blatt02.mw

Blatt 2 und Vorlesung Kapitel 14.2    Kurven im R^n

> restart:

> with(plots):with(linalg):

Warning, the name changecoords has been redefined

Warning, the protected names norm and trace have been redefined and unprotected

14.2 Beispiel 1: Ellipse

> plot([3*cos(t),2*sin(t), t=0..2*Pi],scaling=constrained,thickness=2);

[Plot]

14.2 Beispiel 2: Astroide (vgl. T07)

> plot([(cos(t))^3,(sin(t))^3, t=0..2*Pi],scaling=constrained,thickness=2);

[Plot]

14.2 Beispiel 3: Lissajous-Kurve

> plot([sin(2*t),sin(t), t=0..2*Pi],scaling=constrained,thickness=2);

[Plot]

14.2 Beispiel 4: Schraublinie

> spacecurve([2*cos(t),2*sin(t),t],t=0..6*Pi,thickness=3,axes=normal);

[Plot]

mit Tangente und Bogenlšnge

> a:=2; b:=0.1; t1:=2.5;

a := 2

b := .1

t1 := 2.5

> c:=[a*cos(t),a*sin(t),b*t];

c := [2*cos(t), 2*sin(t), .1*t]

> c1:=diff(c,t); d1:=c1[1]^2+c1[2]^2+c1[3]^2;

c1 := [-2*sin(t), 2*cos(t), .1]

d1 := 4*sin(t)^2+4*cos(t)^2+0.1e-1

> c1:=unapply(c1,t); d1:=unapply(d1,t);

c1 := proc (t) options operator, arrow; [-2*sin(t), 2*cos(t), .1] end proc

d1 := proc (t) options operator, arrow; 4*sin(t)^2+4*cos(t)^2+0.1e-1 end proc

> ct1:=c1(t1);

ct1 := [-1.196944288, -1.602287231, .1]

> p0:=spacecurve(c, t=-0.2..2*Pi, thickness=3, color=blue, axes=normal):

> p1:=spacecurve([a*cos(t),a*sin(t),0], t=0..2*Pi, thickness=2, color=grey):

> p2:=spacecurve([t*cos(t1),t*sin(t1),0], t=0..a, thickness=1, color=grey):

> p3:=spacecurve([a*cos(t1),a*sin(t1),t], t=0..b*t1, thickness=1,color=grey):

> p4:=spacecurve([a*cos(t1)+t*ct1[1],a*sin(t1)+t*ct1[2],b*t1+t*ct1[3]], t=-1..1, thickness=2, color=red):

> Stelle:=spacecurve([a*cos(t1)+0.02*cos(t),a*sin(t1)+0.02*sin(t),b*t1],t=0..2*Pi,thickness=5,color=yellow):

> display(p0,p1,p2,p3,p4,Stelle);

[Plot]

>

> int(sqrt(d1(t)),t=0..t1);

5.006246099

14.2 Beispiel 5:

> spacecurve([t*cos(t),t*sin(t),t],t=0..6*Pi,thickness=3,axes=normal);

[Plot]

14.2 Polardarstellung von ebenen Kurven

> polarplot(1,phi=0..2*Pi,thickness=2);

[Plot]

> polarplot(phi,phi=0..2*Pi,scaling=constrained,thickness=2);

[Plot]

> polarplot(sin(phi),phi=0..Pi,scaling=constrained,thickness=2);

[Plot]

> polarplot(sin(1/2*phi),phi=0..2*Pi,scaling=constrained,thickness=2);

[Plot]

> polarplot(abs(sin(2*phi)),phi=0..2*Pi,scaling=constrained,thickness=2);

[Plot]

Blatt 2 T 8   3D-Kurve auf Kegel

> f:=sqrt((3*x^2+4*y^2)/3);

f := 1/3*(9*x^2+12*y^2)^(1/2)

> c0:=[t^3-3*t,3*t^2,t^3+3*t]; dc0:=diff(c0,t);

c0 := [t^3-3*t, 3*t^2, t^3+3*t]

dc0 := [3*t^2-3, 6*t, 3*t^2+3]

> c:=unapply(c0,t); dc:=unapply(dc0,t);

c := proc (t) options operator, arrow; [t^3-3*t, 3*t^2, t^3+3*t] end proc

dc := proc (t) options operator, arrow; [3*t^2-3, 6*t, 3*t^2+3] end proc

> w:=20; t0:=1.5; ct0:=c(t0); dct0:=dc(t0);

w := 20

t0 := 1.5

ct0 := [-1.125, 6.75, 7.875]

dct0 := [3.75, 9.0, 9.75]

> po:=plot3d(f(x,y),x=-w..w,y=-w..w,view=-w..w, axes=normal, grid=[20,20], style=patchcontour, scaling=constrained, transparency=0.7, tickmarks=[1, 1, 1]):

> pu:=plot3d(-f(x,y),x=-w..w,y=-w..w,view=-w..w, grid=[20,20], style=patchcontour, transparency=0.7):

> p0:=spacecurve([ct0[1]+0.2*cos(t),ct0[2]+0.2*sin(t),ct0[3]], t=0..2*Pi, thickness=7, color=yellow):

> p1:=spacecurve(c(t), t=-3..3, thickness=3, color=blue):

> p2:=spacecurve([ct0[1]+t*dct0[1],ct0[2]+t*dct0[2],ct0[3]+t*dct0[3]], t=-3..3, thickness=2, color=red):

> p3:=spacecurve([ct0[1],ct0[2],ct0[3]+t], t=-50..50, thickness=2, color=green):

> display(po,pu,p0,p1,p2,p3);

[Plot]

Blatt 2 H 4   3D-Kurve auf Paraboloid

> f:=x^2+y^2;

f := x^2+y^2

> c0:=[t*cos(t)-sin(t),t*sin(t)+cos(t),t^2+1]; dc0:=diff(c0,t);

c0 := [t*cos(t)-sin(t), t*sin(t)+cos(t), t^2+1]

dc0 := [-t*sin(t), t*cos(t), 2*t]

> c:=unapply(c0,t); dc:=unapply(dc0,t);

c := proc (t) options operator, arrow; [t*cos(t)-sin(t), t*sin(t)+cos(t), t^2+1] end proc

dc := proc (t) options operator, arrow; [-t*sin(t), t*cos(t), 2*t] end proc

> w:=5; t0:=1.5; ct0:=c(t0); dct0:=dc(t0);

w := 5

t0 := 1.5

ct0 := [-.8913891841, 1.566979682, 3.25]

dct0 := [-1.496242480, .1061058025, 3.0]

> po:=plot3d(f(x,y),x=-w..w,y=-w..w,view=-0.5..w, axes=normal, grid=[20,20], style=patchcontour, scaling=constrained, transparency=0.7, tickmarks=[1, 1, 1]):

> p0:=spacecurve([ct0[1]+0.2*cos(t),ct0[2]+0.2*sin(t),ct0[3]], t=0..2*Pi, thickness=7, color=yellow):

> p1:=spacecurve(c(t), t=0..3, thickness=3, color=blue):

> p2:=spacecurve([ct0[1]+t*dct0[1],ct0[2]+t*dct0[2],ct0[3]+t*dct0[3]], t=-3..3, thickness=2, color=red):

> p3:=spacecurve([ct0[1],ct0[2],ct0[3]+t], t=-10..50, thickness=2, color=green):

> display(po,p0,p1,p2,p3);

[Plot]

Blatt 2 E02 logarithmische Spirale

> polarplot(exp(0.25*phi),phi=-100..Pi,scaling=constrained,thickness=2);

[Plot]

>