Mathematical Experiment 13

1 2

Construct the envelope of the light rays emitted from a radiant point source
at infinity after reflection by the semi-circle.

做法：

> m:=[[-2,sin(t)],[cos(t),sin(t)],[cos(t)-cos(2*t),sin(t)-sin(2*t)]];
> t:=n*2*Pi/100;
> plot([m$n=-25..25],color=brown,scaling=constrained,axes=none);

Construct the envelope of the light rays emitted from a radiant point source
at the circumference after reflection by the same circle.

做法：

> m:=[[1,0],[cos(t),sin(t)],[cos(2*t),sin(2*t)]];
> t:=n*2*Pi/100;
> plot([m$n=0..100],color=blue,scaling=constrained,axes=none);
　

Construct the reflections of a light ray trapped inside a reflective ellipse.
There are two cases.

做法：

1.
> with(linalg):
> a:=5;
> b:=3;
> r:=1.34;
> s:=2.95;
> ll:=[a*cos(r),b*sin(r)];
> l:=[a*cos(s),b*sin(s)];
> m:=[l,ll];
> for k to 200 do u:=ll-l: n:=[-b*ll[1]/a,-a*ll[2]/b]:
     v:=u-2*dotprod(u , n)*n/dotprod(n,n):
      t:=-2*(b^2*v[1]*ll[1]+a^2*ll[2]*v[2])/(b^2*v[1]^2+a^2*v[2]^2):
    lll:=ll+t*v:m:=[op(m),lll]:l:=ll:ll:=lll:od:
> plot(m,scaling=constrained,axes=none);

2.
> with(linalg):
> a:=5;
> b:=3;
> r:=1.34;
> s:=5.9;
> ll:=[a*cos(r),b*sin(r)];
> l:=[a*cos(s),b*sin(s)];
> m:=[l,ll];
> for k to 200 do u:=ll-l: n:=[-b*ll[1]/a,-a*ll[2]/b]:
     v:=u-2*dotprod(u , n)*n/dotprod(n,n):
      t:=-2*(b^2*v[1]*ll[1]+a^2*ll[2]*v[2])/(b^2*v[1]^2+a^2*v[2]^2):
    lll:=ll+t*v:m:=[op(m),lll]:l:=ll:ll:=lll:od:
> plot(m,scaling=constrained,axes=none);

Construct the ellipses

x = (1-u) cos t

y = u sin t

0 < t < 2p

as u ranges over 20 equally spaces nodes in [0,1].

Construct an animation displaying various ellipses tangent to the astroid.

Construct the line segments joining points of the curve with the
corresponding center of curvature for each of the following: the nephroid,
the astroid, the deltoid and the cardioid.

做法：

The main part is

> k:=3;
> x:=k*cos(t)-cos(k*t);
> y:=k*sin(t)-sin(k*t);
> x1:=diff(x,t);
> y1:=diff(y,t);
> x2:=diff(x1,t);
> y2:=diff(y1,t);
> f:=(x1^2+y1^2)/(x1*y2-y1*x2+0.000001);
> xc:=x-f*y1;
> yc:=y+f*x1;
> m:=[[x,y],[xc,yc]];
> t:=n*2*Pi/100;
> plot([m$n=1..100],color=red,scaling=constrained,axes=none);

When k:=-3 ------>

When k:=-2 ------>

When k:=2 ------>