Cissoid of Diocles

In geometry, the cissoid of Diocles (; named for 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 curve 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. 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 t = tan θ in the above equation. Then are parametric equations for the cissoid. Converting the polar form to Cartesian coordinates produces A compass-and-straightedge construction of various points on the cissoid proceeds as follows. Given a line L and a point O not on L, construct the line L' through O parallel to L. Choose a variable point P on L, and construct Q, the orthogonal projection of P on L', then R, the orthogonal projection of Q on OP. Then the cissoid is the locus of points R. To see this, let O be the origin and L the line x = 2a as above. Let P be the point (2a, 2at); then Q is (0, 2at) and the equation of the line OP is y = tx. The line through Q perpendicular to OP is To find the point of intersection R, set y = tx in this equation to get which are the parametric equations given above.