Kurven sehen und verstehen

Haftendorn April 2017, http://www.kurven-sehen-und-verstehen.de

Inversion der Steiner-Kurven, Afg 9.15

Inversion am Kreis um M=(a,b), Radius k,

k=.

invert={x->(((k^2)x)/((x^2)+(y^2))),y->(((k^2)y)/((x^2)+(y^2)))}

invertM={x->(k^2) ((x-a)/((((x-a))^2)+(((y-b))^2)))+a, y->(k^2) ((y-b)/((((x-a))^2)+(((y-b))^2)))+b}

{a=.,b=.,k=.};
kurve=y==x /.invertM

FindInstance[b+(((k^2) (-b+y))/((((-a+x))^2)+(((-b+y))^2)))==a+(((k^2) (-a+x))/((((-a+x))^2)+(((-b+y))^2))),{a,b,k,x,y}]

{a=1,b=2,k=2};ContourPlot[{y==x,(((-a+x))^2)+(((-b+y))^2)==(k^2),2+(((k^2) (-2+y))/((((-1+x))^2)+(((-2+y))^2)))==1+(((k^2) (-1+x))/((((-1+x))^2)+(((-2+y))^2)))},{x,-2,4},{y,-1,4},
AspectRatio->Automatic, Axes->True]
{a=.,b=.,k=.};

(*Zeichnung passt*)

Steinerkurve, invertieren an M=(a,b), Radius k

gemäß Afg 9.15 afg9.15-inv-steiner.ggb

steiner=(9 r^2+12 r x+x^2+y^2)^2==4 r (3 r+2 x)^3

invsteiner=steiner/.invertM //FullSimplify

{a=-0.6,b=0.3,r=1,k=2};

ContourPlot[{(((-a+x))^2)+(((-b+y))^2)==(k^2),invsteiner//Evaluate,steiner//Evaluate},{x,-3,4},{y,-3,4},
AspectRatio->Automatic, Axes->True,ContourStyle->Thickness[0.02]]
{a=.,b=.,k=.};

{a=-1,b=0,r=1,k=2};

ContourPlot[{(((-a+x))^2)+(((-b+y))^2)==(k^2),invsteiner//Evaluate,steiner//Evaluate},{x,-3,4},{y,-3,4},
AspectRatio->Automatic, Axes->True,ContourStyle->Thickness[0.02]]
{a=.,b=.,k=.};

{a=0,b=0,r=1,k=2};

ContourPlot[{(((-a+x))^2)+(((-b+y))^2)==(k^2),invsteiner//Evaluate,steiner//Evaluate},{x,-4,4},{y,-4,4},
AspectRatio->Automatic, Axes->True,ContourStyle->Thickness[0.02]]
{a=.,b=.,k=.};

{a=1.5,b=1.3,r=1,k=2};

ContourPlot[{(((-a+x))^2)+(((-b+y))^2)==(k^2),invsteiner//Evaluate,steiner//Evaluate},{x,-3,4},{y,-4,4},
AspectRatio->Automatic, Axes->True,ContourStyle->Thickness[0.02]]
{a=.,b=.,k=.};