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ray_bounces.cdf
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ray_bounces.cdf
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(* Content-type: application/vnd.wolfram.cdf.text *)
(*** Wolfram CDF File ***)
(* http://www.wolfram.com/cdf *)
(* CreatedBy='Mathematica 10.0' *)
(*************************************************************************)
(* *)
(* The Mathematica License under which this file was created prohibits *)
(* restricting third parties in receipt of this file from republishing *)
(* or redistributing it by any means, including but not limited to *)
(* rights management or terms of use, without the express consent of *)
(* Wolfram Research, Inc. For additional information concerning CDF *)
(* licensing and redistribution see: *)
(* *)
(* www.wolfram.com/cdf/adopting-cdf/licensing-options.html *)
(* *)
(*************************************************************************)
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Pattern[$CellContext`n,
Blank[]]] :=
Module[{$CellContext`a0, $CellContext`caustic}, $CellContext`a0 =
If[$CellContext`a < 1,
1/$CellContext`a, $CellContext`a]; $CellContext`caustic = If[
EvenQ[$CellContext`n],
$CellContext`getCausticEvenN[$CellContext`a0, $CellContext`n],
$CellContext`getCausticOddN[$CellContext`a0, $CellContext`n]];
If[$CellContext`a < 1,
$CellContext`rotCaustic[$CellContext`caustic], \
$CellContext`caustic]], $CellContext`getCausticEvenN[
Pattern[$CellContext`a,
Blank[]],
Pattern[$CellContext`n,
Blank[]]] :=
Module[{$CellContext`mod4, $CellContext`ts, $CellContext`ps, \
$CellContext`p1, $CellContext`left, $CellContext`norms, $CellContext`tsyms, \
$CellContext`min, $CellContext`p2, $CellContext`f1, $CellContext`f2, \
$CellContext`f2p, $CellContext`pb, $CellContext`causticAxes, \
$CellContext`poly}, $CellContext`mod4 =
Mod[$CellContext`n, 4] == 0; $CellContext`p1 = {$CellContext`a, 0};
If[$CellContext`n >
4, $CellContext`ts = $CellContext`getInitTsEven[$CellContext`a, \
$CellContext`n]; $CellContext`ps = $CellContext`getEllPs[$CellContext`a, \
$CellContext`ts]; $CellContext`left = If[$CellContext`mod4, {0, 1},
$CellContext`flipX[
Last[$CellContext`ps]]]; $CellContext`norms =
Map[$CellContext`norm,
Map[$CellContext`ellGrad[$CellContext`a,
Part[#, 1],
Part[#, 2]]& , $CellContext`ps]]; $CellContext`tsyms = Table[
Symbol[
StringJoin["t",
ToString[$CellContext`i]]], {$CellContext`i,
Length[$CellContext`ps]}]; $CellContext`min = Quiet[
FindMinimum[
$CellContext`getRelaxErrEven[$CellContext`a, \
$CellContext`tsyms, $CellContext`mod4],
MapThread[{#, #2}& , {$CellContext`tsyms, $CellContext`ts}]]]; \
$CellContext`p2 = Part[
$CellContext`getEllPs[$CellContext`a,
ReplaceAll[$CellContext`tsyms,
Part[$CellContext`min, 2]]], 1], $CellContext`ps = {{0,
1}}; $CellContext`norms = {{-1, 0}}; $CellContext`p2 = {0,
1}; $CellContext`left = {-$CellContext`a,
0}; $CellContext`min = {0, {}};
Null]; {$CellContext`f1, $CellContext`f2} = \
$CellContext`getFoci[$CellContext`a]; $CellContext`f2p = \
$CellContext`reflAboutLine[$CellContext`f2, $CellContext`p1, $CellContext`p2]; \
$CellContext`pb = $CellContext`interRays[$CellContext`f1, $CellContext`f2p - \
$CellContext`f1, $CellContext`p1, $CellContext`p2 - $CellContext`p1]; \
$CellContext`causticAxes = First[
Select[
$CellContext`getConfocalEll[
