Is this Cornu spiral positively oriented or not?.

Solution 1:

By accepted mathematical convention directions of x and y axes are taken as positive to the right and positive upwards from a reference origin respectively.

Slope $ \phi = \tan^{-1} dy/dx $

So accordingly such a convention is made to apply to next derivative as well. Curvature is defined as the rate of slope / rotation angle change or rate of change of tangent direction to a curve with respect to arc. It means that the counterclockwise (CCW) rotation adds to curvature and clockwise rotation decreases curvature.

In Cornu's spiral $ \kappa = d\phi/ds = a s, \phi = a s^2 /2 $ where $a$ is a constant. At origin curvature is zero.

If $a>0$ and has zero slope tangent to x-axis it curls up CCW in quadrant 1 with increasing curvature. And if tangent to y-axis it curls up CCW in quadrant 2. This is as you have sketched.

Similarly if $a<0, \, \phi = a s^2 /2 $ and has zero slope tangent to x-axis it curls down CW into quadrant 4 with high negative curvature. And if tangent to y-axis it curls to right CW in quadrant 1. Like the Cornu/Euler spiral here.