Polar-Kartesisch- Koppelung

Prof. Dr. Dörte Haftendorn: Mathematik mit MuPAD 4 (es ex. in Version 3),   Mrz. 06 Update 10.01.07

Web:  www.mathematik-verstehen.de             www.uni-lueneburg.de/ing-math

Archimedische Spirale:

r:=t->t/4:r(t);

archi:=plot::Polar([r(t),t],t=0..ende,ende=0..20,

        LineWidth=1, Mesh=400, LineColor=RGB::Red):

archikart:=plot::Curve2d([t,r(t)],t=0.01..0.01+ende,

        LineWidth=1, Mesh=400,ende=0..20, LineColor=RGB::Green):

radius:=plot::Line2d([0,0],[r(t)*cos(t),r(t)*sin(t)],t=0.01..0.01+20):

radiusordi:=plot::Line2d([t,0],[t,r(t)],t=0.01..0.01+20):

plot(archi,radius,archikart,radiusordi,

     AnimationStyle=BackAndForth):

math

MuPAD graphics

image

Polarparabel

r:=t->t^2/4:r(t);

archi:=plot::Polar([r(t),t],t=-8..ende,ende=-8..8,

         LineWidth=1, LineColor=RGB::Red, Mesh=400):

archikart:=plot::Curve2d([t,r(t)],t=-8..ende,ende=-8..8,

          LineWidth=1, Mesh=400,LineColor=RGB::Green):

radius:=plot::Line2d([0,0],[r(t)*cos(t),r(t)*sin(t)],t=-8..8):

radiusordi:=plot::Line2d([t,0],[t,r(t)],t=-8..8):

plot(archi,radius,archikart,radiusordi,

        AnimationStyle=BackAndForth):

math

math

math

MuPAD graphics

image

Polarhyperbel

r:=t->1/t:   r(t);

archi:=plot::Polar([r(t),t],t=0.01..0.01+ende,ende=0..8,

         LineWidth=1, LineColor=RGB::Red, Mesh=400):

archikart:=plot::Curve2d([t,r(t)],t=0.01..0.01+ende,ende=0..8,

         LineWidth=1, Mesh=400, LineColor=RGB::Green):

radius:=plot::Line2d([0,0],[r(t)*cos(t),r(t)*sin(t)],t=0.01..0.01+8):

radiusordi:=plot::Line2d([t,0],[t,r(t)],t=0.01..0.01+8):

plot(archi,radius,archikart,radiusordi,ViewingBox=[-1..8.1,-1..4],

        Scaling=Constrained, AnimationStyle=BackAndForth):

math

MuPAD graphics

image

limit(1/t*sin(t), t = 0)

math

plotfunc2d(1/t*sin(t))

MuPAD graphics

r:=t->10*sin(t)/t:  r(t);

archi:=plot::Polar([r(t),t],t=0..ende,ende=0..20,

    LineWidth=1,LineColor=RGB::Red, Mesh=400):

archikart:=plot::Curve2d([t,r(t)],t=0..ende,ende=0..20,

    LineWidth=1, Mesh=400, LineColor=RGB::Green):

radius:=plot::Line2d([0,0],[r(t)*cos(t),r(t)*sin(t)],t=0..20):

radiusordi:=plot::Line2d([t,0],[t,r(t)],t=0..20):

plot(archi,radius,archikart,radiusordi,

     AnimationStyle=BackAndForth):

math

MuPAD graphics

image

Besondere Muschel

r:=t->t*cos(t):  r(t);

archi:=plot::Polar([r(t),t],t=0..ende,ende=0..20,

    LineWidth=1,LineColor=RGB::Red, Mesh=400):

