Nephroid::usage="It is the geometric locus of a point P(x, y) located in a circumference (generatrix) of radius r that rolls tangentially outside of a fixed circumference (directrix) of radius 2r. It is considered an epicycloid of 2 cusps. The name nephroid means kidney shaped. References: AI; Personal.";
Development:
Coordinates of the center of the mobile circumference:
x
c
y
c
Let α the angle that has rotated the movable circumference on its own center. The arc traveled by the fixed circumference must be equal to the arc traveled in the movable circumference:
2rθ==rα;α==2θ;
Due to the direction of the turn, the net angle at which the point P moves with respect to its center is:
θ+α==3θ;
For the point P to begin exactly at the internal point of contact (on the X axis), the deviations of the point P from its center substract in the opposite direction, resulting in:
Δx==-rCos[3θ];Δy==-rSin[3θ];
Parametric equations of the Nephroid:
η[t_]:={3rCos[t]-rCos[3t],3rSin[t]-rSin[3t]};
Animation:
In[]:=
Manipulate[r=1;C1[t_]:={3rCos[t],3rSin[t]};O1={0,0};η[t_]:={3rCos[t]-rCos[3t],3rSin[t]-rSin[3t]};Show[{ParametricPlot[η[θ],{θ,0,t},PlotStyle->{Red,Thick},MaxRecursion->3],Graphics[{Red,PointSize[0.01],Point[η[t]],Text[Style["P",14,Bold],η[t],{0,-1.2}],Black,Point[C1[t]],Text[Style["C",14,Bold],C1[t],{0,-1.2}],Point[O1],Text[Style["O",14,Bold],O1,{0,-1.2}],Thick,Line[{C1[t],η[t]}],Circle[C1[t],r],{Blue,Circle[O1,2r]}}]},AspectRatioAutomatic,AxesTrue,AxesOrigin{0,0},AxesLabel{x,y},PlotRange5,ImageSize700],Style["Nephroid",Bold,Large],{{t,0.00002,"Value (t)"},0.000001,2π,0.00001},ControlPlacementTop]
Out[]=
