KM03_ex5.mw

> restart:with(plots):

Warning, the name changecoords has been redefined

EXERCICE 5

> a:=5:b:=3:F:=(x,y)->x^2/a^2+y^2/b^2-1;

F := proc (x, y) options operator, arrow; x^2/a^2+y^2/b^2-1 end proc

> P:=[0,0];theta:=Pi/4;

P := [0, 0]

theta := 1/4*Pi

>

> d:=[cos(theta),sin(theta)]:trajet:=P:#d:=[4/5,3/5];

> for i from 1 to 5 do
Imp:=impact_ellipse(P,d);
P:=Imp[1]:d:=Imp[2]:trajet:=trajet,P;
od:

>

> trajet;

[0, 0], [15/34*17^(1/2)*2^(1/2), 15/34*17^(1/2)*2^(1/2)], [3230865/6195106*17^(1/2)*2^(1/2), -2530095/6195106*17^(1/2)*2^(1/2)], [-58382653940122065/176611083474241186*17^(1/2)*2^(1/2), 83842080837752...[0, 0], [15/34*17^(1/2)*2^(1/2), 15/34*17^(1/2)*2^(1/2)], [3230865/6195106*17^(1/2)*2^(1/2), -2530095/6195106*17^(1/2)*2^(1/2)], [-58382653940122065/176611083474241186*17^(1/2)*2^(1/2), 83842080837752...

>

> Boule:=plot([trajet],style=line,thickness=2):

> Billard:=implicitplot(F(x,y)=0,x=-5..5,y=-5..5,color=blue):

>

> display({Boule, Billard});

[Plot]

> b:=3:P:=[-5,0];

P := [-5, 0]

> d:=[5/sqrt(34),3/sqrt(34)];trajet:=P:

d := [5/34*34^(1/2), 3/34*34^(1/2)]

> for i from 1 to 150 do
Imp:=impact_ellipse(P,d);
P:=evalf(Imp[1]):d:=evalf(Imp[2]):trajet:=trajet,P;
od:

>

> Boule:=plot([trajet],style=line,thickness=1):

> Billard:=implicitplot(F(x,y)=0,x=-5..5,y=-5..5,color=blue,thickness=1):

> display({Boule, Billard},axes=none);

[Plot]

Voici d'autres rebonds possibles depuis des configurations initiales distinctes

> display({Boule, Billard},axes=none);

[Plot]

>

> display({Boule, Billard},axes=none);

[Plot]

Question b: Billard Circulaire

> P=[-5,0]:theta=Pi/3:

> display({Boule, Billard},axes=none);

[Plot]

> P=[-5,0]:theta=Pi/6:

> display({Boule, Billard},axes=none);

[Plot]

> P=[-5,0]:theta=1:

> display({Boule, Billard},axes=none);

[Plot]

> P=[-3,0]:theta=Pi/6:

> display({Boule, Billard},axes=none);

[Plot]

>