/*kreis:=plot::Polar([10*cos(t),t],t=0..ende,ende=0..20,

       LineWidth=0.5,LineColor=[1,0,1], Mesh=400):*/

archikart:=plot::Curve2d([t,r(t)],t=0..ende,ende=0..20,

    LineWidth=1, Mesh=400, LineColor=RGB::Green):

radius:=plot::Line2d([0,0],[r(t)*cos(t),r(t)*sin(t)],t=0..20):

radiusordi:=plot::Line2d([t,0],[t,r(t)],t=0..20):

plot(archi,radius,archikart,radiusordi,

     AnimationStyle=BackAndForth):

math

MuPAD graphics

Kardioide

r:=t->cos(t)+1:  r(t);

archi:=plot::Polar([r(t),t],t=0..ende,ende=0..2*PI,

       LineWidth=1, LineColor=RGB::Red, Mesh=400):

archikart:=plot::Curve2d([t,r(t)],t=0..ende,ende=0..2*PI,

       LineWidth=1, Mesh=400, LineColor=RGB::Green):

radius:=plot::Line2d([0,0],[r(t)*cos(t),r(t)*sin(t)],t=0..2*PI):

radiusordi:=plot::Line2d([t,0],[t,r(t)],t=0..2*PI):

plot(archi,radius,archikart,radiusordi,

     AnimationStyle=BackAndForth):

 

math

MuPAD graphics

image

Andere Pascalsche Schnecke

r:=t->cos(t)+1/2:   r(t);

archi:=plot::Polar([r(t),t],t=0..ende,ende=0..2*PI,

           LineWidth=1, LineColor=RGB::Red, Mesh=400):

archikart:=plot::Curve2d([t,r(t)],t=0..ende,ende=0..2*PI,

            LineWidth=1, Mesh=400, LineColor=RGB::Green):

radius:=plot::Line2d([0,0],[r(t)*cos(t),r(t)*sin(t)],

            t=0..2*PI,LineWidth=0.5):

radiusbetrag:=plot::Line2d([0,0],[abs(r(t))*cos(t),abs(r(t))*sin(t)],

            t=0..2*PI,LineColor=[1,0,1],LineWidth=0.5):

radiusordi:=plot::Line2d([t,0],[t,r(t)],t=0..2*PI,LineWidth=0.5):

plot(archi,radiusbetrag,radius,archikart,radiusordi,

     AnimationStyle=BackAndForth):

math

MuPAD graphics

image

Berechnung der Tangenten für die Pascalsche Schnecke im Ursprung

solve(cos(t)+1/2=0,t)

math

tang1:=plot::Function2d(2*PI/3*x,x=-1..1):

tang2:=plot::Function2d(-2*PI/3*x,x=-1..1)

math

r:=t->cos(t)+1/2:   r(t);

archi:=plot::Polar([r(t),t],t=0..ende,ende=0..2*PI,

          LineWidth=1, LineColor=RGB::Red, Mesh=400):

archikart:=plot::Curve2d([t,r(t)],t=0..ende,ende=0..2*PI,

          LineWidth=1, Mesh=400, LineColor=RGB::Green):

radius:=plot::Line2d([0,0],[r(t)*cos(t),r(t)*sin(t)],t=0..2*PI, LineWidth=0.5):

radiusbetrag:=plot::Line2d([0,0],[abs(r(t))*cos(t),abs(r(t))*sin(t)],t=0..2*PI,

            LineColor=[1,0,1], LineWidth=0.5):

radiusordi:=plot::Line2d([t,0],[t,r(t)],t=0..2*PI, LineWidth=0.5):

plot(archi,tang1,tang2,radiusbetrag,radius,archikart,radiusordi,

     AnimationStyle=BackAndForth):

math

MuPAD graphics

image

Blume

r:=t->cos(2*t): r(t);

archi2:=plot::Polar([r(t),t],t=0..ende,ende=0..2*PI,

         LineWidth=1,LineColor=[1,0,0], Mesh=400):

archikart2:=plot::Curve2d([t,r(t)],t=0..ende,ende=0..2*PI,

         LineWidth=1, Mesh=400, LineColor=RGB::Green):

radius2:=plot::Line2d([0,0],[r(t)*cos(t),r(t)*sin(t)],t=0..2*PI,

        LineWidth=0.5):

radiusbetrag2:=plot::Line2d([0,0],[abs(r(t))*cos(t),abs(r(t))*sin(t)],

                                    t=0..2*PI,LineColor=[0,1,0]):

radiusordi2:=plot::Line2d([t,0],[t,r(t)],t=0..2*PI):

plot(archi2,radiusbetrag2,radius2,archikart2,radiusordi2,LineWidth=0.5,

     AnimationStyle=BackAndForth):

math

MuPAD graphics

image

dieselbe Viererblume pur

plot(archi2);

