Cissoid of Diocles

Cissoids

The cissoid of Diocles is a special 3rd order curve that was described by the Greek mathematician Diocles (around 200 BC) in order to use this tool to solve the problem of doubling the cube (also known as the Deli problem). (This construction task cannot be accomplished with compasses and ruler alone.) The name comes from the Greek word κισσοειδής ( kissoeidēs ) for ivy-shaped.

Equations of the cissoids

• Cartesian coordinates: ${\displaystyle y^{2}\,(2a-x)-x^{3}=0}$
• Parametric equation: ${\displaystyle x={\frac {2at^{2}}{1+t^{2}}};\qquad y={\frac {2at^{3}}{1+t^{2}}}}$
• Polar coordinates: ${\displaystyle r={\frac {2a}{\cos(\varphi )}}-2a\cos(\varphi )=2a\sin \varphi \tan \varphi }$

Properties of the cissoids

Cissoids as the base point curve

• The points of the cissoids are characterized by the following geometrical property: A circle with radius  a, a point S on this circle and the tangent that touches this circle at the point opposite S are given. If one designates the intersection of the straight line SP with the circle as K and the intersection of SP with the above-mentioned circle tangent as A for any point P of the cissoids , then the line lengths and are equal (this property follows directly from the definition of general cissoids ) . ${\displaystyle {\overline {SP}}}$ ${\displaystyle {\overline {KA}}}$
• The straight line of the equation is the asymptote of the curve. ${\displaystyle x\,=2a}$
• The area bounded by the cissoid and its asymptote has the area . ${\displaystyle 3\,\pi a^{2}}$
• The cissoid also results as the base point curve of a parabola if you choose its apex as the reference point.

literature

• Dörte Haftendorn: Exploring and understanding curves: With GeoGebra and other tools . Springer, 2016, ISBN 9783658147495 , pp. 64–67, 74–78, 258–261
• Eugene V. Shikin: Handbook and Atlas of Curves . CRC Press, 1996, ISBN 9780849389634 , pp. 110-118
• Jan van Maanen: From Quadrature to Integration: Thirteen Years in the Life of the Cissoid . In: The Mathematical Gazette , Vol. 75, No. 471 (March, 1991), pp. 1-15 ( JSTOR )

Web links

Commons : Cissoid  - collection of images, videos and audio files

• Eric W. Weisstein : Cissoid of Diocles . In: MathWorld (English).
• John J. O'Connor, Edmund F. Robertson :  Cissoid. In: MacTutor History of Mathematics archive .
• Xah Lee: Cissoid of Diocles . (English)
• general kissoids
• Cissoids to line and circle