Post a Comment On: C0DE517E

No big news yet, I've been studying a little bit analytic methods for shading and occlusion, I can't report anything really now, even because I'm not yet satisfied with what I've done.

But I'd like to share this link: http://www.me.utexas.edu/~howell/tablecon.html, differential element to finite area is what you’ll need.

Also, you might find this useful, if you're starting to play around with spherical harmonics, it's a small snipped of what I've been doing, with Mathematica:
(* analytic solution for real spherical harmonics test *)

shIndices[level_] := (Range[-#1, #1] & ) /@ Range[0, level]
shGetNormFn[l_, m_] := Sqrt[((2*l + 1)*(l - m)!)/(4*Pi*(l + m)!)]
shGetFn[l_, m_] :=
Piecewise[{{shGetNormFn[l, 0]*LegendreP[l, 0, Cos[\[Theta]]],
m == 0}, {Sqrt[2]*shGetNormFn[l, m]*Cos[m*\[Phi]]*
LegendreP[l, m, Cos[\[Theta]]],
m > 0}, {Sqrt[2]*shGetNormFn[l, -m]*Sin[(-m)*\[Phi]]*
LegendreP[l, -m, Cos[\[Theta]]], m <>
shFunctions[level_] :=
MapIndexed[
Function[{list, currlevel}, (shGetFn[currlevel - 1, #1] & ) /@
list], shIndices[level]]
shGenCoeffs[shfns_, fn_] :=
Map[Integrate[#1*fn[\[Theta], \[Phi]]*Sin[\[Theta]], {\[Theta], 0,
Pi}, {\[Phi], 0, 2*Pi}] & , shfns, {2}]
shReconstruct[shfns_, shcoeffs_] :=
Simplify[Plus @@ (Flatten[shcoeffs]*Flatten[shfns]),
Assumptions -> {Element[\[Theta], Reals],
Element[\[Phi], Reals], \[Theta] >= 0, \[Phi] >= 0, \[Theta] <=
Pi, \[Phi] <= 2*Pi}]

shIsZonal[shcoeffs_, level_] :=
Plus @@ (Flatten[shIndices[level]] Flatten[shcoeffs]) == 0
shGetSymConvolveNorm[level_] :=
MapIndexed[
Function[{list, currlevel},
Table[Sqrt[(4 \[Pi])/(2 currlevel + 1)], {Length[list]}]],
shIndices[level]]
shGetSymCoeffs[shcoeffs_] :=
Table[#1[[Ceiling[Length[#1]/2]]], {Length[#1]}] & /@ shcoeffs
shSymConvolve[shcoeffs_, shsymkerncoeffs_,
level_] := (Check[shIsZonal[shsymkerncoeffs], err];
shGetSymConvolveNorm[level] shcoeffs shGetSymCoeffs[
shsymkerncoeffs])

(* tests.... *)

testnumlevels = 2
testfn[a_, b_] :=
Cos[a]^10*UnitStep[Cos[a]] (*symmetric on the z axis*)
(*testfn[a_,b_]:= (a/Pi)^4*)
shfns = shFunctions[testnumlevels]
testfncoeffs = shGenCoeffs[shfns, testfn]
shIsZonal[testfncoeffs, testnumlevels]
testfnrec = {\[Theta], \[Phi]} \[Function]
Evaluate[shReconstruct[shfns, testfncoeffs]]
SphericalPlot3D[{testfn[\[Theta], \[Phi]],
testfnrec[\[Theta], \[Phi]]}, {\[Theta], 0, Pi}, {\[Phi], 0, 2 Pi},
Mesh -> False, PlotRange -> Full]

testfn2[a_, b_] := UnitStep[Cos[a] Sin[b]](*asymmetric*)
testfn2coeffs = shGenCoeffs[shfns, testfn2]
testfn3coeffs =
shSymConvolve[testfn2coeffs, testfncoeffs, testnumlevels]
testfn2rec = {\[Theta], \[Phi]} \[Function]
Evaluate[shReconstruct[shfns, testfn2coeffs]]
testfn3rec = {\[Theta], \[Phi]} \[Function]
Evaluate[shReconstruct[shfns, testfn3coeffs]]
SphericalPlot3D[{testfn2[\[Theta], \[Phi]],(*testfn2rec[\[Theta],\
\[Phi]],*)testfn3rec[\[Theta], \[Phi]]}, {\[Theta], 0, Pi}, {\[Phi],
0, 2 Pi}, Mesh -> False, PlotRange -> Full]