Mathematical Experiment 13

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

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

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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:=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);

See also: construction with Derive

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].

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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 (xc,yc) with coordinates given by

is called the center of curvature. The circle with center (xc,yc) radius r is called the osculating circle or the circle of curvature. The center of curvature (xc,yc) lies on the normal of the curve at the point (x,y): the line segment joining (x,y) with (xc,yc) 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.

see: synthetic construction

see: synthetic construction

see: synthetic construction

see: synthetic construction

Construct an animation displaying the various positions of the osculating circle of the following: the nephroid, the astroid, the deltoid and the cardioid.