Sqrt[$CellContext`a^2 - 1], $CellContext`pb], And[
Part[#, 1] > 0, Part[#, 2] > 0]& ]]; $CellContext`poly = Chop[
$CellContext`bounceRay[$CellContext`a, $CellContext`p1, \
$CellContext`p2, $CellContext`n - 1]]; {
"a" -> $CellContext`a, "n" -> $CellContext`n,
"causticAB" -> $CellContext`causticAxes,
"poly" -> $CellContext`poly, "error" -> Part[$CellContext`min, 1],
"f1" -> $CellContext`f1, "f2" -> $CellContext`f2,
"f2p" -> $CellContext`f2p, "pb" -> $CellContext`pb,
"guessPoly" -> {$CellContext`p1,
Apply[Sequence, $CellContext`ps], $CellContext`left},
"guessNorms" -> $CellContext`norms}], $CellContext`getInitTsEven[
Pattern[$CellContext`a,
Blank[]],
Pattern[$CellContext`n,
Blank[]]] :=
Module[{$CellContext`slices, $CellContext`tstep, $CellContext`ts}, \
$CellContext`slices = Floor[($CellContext`n/2 + 1)/2]; $CellContext`tstep =
Pi/(2. $CellContext`slices); $CellContext`ts =
Table[$CellContext`i $CellContext`tstep, {$CellContext`i,
1, $CellContext`slices -
1}]; $CellContext`ts], $CellContext`getEllPs[
Pattern[$CellContext`a,
Blank[]],
Pattern[$CellContext`ts,
Blank[]]] := Map[{$CellContext`a Cos[#],
Sin[#]}& , $CellContext`ts], $CellContext`flipX[{
Pattern[$CellContext`x,
Blank[]],
Pattern[$CellContext`y,
Blank[]]}] := {-$CellContext`x, $CellContext`y}, $CellContext`norm[
Pattern[$CellContext`v,
Blank[]]] := $CellContext`v/$CellContext`magn[$CellContext`v], \
$CellContext`magn[
Pattern[$CellContext`v,
Blank[]]] := Sqrt[
$CellContext`magn2[$CellContext`v]], $CellContext`magn2[
Pattern[$CellContext`v,
Blank[]]] :=
Dot[$CellContext`v, $CellContext`v], $CellContext`ellGrad[
Pattern[$CellContext`a,
Blank[]],
Pattern[$CellContext`x,
Blank[]],
Pattern[$CellContext`y,
Blank[]]] := -{$CellContext`x, $CellContext`y $CellContext`a^2}, \
$CellContext`getRelaxErrEven[
Pattern[$CellContext`a,
Blank[]],
Pattern[$CellContext`ts,
Blank[]],
Pattern[$CellContext`mod4,
Blank[]]] :=
Module[{$CellContext`ps, $CellContext`left, $CellContext`psAug, \
$CellContext`norms, $CellContext`bis}, $CellContext`ps = \
$CellContext`getEllPs[$CellContext`a, $CellContext`ts]; $CellContext`left =
If[$CellContext`mod4, {0, 1},
$CellContext`flipX[
Last[$CellContext`ps]]]; $CellContext`psAug = {{$CellContext`a,
0},
Apply[
Sequence, $CellContext`ps], $CellContext`left}; \
$CellContext`norms = Map[$CellContext`norm,
Map[$CellContext`ellGrad[$CellContext`a,
Part[#, 1],
Part[#, 2]]& , $CellContext`ps]]; $CellContext`bis =
Table[$CellContext`normErr[
Part[$CellContext`psAug, $CellContext`i],
Part[$CellContext`psAug, $CellContext`i - 1],
Part[$CellContext`psAug, $CellContext`i + 1],
Part[$CellContext`norms, $CellContext`i -
1]]^2, {$CellContext`i, 2, Length[$CellContext`psAug] - 1}];
Total[$CellContext`bis]], $CellContext`normErr[
Pattern[$CellContext`p,
Blank[]],
Pattern[$CellContext`pl,
Blank[]],
Pattern[$CellContext`pr,
Blank[]],
Pattern[$CellContext`norm,
Blank[]]] := Dot[
$CellContext`getBisector[$CellContext`pl - $CellContext`p, \
$CellContext`pr - $CellContext`p],
$CellContext`perp[$CellContext`norm]], $CellContext`getBisector[
Pattern[$CellContext`u,
Blank[]],
Pattern[$CellContext`v,
Blank[]]] := $CellContext`norm[$CellContext`norm[$CellContext`u] + \
$CellContext`norm[$CellContext`v]], $CellContext`perp[{
Pattern[$CellContext`x,
Blank[]],
Pattern[$CellContext`y,
Blank[]]}] := {-$CellContext`y, $CellContext`x}, \
