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, ]
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, ]
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, ]
PlotEllipseCaustic[a, {i, j}, plotrange, ]
PlotCaustic[{fx,fy}, {i, j}, {t, tmin, tmax}, plotrange, ]
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.