Mathematical Experiment 11

Construct a family of 20 concentric circles:

plot([[n*cos(t),n*sin(t),t=0..2*Pi]$n=1..20], color=red,axes=none);

Construct a family of circles each centered at [cos t, sin t] passing through the point [1,0]:

x:=cos(t)+d*cos(s);y:=sin(t)+d*sin(s);
t:=k*2*Pi/100;
with(linalg):
d:=norm([cos(t),sin(t)]-[1,0],2);
plot([[x,y,s=0..2*Pi]$k=1..100],color=red, scaling=constrained,axes=none);

Construct a family of circles each centered at [cos t, sin t] and tangent to the x-axis:

x:=cos(t)+d*cos(s);y:=sin(t)+d*sin(s);
t:=k*2*Pi/100;
with(linalg):
d:=sin(t);
plot([[x,y,s=0..2*Pi]$k=1..100],color=red, scaling=constrained,axes=none);

Construct a family of circles each centered at [3cos t, 3sin t] and passing through [2cos t + cos 2t, 2sin t - sin 2t]:

x:=3*cos(t)+d*cos(s);y:=3*sin(t)+d*sin(s);
t:=k*2*Pi/100;
with(linalg):
d:=norm([2*cos(t)+cos(-2*t),2*sin(t)+sin(-2*t)]-[3*cos(t),3*sin(t)],2);
plot([[x,y,s=0..2*Pi]$k=1..100],color=red, scaling=constrained,axes=none);

Construct a family of circles each centered at [4cos t, 4sin t] and passing through [3cos t + cos 3t, 3sin t - sin 3t]:

x:=4*cos(t)+d*cos(s);y:=4*sin(t)+d*sin(s);
t:=k*2*Pi/100;
with(linalg):
d:=norm([3*cos(t)+cos(-3*t),3*sin(t)+sin(-3*t)]-[4*cos(t),4*sin(t)],2);
plot([[x,y,s=0..2*Pi]$k=1..100], color=red,scaling=constrained,axes=none);

Construct a family of ellipses enveloping the astroid:

plot([[(1-k/25)*cos(t),k/25*sin(t),t=0..2*Pi]$k=0..25], color=red,scaling=constrained,axes=none);

Construct the line segments joining
(cos(t),0) with (0,sin(t)) as t ranges
over [0,2p].

> x:=cos(t);
> y:=sin(t);
> m:=[[x,0],[0,y]];
> t:=n*Pi/50;
> plot([m$n=1..100],color=green,
scaling=constrained,axes=none);

@

Construct the velocity vector field
along a constant motion around a circle.

Construct the line segments joining
 [cos(t),sin(t)] with [cos(2t),sin(2t)]
 as t ranges over [0,2p].

Construct the line segments
joining [cos(t),sin(t)] with
[cos(3t),sin(3t)] as t ranges
over [0,2p].

Construct this pattern: