Cissoids

Cissoids (red) of the curves (green) and (blue) with respect to the pole ${\displaystyle C_{1}}$ ${\displaystyle C_{2}}$ ${\displaystyle O}$

A cissoid or ivy curve is a plane curve that is defined using two other curves and a point. The definition allows many different curve shapes, so that many other flat curves can be understood as zissoids. One of the oldest examples of a cissoid is the cissoid of Diocles , which has been known since ancient times .

definition

There are two curves and and a point called the pole . At a point on the straight line intersects the curve in . Now you add the vector to the pole and get the point . The cissoid of the curves and with respect to the pole is now defined as the geometric location of all points A, which are obtained when the point moves along the curve . ${\displaystyle C_{1}}$ ${\displaystyle C_{2}}$ ${\displaystyle O}$ ${\displaystyle P_{1}}$ ${\displaystyle C_{1}}$ ${\displaystyle OP_{1}}$ ${\displaystyle C_{2}}$ ${\displaystyle P_{2}}$ ${\displaystyle {\overrightarrow {P_{1}P_{2}}}}$ ${\displaystyle O}$ ${\displaystyle A}$ ${\displaystyle C_{1}}$ ${\displaystyle C_{2}}$ ${\displaystyle O}$ ${\displaystyle P_{1}}$ ${\displaystyle C_{1}}$

Interchanging the curves and in the definition above leads to a point reflection of the original cissoids at their pole . ${\displaystyle C_{1}}$ ${\displaystyle C_{2}}$ ${\displaystyle O}$

If the curves and are described by the polar equations and (with the pole at the origin), the polar equation for the associated cissoids results . It should be noted that the variable of the zissoids is signed or oriented in contrast to that of the two curves. ${\displaystyle C_{1}}$ ${\displaystyle C_{2}}$ ${\displaystyle r=c_{1}(\varphi )}$ ${\displaystyle r=c_{2}(\varphi )}$ ${\displaystyle O}$ ${\displaystyle r=c_{2}(\varphi )-c_{1}(\varphi )}$ ${\displaystyle r}$

Circle-straight-cissoids

Cissoid of Diocles (red) with circle (green) and straight line (blue) as well as pole ${\displaystyle C_{1}}$ ${\displaystyle C_{2}}$ ${\displaystyle O}$

Cissoids in which a circle is selected for the curve and a straight line for the curve are called circle-straight cissoids. The cissoid of Diocles is a special circle-straight-cissoid in which the pole lies on the circle and the straight line is the tangent to the circle whose point of contact is opposite the pole. This means that the segment is a diameter of the circle and is perpendicular to the straight line. ${\displaystyle C_{1}}$ ${\displaystyle C_{2}}$ ${\displaystyle O}$ ${\displaystyle B}$ ${\displaystyle OB}$

literature

• Dörte Haftendorn: Exploring and understanding curves: With GeoGebra and other tools . Springer, 2016, ISBN 9783658147495 , pp. 64–75, 258-61
• Eugene V. Shikin: Handbook and Atlas of Curves . CRC Press, 1996, ISBN 9780849389634 , pp. 110-118

Web links

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

• Eric W. Weisstein : Cissoid . In: MathWorld (English).
• Cisoids on mathcurve.com
• Cisoids on 2dcurvess.com

Individual evidence

1. ↑ Other common spellings are Cissoide or Kissiode . All three variants are derived from the Greek word ( Greek κισσοειδής kissoeidēs ) for "ivy-shaped".
2. ↑ ^{a b c d} Dörte Haftendorn: Exploring and understanding curves: With GeoGebra and other tools . Springer, 2016, ISBN 9783658147495 , pp. 64–75, 258-61