Torus Curves With Vanishing Curvature

Let T be the standard torus of revolution in R^3 with radii b and 1, 0<b<1. Let \alpha be a (p,q) torus curve on T. We show that there are points of zero curvature on \alpha for only one value of the variable radius of T, b=p^2/(p^2+q^2). The curve \alpha has non-vanishing curvature for all other values of b. Moreover, for this value of b, there are exactly q points of zero curvature on \alpha.