Dürer's Conchoid

Author

Eric W. Weisstein
February 17, 2012

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneCurves/DuerersConchoid.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/DuerersConchoid.html.

©2012 Wolfram Research, Inc. except for portions noted otherwise

Definition

curve[a_,b_][x_,y_]:=2 y^2(x^2+y^2)-2 b y^2(x+y)+(b^2-3 a^2)y^2-a^2 x^2+2 a^2 b(x+y)+a^2(a^2-b^2)

Equation

AlgebraicDegree[curve[a,b][x,y],{x,y}]

Plot

V5

<<Graphics`ImplicitPlot`

MakeBoxes[ylabel[a_,b_],TraditionalForm]:=RowBox[{RowBox[{"a","==",MakeBoxes[a,TraditionalForm]}],",",RowBox[{"b","==",MakeBoxes[b,TraditionalForm]}]}]

Show[GraphicsArray[List/@Block[{$DisplayFunction=Identity},
Function[{a,b},ImplicitPlot[2 y^2(x^2+y^2)-2 b y^2(x+y)+(b^2-3 a^2)y^2-a^2 x^2+2 a^2 b(x+y)+a^2(a^2-b^2)==0,{x,-10,10},
PlotLabel->TraditionalForm[ylabel[a,b]],Ticks->None,PlotStyle->Red]]@@@{{3,1},{3,3},{3,5}}
]]]

Show[
Block[{$DisplayFunction=Identity},
Table[ImplicitPlot[2 y^2(x^2+y^2)-2 b y^2(x+y)+(b^2-3 a^2)y^2-a^2 x^2+2 a^2 b(x+y)+a^2(a^2-b^2)==0/.a->1,{x,-5,5},AspectRatio->Automatic,Ticks->None,PlotStyle->Hue[b/2]],
{b,.05,2,.05}]]]

V6

ContourPlot[curve[##][x,y]==0,{x,-10,10},{y,-5,5},AspectRatio->Automatic]&@@@{{3,1},{3,3},{3,5},{1,0},{1,1/2},{2,.1}}

Special Cases

Show[GraphicsArray[List/@Block[{$DisplayFunction=Identity},
Function[{a,b},ImplicitPlot[2 y^2(x^2+y^2)-2 b y^2(x+y)+(b^2-3 a^2)y^2-a^2 x^2+2 a^2 b(x+y)+a^2(a^2-b^2)==0,{x,-5,5},
PlotLabel->TraditionalForm[ylabel[a,b]],Ticks->None,PlotStyle->Red]]@@@{{1,0},{.0001,1},{1,1}}
]]]

Show[GraphicsArray[List/@Block[{$DisplayFunction=Identity},
Function[{a,b},ImplicitPlot[2 y^2(x^2+y^2)-2 b y^2(x+y)+(b^2-3 a^2)y^2-a^2 x^2+2 a^2 b(x+y)+a^2(a^2-b^2)==0,{x,-5,5},
PlotLabel->TraditionalForm[ylabel[a,b]],PlotStyle->Red]]@@@({{1,0},{.0001,1},{1,1}})
]]]

Properties

Implicit Curvature

FullSimpify[ImplicitCurvature[2 y^2(x^2+y^2)-2 b y^2(x+y)+(b^2-3 a^2)y^2-a^2 x^2+2 a^2 b(x+y)+a^2(a^2-b^2),{x,y}],a>0&&b>0]

Slope

Slope[2 y^2 (x^2+y^2)-2 b y^2 (x+y)+(b^2-3 a^2) y^2-a^2 x^2+2 a^2 b (x+y)+a^2 (a^2-b^2)==0,{x,y}]//FullSimplify