}P_{2}=\angle P_{3}P_{1}P_{2}}$
${\displaystyle 3\varphi =180^{\circ }-(180^{\circ }-4\varphi )-\varphi }$

For a point on the unit circle , the straight line connecting to the origin is constructed . Then one determines on the x-axis that of the different point which has the distance 1 from. Finally, one then determines the point on the straight line from different that has the distance 1 from. The cycloid of Ceva is now the locus of , which is obtained by rotating the point and thus also the straight line around the origin . ${\displaystyle P_{1}}$ ${\displaystyle OP_{1}}$ ${\displaystyle O}$ ${\displaystyle O}$ ${\displaystyle P_{2}}$ ${\displaystyle P_{1}}$ ${\displaystyle P_{1}}$ ${\displaystyle P_{3}}$ ${\displaystyle OP_{1}}$ ${\displaystyle P_{2}}$ ${\displaystyle P_{3}}$ ${\displaystyle P_{1}}$ ${\displaystyle OP_{1}}$ ${\displaystyle O}$

The locus consists of four axially symmetrical loops lying at the origin, the two loops on the x-axis being significantly larger than the two on the y-axis. If you only use a ray instead of the straight line , the two small loops on the y-axis are omitted. ${\displaystyle OP_{1}}$ ${\displaystyle OP_{1}}$

Due to the construction, the angle between the straight line and the x-axis is exactly one third of the angle between the line and the x-axis (see drawing). Due to this property, the curve can be used as a trisectrix. ${\displaystyle OP_{1}}$ ${\displaystyle P_{2}P_{3}}$

Substituting the method of construction for points , and for more points away, the result is odd as loci of the Sektrizen of Ceva . ${\displaystyle P_{1}}$ ${\displaystyle P_{2}}$ ${\displaystyle P_{3}}$ ${\displaystyle P_{k}}$ ${\displaystyle k}$ ${\displaystyle P_{k}}$

Equation and parametric form

The following equation in polar coordinates can be derived from the geometric definition with the help of the law of cosines :

${\displaystyle r=1+2\cos(2\varphi )}$.

The following representation is obtained as a parameter curve in Cartesian coordinates: ${\displaystyle \gamma :[0,2\pi ]\rightarrow \mathbb {R} ^{2}}$

${\displaystyle \gamma (t)={\begin{pmatrix}x(t)\\y(t)\end{pmatrix}}={\begin{pmatrix}2\cos(t)+\cos(3t)\\\sin(3t)\end{pmatrix}}}$.

In addition, the following equation results in Cartesian coordinates, whereby the cycloid of Ceva is an algebraic curve of the sixth degree:

${\displaystyle (x^{2}+y^{2})^{3}=(3x^{2}-y^{2})^{2}}$.

Angular trisection

Trisection of acute angles with the cycloid from Ceva

Angular trisection of obtuse angles with the cycloid of Ceva

The angular property of Ceva's cycloids described above provides the following construction for dividing an angle into three. At a given angle , you first extend the leg and draw the cycloid on the extension as the x-axis. Then, on the other leg, draw the line with the length 1 and draw the parallel to through the point . This cuts the cycloid at that point . Now you connect the point with the center of the cycloid (origin of the coordinate system), then form the line with the extension of an angle whose angular dimension is exactly one third of the angular dimension of the starting angle . Please note that the parallel in the case of acute or obtuse angles always intersects the cycloid in two points and so two points are initially available for determining . If it is an acute angle ( ), then the intersection point closer to the angle is selected as . In the case of an obtuse angle ( ), on the other hand, the point of intersection further away is chosen as . ${\displaystyle \angle CBA}$ ${\displaystyle AB}$ ${\displaystyle AB}$ ${\displaystyle BC}$ ${\displaystyle BD}$ ${\displaystyle AB}$ ${\displaystyle D}$ ${\displaystyle P_{3}}$ ${\displaystyle P_{3}}$ ${\displaystyle O}$ ${\displaystyle OP_{3}}$ ${\displaystyle AB}$ ${\displaystyle \angle CBA}$ ${\displaystyle P_{3}}$ ${\displaystyle \angle CBA<90^{\circ }}$ ${\displaystyle P_{3}}$ ${\displaystyle \angle CBA>90^{\circ }}$ ${\displaystyle P_{3}}$

Historical

Tommaso Ceva (1648–1736), brother of Giovanni Ceva (1647–1734), described the curve in his work Opuscula mathematica , published in 1699, and referred to it there as cycloidum anomalarum . The angle property or the mathematical idea on which the curve construction is based goes back to Archimedes (287–212 BC), who used it to divide the angle into three with the help of a marked ruler .

literature

• Gino Loria: Special Algebraic and Transcendent Plane Curves: Theory and History . Teubner, 1902, pp. 324-325
• Eugene V. Shikin: Handbook and Atlas of Curves . CRC Press, 1996, ISBN 9780849389634 , p. 315
• Robert C. Yates: The Trisection Problem . National Mathematics Magazine, Volume 15, No. 4 (Jan., 1941), pp. 191-202 ( JSTOR )
• Robert C. Yates: The Trisection Problem . Classics in Mathematics Education Series Volume 3, The National Teachers of Mathematics, Education Resources Information Center, 1971, pp. 39–40 ( online copy )
• Laszlo Nemeth: Sectrix Curves on the Sphere . KOG 19, December 2015, pp. 42–47
• Tommaso Ceva: Opuscula mathematica . Milan, 1699, p. 31 ( online copy )

Web links

Commons : Cycloid of Ceva  - collection of images, videos and audio files

• Eric W. Weisstein : Cycloid of Ceva . In: MathWorld (English).
• Trisection using Special Curves
• The angular trisection (construction with additional tools)
• Ceva Trisectrix and Sectrix on mathcurve.com