$CellContext`getFoci[
Pattern[$CellContext`a,
Blank[]]] := Module[{$CellContext`c},
If[$CellContext`a <
1, $CellContext`c =
Sqrt[1 - $CellContext`a^2]; {{0, $CellContext`c}, {
0, -$CellContext`c}}, $CellContext`c =
Sqrt[$CellContext`a^2 - 1]; {{-$CellContext`c,
0}, {$CellContext`c, 0}}]], $CellContext`reflAboutLine[
Pattern[$CellContext`p,
Blank[]],
Pattern[$CellContext`l1,
Blank[]],
Pattern[$CellContext`l2,
Blank[]]] := $CellContext`refl[$CellContext`p - $CellContext`l1, \
$CellContext`l2 - $CellContext`l1] + $CellContext`l1, $CellContext`refl[
Pattern[$CellContext`v,
Blank[]],
Pattern[$CellContext`n,
Blank[]]] := (2
Dot[$CellContext`v, $CellContext`n]) \
($CellContext`n/$CellContext`magn2[$CellContext`n]) - $CellContext`v, \
$CellContext`interRays[
Pattern[$CellContext`p1,
Blank[]],
Pattern[$CellContext`n1,
Blank[]],
Pattern[$CellContext`p2,
Blank[]],
Pattern[$CellContext`n2,
Blank[]]] :=
Module[{$CellContext`m, $CellContext`b, $CellContext`sols}, \
$CellContext`m = Transpose[{$CellContext`n1, $CellContext`n2}]; If[
$CellContext`negl[
Det[$CellContext`m]], $CellContext`p1, $CellContext`b = \
$CellContext`p2 - $CellContext`p1; $CellContext`sols =
LinearSolve[$CellContext`m, $CellContext`b]; \
$CellContext`ray[$CellContext`p1, $CellContext`n1,
Part[$CellContext`sols, 1]]]], $CellContext`negl[
Pattern[$CellContext`v,
Blank[]]] := Abs[$CellContext`v] < 10^(-18), $CellContext`ray[
Pattern[$CellContext`p0,
Blank[]],
Pattern[$CellContext`phat,
Blank[]],
Pattern[$CellContext`d,
Blank[]]] := $CellContext`p0 + $CellContext`phat $CellContext`d, \
$CellContext`getConfocalEll[
Pattern[$CellContext`c,
Blank[]], {
Pattern[$CellContext`x0,
Blank[]],
Pattern[$CellContext`y0,
Blank[]]}] = {{
Root[$CellContext`c^2 $CellContext`x0^2 + (-$CellContext`c^2 - \
$CellContext`x0^2 - $CellContext`y0^2) #^2 + #^4& , 1], -
Sqrt[-$CellContext`c^2 +
Root[$CellContext`c^2 $CellContext`x0^2 + (-$CellContext`c^2 - \
$CellContext`x0^2 - $CellContext`y0^2) #^2 + #^4& , 1]^2]}, {
Root[$CellContext`c^2 $CellContext`x0^2 + (-$CellContext`c^2 - \
$CellContext`x0^2 - $CellContext`y0^2) #^2 + #^4& , 1],
Sqrt[-$CellContext`c^2 +
Root[$CellContext`c^2 $CellContext`x0^2 + (-$CellContext`c^2 - \
$CellContext`x0^2 - $CellContext`y0^2) #^2 + #^4& , 1]^2]}, {
Root[$CellContext`c^2 $CellContext`x0^2 + (-$CellContext`c^2 - \
$CellContext`x0^2 - $CellContext`y0^2) #^2 + #^4& , 4], -
Sqrt[-$CellContext`c^2 +
Root[$CellContext`c^2 $CellContext`x0^2 + (-$CellContext`c^2 - \
$CellContext`x0^2 - $CellContext`y0^2) #^2 + #^4& , 4]^2]}, {
Root[$CellContext`c^2 $CellContext`x0^2 + (-$CellContext`c^2 - \
$CellContext`x0^2 - $CellContext`y0^2) #^2 + #^4& , 4],
Sqrt[-$CellContext`c^2 +
Root[$CellContext`c^2 $CellContext`x0^2 + (-$CellContext`c^2 - \
$CellContext`x0^2 - $CellContext`y0^2) #^2 + #^4& ,
4]^2]}}, $CellContext`bounceRay[
Pattern[$CellContext`a$,
Blank[]],
Pattern[$CellContext`p1$,
Blank[]],
Pattern[$CellContext`p2$,
Blank[]],
Pattern[$CellContext`bounces$,
Blank[]]] :=
Module[{$CellContext`p1v$, $CellContext`p2v$, $CellContext`p3v$}, \
$CellContext`p1v$ = $CellContext`p1$; $CellContext`p2v$ = $CellContext`p2$; \
{$CellContext`p1$, $CellContext`p2$,
Apply[Sequence,
Table[$CellContext`p3v$ = \
$CellContext`getInterRefl[$CellContext`a$, $CellContext`p1v$, \
$CellContext`p2v$]; $CellContext`p1v$ = $CellContext`p2v$; $CellContext`p2v$ = \
