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Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 ConchoidCurve

Source Notebook

Compute a conchoid curve

Contributed by: Jan Mangaldan

ResourceFunction["ConchoidCurve"][c,p,d]

computes one branch of the conchoid of the curve c with respect to the fixed point p and distance d.

ResourceFunction["ConchoidCurve"][c,p,d,-1]

computes the other branch of the conchoid.

Details

The conchoid of a curve with respect to a fixed point is the locus of points on a variable line passing through the fixed point and intersecting the curve, such that their distance from the intersection point is equal to a given distance d.

Examples

Basic Examples (2) 

Conchoid of a circle:

In[1]:=
c1 = ResourceFunction["ConchoidCurve"][{a Cos[t], a Sin[t]}, {h, k}, d]
Out[1]=

Plot one branch of the conchoid:

In[2]:=
plot1 = ParametricPlot[ Evaluate[c1 /. {a -> 1, h -> 3/2, k -> 0, d -> 2}], {t, -\[Pi], \[Pi]}]
Out[2]=

Plot the other branch:

In[3]:=
c2 = ResourceFunction[ "ConchoidCurve"][{a Cos[t], a Sin[t]}, {h, k}, -1];
In[4]:=
plot2 = ParametricPlot[ Evaluate[c2 /. {a -> 1, h -> 3/2, k -> 0, d -> 2}], {t, -\[Pi], \[Pi]}, PlotStyle -> ColorData[97, 2]]
Out[4]=

Show both branches together:

In[5]:=
Show[plot1, plot2, PlotRange -> All]
Out[5]=

Find the implicit Cartesian equation of the conchoid:

In[6]:=
First[GroebnerBasis[ MapAll[TrigExpand, Thread[{x, y} == c1] /. t -> 2 ArcTan[u]], {x, y}, u]] == 0 // FullSimplify
Out[6]=

The conchoid of Nicomedes is the conchoid of a line with respect to the origin:

In[7]:=
Simplify[ResourceFunction["ConchoidCurve"][{a, a Tan[t]}, {0, 0}, d], a > 0 && -\[Pi]/2 < t < \[Pi]/2]
Out[7]=
In[8]:=
ParametricPlot[Evaluate[% /. {d -> 1, a -> 1/4}], {t, -\[Pi], \[Pi]}]
Out[8]=

Version History

  • 1.0.0 – 09 March 2021

Source Metadata

  • Citation:
    • Lawrence J.D., A Catalog of Special Plane Curves. New York: Dover, pp. 49-51, 1972.
    • Lockwood E. H., "Conchoids." Ch. 14 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 126-129, 1967.

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