Conchoid

conchoid5Hypocycloid
Conchoids of a 5 cusped epicycloid with pole at center.

Mathematica Notebook for This Page.

History

A generalization of methods of conchoid of Nicomedes, invented by Nicomedes (~225 BCE) to trisect a angle. Who first generalized this? Why?

Description

Conchoid is a way of deriving a new curve based on a given curve, a fixed point, and a positive constant k. Curves generated this way are sometimes called general conchoid because this method is a generalization of conchoid of Nicomedes.

Step-by-step description:

  1. Given a curve c, a point O not on l, and a constant k.
  2. Draw a line m passing O and any point P on C.
  3. Mark points Q1 and Q2 on m such that distance[Q1,P]==distance[Q2,p]==k.
  4. The locus of Q1 and Q2 for variable points P on c is the conchoid of c with respect to O and offset k.

The point O is called the pole.

conchoidSinGen conchoidSinGen
The conchoid curve (red dots) of sine curve, with pole at {-3,3} and k:=2.

Formula

Conchoids has two branches. We associate the +Abs[k] offset to the branch that lies on the side of the given curve opposite the pole, and the -Abs[k] offset corresponds to the branch that lies on the same side as the pole. Often, it is better to define conchoid with a signed offset, so that a conchoid with a signed constant k has just one branch. If we want both branches, we say a conchoid with offsets k and -k. In the following formulas, k is a signed constant.

If a curve is given in polar form r==f[θ], then its conchoid with respect to the pole and offset k is r==f[θ]+k.

The formula for a conchoid of a point {x,y} with pole at {a,b} and offset k in Cartesian coordinate is: {x + (k*(-a + x))/Sqrt[(-a + x)^2 + (-b + y)^2], y + (k*(-b + y))/Sqrt[(-a + x)^2 + (-b + y)^2]}.

The formula can be easily derived using vectors. Suppose V:={x,y} is a vector. We want to find its conchoid point with respect to the origin and offset k. Immediatly, the conchoid points are located in the direction of V with distance (length[V]+k), where k can be both positive or negative. We multiply the unit vector (V/length[V]) by the required length: (V/length[V])*(length[V]+k). This simplifies to V+(k*V)/length[V] or {x,y}*(1+k/Sqrt[x^2+y^2]). If the pole O is at {a,b}, the we can subsitute V by V-O, and translate the whole thing back by adding {a,b}, resulting the general formula.

Properties

Examples

Verbatim from E.H.Littlewoods, 1961:

Related Web Sites

see Generic Reference Page.

Robert Yates: Curves and Their Properties.

Conchoid (mathematics).

The MacTutor History of Mathematics archive

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