page_title: Cissoid of Diocles
section_title: Construction by double projection
hierarchical_section_title: Cissoid of Diocles / Construction and equations / Construction by double projection
context_page_description: 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 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.
  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.
context_section_description: 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.
  While this construction produces arbitrarily many points on the cissoid, it cannot trace any continuous segment of the curve.
caption: Mechanism to generate the cissoid