3.2.1 Geometric interpretation
The vanishing of these angle sums has a simple geometric interpretation. The angles \((2k+1)\pi /(2N)\) for \(k = 0, \ldots , N-1\) correspond to equally-spaced points on the upper half of the unit circle (when \(N\) is even, they avoid the points \(\pm 1\)). Their symmetry about the imaginary axis ensures that horizontal projections cancel. This is precisely the discrete analog of the orthogonality integral \(\int _0^\pi \cos (m\theta ) \, d\theta = 0\) for \(m {\gt} 0\).