Construct the envelope of the light rays emitted from a
radiant point source at infinity after reflection by the semicircle.
¡@ ¡@ 
Construct the envelope of the light rays emitted from a
radiant point source at the circumference after reflection by the same
circle. ¡@ ¡@ 
Construct the reflections of a light ray trapped inside a reflective ellipse. There are two cases.  
> restart;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:=lll: n:=[b*ll[1]/a,a*ll[2]/b]:v:=u2*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); See also: construction with Derive 

Construct the ellipses x = (1u) 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. 
For a plane curve (x(t),y(t)) its curvature is given by
The quantity
is called the radius of curvature. The point (x_{c},y_{c}) with coordinates given by
is called the center of curvature. The circle with center (x_{c},y_{c}) radius r is called the osculating circle or the circle of curvature. The center of curvature (x_{c},y_{c}) lies on the normal of the curve at the point (x,y): the line segment joining (x,y) with (x_{c},y_{c}) is perpendicular to the tangent at (x,y).
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.
Construct an animation displaying the various positions of the osculating circle of the following: the nephroid, the astroid, the deltoid and the cardioid.


