Fresnel Integrals and Cornu Spiral

The Fresnel integrals, denoted by C(x) and S(x), are special functions named after Augustin-Jean Fresnel. They arose in the description of near-field Fresnel diffraction phenomena and are fundamental in optics. When plotted parametrically as (C(x), S(x)), they trace out the Cornu spiral (also known as the Euler spiral or clothoid).

Experiment

Explore the path of the Cornu spiral by controlling the upper bound of the integration, x.

1. Open the Fresnel Integrals and Cornu Spiral module.
2. Select a value from the "Probe x" dropdown, for example 0.5 or 2.
3. Observe the calculated values for C(x) and S(x) along with the derivatives C'(x) and S'(x).
4. Watch the parametric plot on the right update to highlight the portion of the Cornu spiral from 0 to the chosen x, complete with the tangent vector indicating the instantaneous direction.