$CellContext`p3v$; $CellContext`p3v$, {FE`i$$1325, $CellContext`bounces$}]]}],
Attributes[$CellContext`a$] = {Temporary},
Attributes[$CellContext`p1$] = {Temporary},
Attributes[$CellContext`p2$] = {Temporary},
Attributes[$CellContext`bounces$] = {Temporary},
Attributes[$CellContext`p1v$] = {Temporary},
Attributes[$CellContext`p2v$] = {Temporary},
Attributes[$CellContext`p3v$] = {Temporary}, $CellContext`getInterRefl[
Pattern[$CellContext`a,
Blank[]],
Pattern[$CellContext`pfrom,
Blank[]],
Pattern[$CellContext`pto,
Blank[]]] :=
Module[{$CellContext`norm, $CellContext`theRefl, $CellContext`pnext}, \
$CellContext`norm = $CellContext`ellGrad[$CellContext`a,
Part[$CellContext`pto, 1],
Part[$CellContext`pto,
2]]; $CellContext`theRefl = \
$CellContext`refl[$CellContext`pfrom - $CellContext`pto, $CellContext`norm]; \
$CellContext`pnext = Part[
$CellContext`ellInterRayUnprot[$CellContext`a, $CellContext`pto, \
$CellContext`theRefl], 2]; $CellContext`pnext], $CellContext`ellInterRayUnprot[
Pattern[$CellContext`a,
Blank[]],
Pattern[$CellContext`p,
Blank[]],
Pattern[$CellContext`n,
Blank[]]] :=
Module[{$CellContext`x, $CellContext`y, $CellContext`nx, \
$CellContext`ny, $CellContext`a2, $CellContext`c2, $CellContext`c1, \
$CellContext`c0, $CellContext`ss}, {$CellContext`x, $CellContext`y} = \
$CellContext`p; {$CellContext`nx, $CellContext`ny} = $CellContext`n; \
$CellContext`a2 = $CellContext`a $CellContext`a; $CellContext`c2 = \
$CellContext`nx^2 + $CellContext`a2 $CellContext`ny^2; $CellContext`c1 =
2 ($CellContext`nx $CellContext`x + ($CellContext`a2 \
$CellContext`ny) $CellContext`y); $CellContext`c0 = $CellContext`x^2 + \
$CellContext`a2 (-1 + $CellContext`y^2); $CellContext`ss = \
$CellContext`quadRootsUnprot[$CellContext`c2, $CellContext`c1, \
$CellContext`c0];
Map[$CellContext`ray[$CellContext`p, $CellContext`n, #]& , \
$CellContext`ss]], $CellContext`quadRootsUnprot[
Pattern[$CellContext`a,
Blank[]],
Pattern[$CellContext`b,
Blank[]],
Pattern[$CellContext`c,
Blank[]]] :=
Module[{$CellContext`det = $CellContext`b^2 - (
4 $CellContext`a) $CellContext`c, $CellContext`sqrtDet}, \
$CellContext`sqrtDet =
Sqrt[$CellContext`det]; {-$CellContext`b - $CellContext`sqrtDet, \
-$CellContext`b + $CellContext`sqrtDet}/(2 $CellContext`a)], FE`i$$1325 =
31, $CellContext`getCausticOddN[
Pattern[$CellContext`a,
Blank[]],
Pattern[$CellContext`n,
Blank[]]] :=
Module[{$CellContext`ts, $CellContext`ps, $CellContext`p1, \
$CellContext`lower, $CellContext`norms, $CellContext`tsyms, $CellContext`min, \
$CellContext`p2, $CellContext`f1, $CellContext`f2, $CellContext`f2p, \
$CellContext`pb, $CellContext`causticAxes, $CellContext`poly}, \
$CellContext`ts = $CellContext`getInitTs2[$CellContext`a, $CellContext`n]; \
$CellContext`ps = $CellContext`getEllPs[$CellContext`a, $CellContext`ts]; \
$CellContext`lower = $CellContext`flipY[
Last[$CellContext`ps]]; $CellContext`p1 = {$CellContext`a,
0}; $CellContext`norms = Map[$CellContext`norm,
Map[$CellContext`ellGrad[$CellContext`a,
Part[#, 1],
Part[#, 2]]& , $CellContext`ps]]; $CellContext`tsyms = Table[
Symbol[
StringJoin["t",
ToString[$CellContext`i]]], {$CellContext`i,
Length[$CellContext`ps]}]; $CellContext`min = Quiet[
FindMinimum[
$CellContext`getRelaxErr[$CellContext`a, $CellContext`tsyms],
MapThread[{#, #2}& , {$CellContext`tsyms, $CellContext`ts}]]]; \
