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I'm reading "On Lattices, Learning with Errors, Random Linear Codes, and Cryptography" by O. Regev.

I'm having trouble with understanding graphs in Figure 1.

By the definition of $\overline{\psi}_{\alpha}$, its support is $[0,1)$.

However, graphs in Figure 1 use support as $[-1,1] \times [-1,1]$.

Can anyone explain me about this?

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The points of $\mathbb Z/q\mathbb Z$ are mapped to $\mathbb R/\mathbb Z$ and plotted in a circle. The plot shows \begin{align*} x &= \sin (2\pi i/q), \\ y &= \cos (2\pi i/q), \\ p &= \Psi_\alpha(i/q). \end{align*} for each $i$ in the least nonnegative residues modulo $q$.

The following gnuplot script gives a similar graph, for $q = 31$:

set parametric
set samples 31
psi(alpha, r) = exp(-pi*(r/alpha)**2) + exp(-pi*((r - 1)/alpha)**2)
splot [i=0:30] sin(2*pi*i/31), cos(2*pi*i/31), psi(0.1, i/31) with points
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