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 BC) 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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Plane Curves

Ancient

  1. Conic Sections
  2. Parabola
  3. Hyperbola
  4. Ellipse
  5. Cissoid
  6. Conchoid
  7. Quadratrix
  8. Archimedean Spiral
  9. Equiangular Spiral
  10. Lituus
  11. Cornu Spiral

Cyclodal

  1. Epitrochoid
  2. Hypotrochoid
  3. Epicycloid and Hypocycloid
  4. Rose Curve
  5. Astroid
  6. Deltoid
  7. Nephroid
  8. Cardioid
  9. Trochoid
  10. Cycloid

Calculus Era

  1. Cassinian Oval
  2. Cross Curve
  3. Folium of Descartes
  4. Piriform
  5. Semicubic Parabola
  6. Tractrix
  7. Trisectrix
  8. Trisectrix of Maclaurin
  9. Lemniscate of Bernoulli
  10. Lemniscate of Gerono
  11. Limacon Of Pascal
  12. Witch of Agnesi
  13. Sine Curve
  14. Catenary
  15. Bezier Curve

Methods

  1. Caustics
  2. Cissoid
  3. Conchoid
  4. Envelope
  5. Evolute
  6. Involute
  7. Geometric Inversion
  8. Orthoptic
  9. Parallel Curve
  10. Pedal Curve
  11. Radial Curve
  12. Roulette

Math of Curves

  1. Geometry: Coordinate Systems for Plane Curves
  2. Coordinate Transformation
  3. Vectors
  4. Naming and Classification of Curves
  1. Cusp
  2. Curvature