ConchoidOfNicomedes::usage="Let r be a straight line, O a point that does not belong to r and k (k is a constant). From O a straight line is drawn that intersects to r at a point P. On this new line you have to find the points and such as d(P, ) = d(P, ) = k . The conchoid of Nicomedes is the geometric locus described by the points Q1 and Q2 when P goes through the line r. References: Carrillo de Albornoz Torres, Agustín. Lugares geométricos. Concoide de Nicomedes. Geogebra (Text and figure); AI Assistant; Personal.";
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Development:
Cartesian equation of the conchoid of Nicomedes:
Let the line r: y = b.
P={t,b};O={0,0};
Straight line that passes through ,, P and O:
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t
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b
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Note: The coordinates of the points O, P and (or) are on the same line their coordinates are proportional.
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Then :
d(P,)=d(P,)=k
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2
x-
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(y-b)
2
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(y-b)
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xy-bx
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y-b
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(y-b)
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Animation:
In[]:=
Manipulate[b=2;k=4;O1={0,0};P[t_]:={t,b};u[t_]:=Normalize[P[t]-O1];Q1[t_]:=P[t]+ku[t];Q2[t_]:=P[t]-ku[t];Show[{ParametricPlot[Q1[t],{t,-10,t1},PlotStyle->{Red,Thick}],ParametricPlot[Q2[t],{t,-10,t1},PlotStyle->{Red,Thick}],ParametricPlot[Callout[{x,b},"y = b",{Scaled[0.75],Below},LabelStyle{14,Bold,Black}],{x,-15,15},PlotStyle->Blue],Graphics[{Black,PointSize[0.01],Point[O1],Text[Style["O",14,Bold],O1,{0,-1.2}],Point[Q1[t1]],Text[Style["",14,Bold],Q1[t1],{0,-1.2}],Point[Q2[t1]],Text[Style["",14,Bold],Q2[t1],{0,-1.2}],Point[P[t1]],Text[Style["P",14,Bold],P[t1],{0,-1.2}],{DarkYellow,Circle[P[t1],k]},InfiniteLine[{Q1[t1],Q2[t1]}]}]},AspectRatioAutomatic,AxesTrue,AxesOrigin{0,0},AxesLabel{x,y},PlotRange{{-15,15},{-5,10}},ImageSize700],Style["Conchoid of Nicomedes",Bold,Large],{{t1,-10+0.00002,"Value ()"},-10+0.000001,10,0.00001},ControlPlacementTop]
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