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Thermodynamics

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Fresnel Integrals

For any point on the Cornu’s spiral, the x and y co-ordinates are given by the two integrals known as Fresnel’s integrals. Consider the point P on the spiral. The distance of the point P along the curve from the origin is v. The tangent to the curve at P makes an angle δ with the x-axis. δ corresponds to the phase change from O to P. For a small displacement dv of the point along the curve, let the corresponding changes in the co-ordinates of the point be dx and dy.

Then, dx = dv cos δ

dy = dv sin δ

substituting the value of δ from equation (iv)

the co-ordinates x and y of the Cornu’s spiral are given by

These are called Fresnel’s integrals.