Given the phase Ψ0 of a ray at the point (qx, qy) where the ray intersects a plane, the phase Ψ at a nearby point (qx+Δx, qy+Δy) is computed as follows. Without loss of generality assume that the x- and y-axes are oriented so that they are parallel to the projections of the principal curvature directions onto the cut plane. For a spherical wavefront, the orientations of these axes may be determined arbitrarily as long as they are orthogonal. Let ni,x and ni,y denote the x- and y- components, respectively, of the normalized ray direction vector. Let r1 and r2 denote the principal radii of curvature in the directions parallel to the x- and y-axes, respectively. For an incident ray with wave vector magnitude k, the phase at (qx+Δx, qy) is

Where Δr1 is the change in the principal radius of curvature of the wavefront when moving between the two points.

This equation yields an indeterminate form in the limit of infinite radii of curvature, making it unsuitable for computing interference patterns involving plane waves. The equation can be made more robust for plane waves by applying a Taylor series approximation about Δx=0,

Discarding terms of order Δx3 or higher yields

Similarly, the change in principal radius of curvature due to a small perturbation in the y-direction is

The rationale for discarding higher-order terms is that they all involve division by higher powers of r1 and r2. Therefore this series approximation is only valid when the perturbations in the x- and y-directions are much smaller than the principal radii of curvature. To assume otherwise would mean that the incident wavefronts subtend very large solid angles.

Combining the perturbations in the two principal radii of curvature yields an expression for the phase at any point (qx+Δx, qy+Δy) on the cut plane,