$CellContext`p2 = Part[
$CellContext`getEllPs[$CellContext`a,
ReplaceAll[$CellContext`tsyms,
Part[$CellContext`min, 2]]],
1]; {$CellContext`f1, $CellContext`f2} = \
$CellContext`getFoci[$CellContext`a]; $CellContext`f2p = \
$CellContext`reflAboutLine[$CellContext`f2, $CellContext`p1, $CellContext`p2]; \
$CellContext`pb = $CellContext`interRays[$CellContext`f1, $CellContext`f2p - \
$CellContext`f1, $CellContext`p1, $CellContext`p2 - $CellContext`p1]; \
$CellContext`causticAxes = First[
Select[
$CellContext`getConfocalEll[
Sqrt[$CellContext`a^2 - 1], $CellContext`pb], And[
Part[#, 1] > 0, Part[#, 2] > 0]& ]]; $CellContext`poly = Chop[
$CellContext`bounceRay[$CellContext`a, $CellContext`p1, \
$CellContext`p2, $CellContext`n - 1]]; {
"a" -> $CellContext`a, "n" -> $CellContext`n,
"causticAB" -> $CellContext`causticAxes,
"poly" -> $CellContext`poly, "error" -> Part[$CellContext`min, 1],
"f1" -> $CellContext`f1, "f2" -> $CellContext`f2,
"f2p" -> $CellContext`f2p, "pb" -> $CellContext`pb,
"guessPoly" -> {$CellContext`p1,
Apply[Sequence, $CellContext`ps], $CellContext`lower},
"guessNorms" -> $CellContext`norms}], $CellContext`getInitTs2[
Pattern[$CellContext`a,
Blank[]],
Pattern[$CellContext`n,
Blank[]]] :=
Module[{$CellContext`tstep, $CellContext`ts, $CellContext`c, \
$CellContext`xmin, $CellContext`xmax, $CellContext`xstep, $CellContext`xs}, \
$CellContext`c = If[$CellContext`a < 1, 0.25 $CellContext`a,
Sqrt[$CellContext`a^2 -
1]]; $CellContext`xmax = ($CellContext`c + $CellContext`a)/
2; $CellContext`xmin = -$CellContext`xmax; $CellContext`xstep = \
($CellContext`xmax - $CellContext`xmin) (2/($CellContext`n -
1)); $CellContext`xs =
Table[$CellContext`xmax - $CellContext`i $CellContext`xstep, \
{$CellContext`i, 1, ($CellContext`n - 1)/2}]; $CellContext`ts =
Map[ArcCos[#/$CellContext`a]& , $CellContext`xs]; \
$CellContext`ts], $CellContext`flipY[{
Pattern[$CellContext`x,
Blank[]],
Pattern[$CellContext`y,
Blank[]]}] := {$CellContext`x, -$CellContext`y}, \
$CellContext`getRelaxErr[
Pattern[$CellContext`a,
Blank[]],
Pattern[$CellContext`ts,
Blank[]]] :=
Module[{$CellContext`ps, $CellContext`lower, $CellContext`psAug, \
$CellContext`norms, $CellContext`bis}, $CellContext`ps = \
$CellContext`getEllPs[$CellContext`a, $CellContext`ts]; $CellContext`lower = \
$CellContext`flipY[
Last[$CellContext`ps]]; $CellContext`psAug = {{$CellContext`a,
0},
Apply[
Sequence, $CellContext`ps], $CellContext`lower}; \
$CellContext`norms = Map[$CellContext`norm,
Map[$CellContext`ellGrad[$CellContext`a,
Part[#, 1],
Part[#, 2]]& , $CellContext`ps]]; $CellContext`bis =
Table[$CellContext`normErr[
Part[$CellContext`psAug, $CellContext`i],
Part[$CellContext`psAug, $CellContext`i - 1],
Part[$CellContext`psAug, $CellContext`i + 1],
Part[$CellContext`norms, $CellContext`i -
1]]^2, {$CellContext`i, 2, Length[$CellContext`psAug] - 1}];
Total[$CellContext`bis]], $CellContext`rotCaustic[
Pattern[$CellContext`caustic,
Blank[]]] :=
Module[{$CellContext`a}, $CellContext`a =
1/ReplaceAll["a", $CellContext`caustic]; {
"a" -> $CellContext`a, "n" ->
ReplaceAll["n", $CellContext`caustic],
"causticAB" -> $CellContext`a Reverse[
ReplaceAll["causticAB", $CellContext`caustic]],
"poly" -> $CellContext`a Map[$CellContext`perp,
ReplaceAll["poly", $CellContext`caustic]], "error" ->
ReplaceAll["error", $CellContext`caustic],
"f1" -> $CellContext`a $CellContext`perp[
