Cissoid of Diocles
In geometry, the cissoid of Diocles is a cubic plane curve notable for the property that it can be used to construct two mean proportionals to a given ratio. In particular, it can be used to double a cube. It can be defined as the cissoid of a circle and a line tangent to it with respect to the point on the circle opposite to the point of tangency. In fact, the family of cissoids is named for this example and some authors refer to it simply as the cissoid. It has a single cusp at the pole, and is symmetric about the diameter of the circle which is the line of tangency of the cusp. The line is an asymptote. It is a member of the conchoid of de Sluze family of curves and in form it resembles a tractrix.
The word "cissoid" comes from the Greek κισσοειδής kissoeidēs "ivy shaped" from κισσός kissos "ivy" and -οειδής -oeidēs "having the likeness of". The curve is named for Diocles who studied it in the 2nd century BCE.
Let the radius of C be a. By translation and rotation, we may take O to be the origin and the center of the circle to be (a, 0), so A is (2a, 0). Then the polar equations of L and C are:
By construction, the distance from the origin to a point on the cissoid is equal to the difference between the distances between the origin and the corresponding points on L and C. In other words, the polar equation of the cissoid is
Applying some trigonometric identities, this is equivalent to
Let in the above equation. Then
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