Parallel Curves, Evolutes, and Caustics
                       by Adrian Mariano
                   email adrian@u.washington.edu


                                Caustic.m


If light is shined from a point source onto a curved mirror, it
focuses on a curve called the caustic.  The orthotomic is an
intermediate curve used to obtain the caustic.  If gamma is a
parametrized curve, then the orthotomic of gamma relative to the point
S is given by
               2((gamma - S) . N ) N
and the caustic of gamma for the light source at S is the evolute
of the orthotomic.

Orthotomic[{fx,fy}, {i,j}, t, <Options>]

Returns an expression for the orthtomic of {fx,fy} with respect to the
point {x,y}.  The curve {fx,fy} is specified parametrically in t.  The
only option is to disable internal simplification via calls to
TrigReduce and Simplify with Simplify->False.

Caustic[{fx,fy}, {i,j}, t, <Options>]

Returns an expression for the caustic of {fx,fy} relative to the light
source {i,j}.  The only option is to disable internal simplification
with Simplify->False.

EqnEllipse[a, {i, j},  t]

If a caustic is drawn to an ellipse with axes a and 1 from a light
source at {i,j}, the caustic may have asymptotes.  The roots of this
equation of t give the t values for the points at infinity, assuming
the parametrization {a Cos[t], Sin[t]}.

EqnCircle[i, t]

If a caustic is drawn to a unit circle from the point {i,0}, it may
have asymptotes.  The roots of this equation of t give the t values
for the points at infinity, assuming the usual parametrization for the
circle.

CausticBadEqn[{fx,fy}, {i,j}, t]

Given the mirror defined by {fx,fy}, the caustic from {i,j} may have
asymtotes.  This function returns an equation whose roots are the
parameter values for the points at infinity of the specified caustic.

EllipseCaustic[a, {i, j}, t]

Gives the equation for a caustic to the ellipse {a Cos[t], Sin[t]}
from the point {i,j}.

CircleCaustic[i_, t_]

Gives the equation for the caustic to the unit circle from the point
{i,0}.


CircleCausticBad[i_]

Returns a list of parameter values near which the caustic to the unit
circle from {i,0} is not defined.

EllipseCausticBad[a, {i, j}]

Returns a list of parameter values near which the caustic to the
ellipse from {i,j} is not defined.


CausticBad[{fx,fy}, {i,j}, {t, tmin, tmax}]

Returns the points in {tmin, tmax} which give bad sections of the
caustic to {fx,fy} from {i,j}.

PlotCircleCaustic[i, plotrange, <Options>]
PlotEllipseCaustic[a, {i, j}, plotrange, <Options>]
PlotCaustic[{fx,fy}, {i, j}, {t, tmin, tmax}, plotrange, <Options>]

Plot the caustic to a unit circle, ellipse, or arbitrary curve.  This
function will graph the caustics and properly handle asymptotes.  For
caustics to a circle, the light source is at {i,0}.  For arbitrary
caustics, the mirror is defined by {fx,fy}, evaluated over the
specified range of t.

The options are
  ShowSource->False
  Normally the light source is displayed as a dot.  This can be disabled.
 
  DotSize->.015  
  The default size of the light source dot is .015.  This can be changed
  as desired

  ShowCurve->False
  By default, the mirror curve is displayed.  This can be disable.

Any other options are passed on to Show.