ReplaceAll["f1", $CellContext`caustic]],
"f2" -> $CellContext`a $CellContext`perp[
ReplaceAll["f2", $CellContext`caustic]],
"f2p" -> $CellContext`a $CellContext`perp[
ReplaceAll["f2p", $CellContext`caustic]],
"pb" -> $CellContext`a $CellContext`perp[
ReplaceAll["pb", $CellContext`caustic]],
"guessPoly" -> $CellContext`a Map[$CellContext`perp,
ReplaceAll["guessPoly", $CellContext`caustic]],
"guessNorms" -> $CellContext`a Map[$CellContext`perp,
ReplaceAll[
"guessNorms", $CellContext`caustic]]}], $CellContext`toRad[
Pattern[$CellContext`d,
Blank[]]] := Pi ($CellContext`d/180.), $CellContext`toDeg[
Pattern[$CellContext`r,
Blank[]]] :=
180. ($CellContext`r/Pi), $CellContext`getAlphaAlignCaustic[
Pattern[$CellContext`a,
Blank[]],
Pattern[$CellContext`tDeg0,
Blank[]],
Pattern[$CellContext`causticN,
Blank[]]] :=
Module[{$CellContext`c, $CellContext`ns, $CellContext`causticABs, \
$CellContext`match, $CellContext`t, $CellContext`p0, $CellContext`n0, \
$CellContext`n0rot}, $CellContext`c =
Sqrt[$CellContext`a^2 - 1]; $CellContext`ns = {3, 4, 5,
6}; $CellContext`causticABs = Map[ReplaceAll["causticAB",
$CellContext`getCausticBoth[$CellContext`a, #]]& , \
$CellContext`ns]; $CellContext`match = First[
FirstPosition[$CellContext`ns, $CellContext`causticN]]; \
$CellContext`t = $CellContext`toRad[$CellContext`tDeg0]; $CellContext`p0 = \
{$CellContext`a Cos[$CellContext`t],
Sin[$CellContext`t]}; $CellContext`n0 = $CellContext`norm[
$CellContext`ellGrad[$CellContext`a,
Apply[
Sequence, $CellContext`p0]]]; $CellContext`n0rot = \
$CellContext`norm[$CellContext`getTangCW[$CellContext`a, $CellContext`p0,
Apply[Sequence,
Part[$CellContext`causticABs, $CellContext`match]]] - \
$CellContext`p0]; $CellContext`toDeg[
ArcTan[
Dot[$CellContext`n0rot, $CellContext`n0],
Dot[$CellContext`n0rot,
$CellContext`perp[$CellContext`n0]]]]], $CellContext`getTangCW[
Pattern[$CellContext`a,
Blank[]],
Pattern[$CellContext`p1,
Blank[]],
Pattern[$CellContext`acaustic,
Blank[]],
Pattern[$CellContext`bcaustic,
Blank[]]] :=
Module[{$CellContext`tangs, $CellContext`norm}, $CellContext`tangs = \
$CellContext`ellTangentsAB[$CellContext`acaustic, $CellContext`bcaustic, \
$CellContext`p1]; $CellContext`norm = $CellContext`ellGrad[$CellContext`a,
Apply[Sequence, $CellContext`p1]];
If[Dot[Part[$CellContext`tangs, 1] - $CellContext`p1,
$CellContext`perp[$CellContext`norm]] > 0,
Part[$CellContext`tangs, 1],
Part[$CellContext`tangs, 2]]], $CellContext`ellTangentsAB[
Pattern[$CellContext`a,
Blank[]],
Pattern[$CellContext`b,
Blank[]],
Pattern[$CellContext`p1,
Blank[]]] := $CellContext`b \
$CellContext`ellTangents[$CellContext`a/$CellContext`b, \
$CellContext`p1/$CellContext`b], $CellContext`ellTangents =
CompiledFunction[{10, 10., 5468}, {
Blank[Real], {
Blank[Real], 1}}, {{3, 0, 0}, {3, 1, 0}, {3, 2,
1}}, {{2, {2, 0, 1}}, {-1, {2, 0, 3}}, {1, {2, 0, 2}}}, {1, 4, 16,
0, 2}, {{42, "CopyTensor", 3, 1, 0, 3, 1, 1}, {33, 1, 0}, {24, 0, 1,
0}, {32, 0, 0}, {2, 0, 3}, {49}, {3, 1}, {37, 1, 2, 3, 1}, {37, 1,
1, 3, 2}, {16, 0, 0, 3}, {16, 1, 1, 4}, {16, 2, 2, 5}, {16, 1, 4,
6}, {16, 3, 1, 5, 7}, {13, 6, 7, 8}, {16, 3, 5, 7}, {13, 4, 7,
9}, {10, 3, 7}, {13, 5, 7, 10}, {16, 3, 10, 7}, {13, 4, 7, 10}, {
40, 57, 3, 0, 10, 3, 0, 7}, {40, 38, 3, 0, 1, 3, 0, 11}, {16, 7,
11, 7}, {16, 2, 7, 11}, {13, 4, 11, 12}, {