MuPAD graphics

image

Andere Viererblume

r:=t->cos(2*t)^2:  r(t);

archi:=plot::Polar([r(t),t],t=0..ende,ende=0..2*PI,

LineWidth=1, LineColor=RGB::Red, Mesh=400):

archikart:=plot::Curve2d([t,r(t)],t=0..ende,ende=0..2*PI,

LineWidth=1, Mesh=400, LineColor=RGB::Green):

radius:=plot::Line2d([0,0],[r(t)*cos(t),r(t)*sin(t)],t=0..2*PI,

                        LineWidth=0.5):

radiusbetrag:=plot::Line2d([0,0],[abs(r(t))*cos(t),abs(r(t))*sin(t)],

                        t=0..2*PI,LineWidth=0.5,LineColor=[0,1,0]):

radiusordi:=plot::Line2d([t,0],[t,r(t)],t=0..2*PI):

plot(archi,radiusbetrag,radius,archikart,radiusordi,

     AnimationStyle=BackAndForth):

math

MuPAD graphics

image

andere Viererblume pur, sie wird anders durchlaufen als die erste Vierenblume

plot(archi);

MuPAD graphics

image

Beide Blumen zusammen

plot(archi, archi2 ,  AnimationStyle=BackAndForth):

MuPAD graphics

image

Fazit: An der fertigen Kurve sieht man den Durchlaufsinn gar nicht.

Wenn man ihn durch die Animation bemerkt, dann kann man ihn mit der karteischen

Koppelung bestens erklären.

----------------------------------------------

Doppel-Ei

r:=t->cos(1*t)^2:  r(t);

archi:=plot::Polar([r(t),t],t=0..ende,ende=0..2*PI,

LineWidth=1, LineColor=RGB::Red, Mesh=400):

archikart:=plot::Curve2d([t,r(t)],t=0..ende,ende=0..2*PI,

LineWidth=1, Mesh=400, LineColor=RGB::Green):

radius:=plot::Line2d([0,0],[r(t)*cos(t),r(t)*sin(t)],t=0..2*PI,

                        LineWidth=0.5):

radiusbetrag:=plot::Line2d([0,0],[abs(r(t))*cos(t),abs(r(t))*sin(t)],

                        t=0..2*PI,LineWidth=0.5,LineColor=[0,1,0]):

radiusordi:=plot::Line2d([t,0],[t,r(t)],t=0..2*PI):

plot(archi,radiusbetrag,radius,archikart,radiusordi,

     AnimationStyle=BackAndForth):

math

MuPAD graphics

image

r:=t->cos(3*t)^2:  r(t);

archi:=plot::Polar([r(t),t],t=0..ende,ende=0..2*PI,

      LineWidth=1, LineColor=RGB::Red, Mesh=400):

archikart:=plot::Curve2d([t,r(t)],t=0..ende,ende=0..2*PI,

      LineWidth=1, Mesh=400, LineColor=RGB::Green):

radius:=plot::Line2d([0,0],[r(t)*cos(t),r(t)*sin(t)],t=0..2*PI,

      LineWidth=0.5):

radiusbetrag:=plot::Line2d([0,0],[abs(r(t))*cos(t),abs(r(t))*sin(t)],

      t=0..2*PI,LineWidth=0.5,LineColor=[0,1,0]):

radiusordi:=plot::Line2d([t,0],[t,r(t)],t=0..2*PI,LineWidth=0.5):

plot(archi,radiusbetrag,radius,archikart,radiusordi,

                                     AnimationStyle=BackAndForth):

 

math

MuPAD graphics

image