# Can not understand the code to plot Fourier Series approximation of the line

I'm reading the CDF file from Making Formulas… for Everything in the WolframAlpha blog. I am confused by what's happening with In[9] and In[10].

Can anyone tell me what these two functions mean, or point out a book or documentation that I should read for reference? The expression in fourierCoefficientLineSegment looks quite different from the definition of fourier-series.

fourierCoefficientLineSegment[cs_, n_, {{t1_, x1_}, {t2_, x2_}}] :=
With[{a = -(t2 x1 - t1 x2)/(t1 - t2), b = -(x2 - x1)/(t1 - t2)},
If[cs === Cos,
If[n == 0, (t2 - t1) (2 a + b (t1 + t2))/2,
(-(n (a + b t1) Sin[n t1]) + n*(a + b*t2)*Sin[n*t2] -
b Cos[n t1] + b Cos[n t2])/n^2],
If[n == 0, 0,
(n (a + b t1) Cos[n t1] - n (a + b t2)*Cos[n t2] +
b (Sin[n t2] - Sin[n t1]))/n^2]]]

fourierSeriesLine[l_, n_, t_] :=
Module[{ls, L, ts, xs, ys, Xs, Ys },
ls = EuclideanDistance @@@ Partition[l, 2, 1];
L = Total[ls];
ts = 1. Prepend[Accumulate[ls], 0]/L 2 Pi - Pi;
{xs, ys} = Transpose[l];
{Xs, Ys} = Partition[Transpose[{ts, #}], 2, 1] & /@ {xs, ys};
(Total[fourierCoefficientLineSegment[Cos, 0, #] & /@ #]/2 +
Sum[
Total[fourierCoefficientLineSegment[Cos, k, #] & /@ #] Cos[
k t], {k, n}] +
Sum[
Total[fourierCoefficientLineSegment[Sin, k, #] & /@ #] Sin[
k t], {k, n}])/Pi & /@ {Xs, Ys}]

• I didn't see an In[9] on that page? It skips from In[8] to In[11] I believe?
– user1722
Commented Oct 30, 2014 at 14:50
• @barrycarter, Download the .CDF file and open it, you'll see In[9] &In[10]. Commented Oct 30, 2014 at 14:52