47, $CellContext`safeDiv, 3, 0, 12, 3, 0, 8, 3, 0, 11}, {16, 3, 11,
12}, {16, 3, 2, 11}, {19, 7, 13}, {13, 11, 13, 11}, {
47, $CellContext`safeDiv, 3, 0, 11, 3, 0, 9, 3, 0, 13}, {16, 2, 7,
11}, {19, 11, 14}, {13, 4, 14, 11}, {
47, $CellContext`safeDiv, 3, 0, 11, 3, 0, 8, 3, 0, 14}, {16, 3, 14,
11}, {16, 3, 2, 14}, {13, 14, 7, 14}, {
47, $CellContext`safeDiv, 3, 0, 14, 3, 0, 9, 3, 0, 15}, {34, 2, 2,
2, 12, 13, 11, 15, 3, 1}, {1}},
Function[{$CellContext`a, $CellContext`p},
Module[{$CellContext`a2, $CellContext`px, $CellContext`py, \
$CellContext`px2, $CellContext`px3, $CellContext`py2, $CellContext`radicand, \
$CellContext`numFact, $CellContext`denomx, $CellContext`denomy}, \
{$CellContext`px, $CellContext`py} = $CellContext`p; $CellContext`a2 = \
$CellContext`a $CellContext`a; $CellContext`px2 = $CellContext`px \
$CellContext`px; $CellContext`py2 = $CellContext`py $CellContext`py; \
$CellContext`px3 = $CellContext`px $CellContext`px2; $CellContext`denomx = \
$CellContext`px3 + ($CellContext`a2 $CellContext`px) $CellContext`py2; \
$CellContext`denomy = $CellContext`px2 + $CellContext`a2 $CellContext`py2; \
$CellContext`radicand = $CellContext`px2 + $CellContext`a2 ($CellContext`py2 -
1); $CellContext`numFact =
Sqrt[$CellContext`radicand]
Abs[$CellContext`px]; {{$CellContext`a2 \
$CellContext`safeDiv[$CellContext`px2 + $CellContext`py $CellContext`numFact, \
$CellContext`denomx],
$CellContext`safeDiv[$CellContext`a2 $CellContext`py - \
$CellContext`numFact, $CellContext`denomy]}, {$CellContext`a2 \
$CellContext`safeDiv[$CellContext`px2 - $CellContext`py $CellContext`numFact, \
$CellContext`denomx],
$CellContext`safeDiv[$CellContext`a2 $CellContext`py + \
$CellContext`numFact, $CellContext`denomy]}}]],
Evaluate], $CellContext`safeDiv[
Pattern[$CellContext`num,
Blank[]],
Pattern[$CellContext`denom,
Blank[]]] := If[
$CellContext`negl[$CellContext`denom],
0, $CellContext`num/$CellContext`denom], $CellContext`showBasicImpa[
Pattern[$CellContext`a,
Blank[]],
Pattern[$CellContext`ell,
Blank[]],
Pattern[$CellContext`tDeg0,
Blank[]],
Pattern[$CellContext`alphaDeg,
Blank[]],
OptionsPattern[]] :=
Module[{$CellContext`t, $CellContext`p0, $CellContext`n0, \
$CellContext`n0rot, $CellContext`p1, $CellContext`c, $CellContext`gr, \
$CellContext`fs, $CellContext`confocalAxes, $CellContext`ns, \
$CellContext`causticABs, $CellContext`ellCaustics, $CellContext`poly, \
$CellContext`match, $CellContext`norms, $CellContext`vtxTs, \
$CellContext`vtxOrder, $CellContext`edges, $CellContext`normalArrows}, \
$CellContext`fs = $CellContext`getFoci[$CellContext`a]; $CellContext`c =
Sqrt[$CellContext`a^2 - 1]; $CellContext`ns = {3, 4, 5,
6}; $CellContext`causticABs = Map[ReplaceAll["causticAB",
$CellContext`getCausticBoth[$CellContext`a, #]]& , \
$CellContext`ns]; $CellContext`match =
If[OptionValue[$CellContext`alignCaustic] != 0,
First[
FirstPosition[$CellContext`ns,
OptionValue[$CellContext`alignCaustic]]],
1]; $CellContext`t = $CellContext`toRad[$CellContext`tDeg0]; \
$CellContext`p0 = {$CellContext`a Cos[$CellContext`t],
Sin[$CellContext`t]}; $CellContext`n0 = $CellContext`norm[
$CellContext`ellGrad[$CellContext`a,
Apply[Sequence, $CellContext`p0]]]; $CellContext`n0rot = If[
Or[
OptionValue[$CellContext`alignFoci] == 1,
OptionValue[$CellContext`alignFoci] == 2],
$CellContext`norm[Part[$CellContext`fs,
OptionValue[$CellContext`alignFoci]] - $CellContext`p0],
