Pascalsche Schnecken in polar-kartesischer Darstellung

Prof. Dr. Dörte Haftendorn: Mathematik mit MuPAD 4,  Aug. 07 Update 20.08.07

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kreis:=t->cos(t);

r:=t->kreis(t)+k;

math

math

k:=0.5: //Leinenlänge

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

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

pkt:=plotPoint2d([r(t),t],t=0..ende,ende=0..2*PI,PointSize=1.1):

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

        LineWidth=0.5):

pascalkart:=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(pascal,radiusbetrag2,radius2,pascalkart,radiusordi2,LineWidth=0.5,

     AnimationStyle=BackAndForth);

MuPAD graphics

image

Variation der Leinenlänge

k:=1.5: //Leinenlänge

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

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

pkt:=plotPoint2d([r(t),t],t=0..ende,ende=0..2*PI,PointSize=1.1):

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

        LineWidth=0.5):

pascalkart:=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(pascal,radiusbetrag2,radius2,pascalkart,radiusordi2,LineWidth=0.5,

     AnimationStyle=BackAndForth);

MuPAD graphics

image

k:=1: //Leinenlänge

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

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

pkt:=plotPoint2d([r(t),t],t=0..ende,ende=0..2*PI,PointSize=1.1):

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

        LineWidth=0.5):

pascalkart:=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(pascal,radiusbetrag2,radius2,pascalkart,radiusordi2,LineWidth=0.5,

     AnimationStyle=BackAndForth);

MuPAD graphics

image

Ob sich beide Kurven  treffen, ist in einer Extraseite untersucht.

delete k

Animation der Pascalschen Schnecken allein:

pascalk:=plot::Polar([r(t),t],t=0..2*PI,k=-1.5..1.5,

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

kreisg:=plot::Polar([r(t)|k=0,t],t=0..2*PI,

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

plot(pascalk,kreisg)

MuPAD graphics

image