If[OptionValue[$CellContext`alignCaustic] > 0,
$CellContext`norm[$CellContext`getTangCW[$CellContext`a, \
$CellContext`p0,
Apply[Sequence,
Part[$CellContext`causticABs, $CellContext`match]]] - \
$CellContext`p0], $CellContext`n0rot = $CellContext`rot[$CellContext`n0,
$CellContext`toRad[$CellContext`alphaDeg]]]]; $CellContext`p1 =
Part[
$CellContext`ellInterRayUnprot[$CellContext`a, $CellContext`p0, \
$CellContext`n0rot],
2]; $CellContext`poly = $CellContext`bounceRay[$CellContext`a, \
$CellContext`p0, $CellContext`p1,
OptionValue[$CellContext`maxBounce]]; $CellContext`vtxTs =
Map[$CellContext`ArcTan2Pi[
Part[#, 1], $CellContext`a
Part[#, 2]]& , $CellContext`poly]; $CellContext`vtxOrder =
Ordering[$CellContext`vtxTs]; $CellContext`norms = If[
OptionValue[$CellContext`drNormals],
Map[$CellContext`norm[
$CellContext`ellGrad[$CellContext`a,
Apply[Sequence, #]]]& ,
Most[$CellContext`poly]], {}]; $CellContext`normalArrows = If[
OptionValue[$CellContext`drNormals],
MapThread[{#, # + OptionValue[$CellContext`lgt] #2}& , {
Most[$CellContext`poly], $CellContext`norms}], {}]; \
$CellContext`ellCaustics = If[
OptionValue[$CellContext`drCaustics],
Map[$CellContext`plotEllb[
Apply[Sequence, #], {Thick,
Darker[
Green]}]& , $CellContext`causticABs], {}]; $CellContext`edges =
MapThread[{#, #2}& , {
Most[$CellContext`poly],
Rest[$CellContext`poly]}]; $CellContext`gr = Graphics[{
FaceForm[None],
PointSize[Large],
Arrowheads[Medium], {Black,
Point[$CellContext`fs], Dashed,
Line[{{-$CellContext`a, 0}, {$CellContext`a, 0}}],
Line[{{0, -1}, {0, 1}}]},
If[
OptionValue[$CellContext`drPrimeEdges],
Module[{$CellContext`primes, $CellContext`notPrimes}, \
$CellContext`primes = Map[PrimeQ,
Range[
Length[$CellContext`poly] - 1]]; $CellContext`notPrimes =
Map[Not, $CellContext`primes]; {Thick,
Map[{Blue,
Line[{
Part[#, 1],
Part[#, 2]}],
Point[
Part[#, 1]]}& ,
Pick[$CellContext`edges, $CellContext`notPrimes]],
Map[{Red,
Line[{
Part[#, 1],
Part[#, 2]}],
Point[
Part[#, 1]]}& ,
Pick[$CellContext`edges, $CellContext`primes]]}], {}],
If[
OptionValue[$CellContext`drPrimeVertices],
Module[{$CellContext`primes, $CellContext`notPrimes}, \
$CellContext`primes = Map[PrimeQ,
Range[
Length[$CellContext`poly] - 1]]; $CellContext`notPrimes =
Map[Not, $CellContext`primes]; {Thick,
Map[{Blue,
Point[
Part[#, 1]]}& ,
Pick[$CellContext`edges, $CellContext`notPrimes]],
Map[{Red,
Point[
Part[#, 1]]}& ,
Pick[$CellContext`edges, $CellContext`primes]]}], {}],
If[
OptionValue[$CellContext`drPowerTwo],
Module[{$CellContext`pow2s, $CellContext`notPow2s}, \
$CellContext`pow2s = Map[# + 0& , 2^Range[1,
Floor[
Log[
2, Length[$CellContext`poly] -
1]]]]; $CellContext`notPow2s = Complement[
Range[
1, Length[$CellContext`poly] - 1], $CellContext`pow2s]; {
Thick,
Map[{Blue,
Point[
Part[#, 1]]}& ,
Part[$CellContext`edges, $CellContext`notPow2s]],
Map[{
Darker[Green],
Point[
Part[#, 1]],
Line[{
Part[#, 1],
Part[#, 2]}]}& ,
Part[$CellContext`edges, $CellContext`pow2s]]}], {}],
If[
And[
Not[
OptionValue[$CellContext`drPrimeEdges]],
Not[
OptionValue[$CellContext`drPrimeVertices]],
Not[
OptionValue[$CellContext`drPowerTwo]]], {Blue, Thick,
Line[$CellContext`poly],
Point[
Rest[$CellContext`poly]]}, {}],
If[
OptionValue[$CellContext`drNormals], {Thick,
Map[Arrow, $CellContext`normalArrows]}, {}],
If[
OptionValue[$CellContext`drPs],