`transform(v, ε, δ, ϕ, ω)`

Pupil transform function; computes a new set of Zernike wavefront error expansion coefficients under a given set of transformation factors and plots the result.

Available transformations are scaling, translation, & rotation for circular and elliptical exit pupils. These are essentially coordinate transformations in the pupil plane over the wavefront map.

v::Vector{Float64}: vector of full Zernike expansion coefficients ordered in accordance with the ANSI / OSA single index standard. This is the v vector returned by wavefront(ρ, θ, OPD, n_max);
ε::Float64: scaling factor {0 ≤ ε ≤ 1};
δ::ComplexF64: translational complex coordinates (displacement of the pupil center in the complex plane);
ϕ::Float64: rotation of the pupil in radians (mod 2π), defined positive counter-clockwise from the horizontal x-axis;
ω::NTuple{2, Float64}: elliptical pupil transform parameters; 2-tuple where ω[1] is the ratio of the minor radius to the major radius of the ellipse and ω[2] is the angle defined positive counter-clockwise from the horizontal coordinate axis of the exit pupil to the minor axis of the ellipse.

The order the transformations are applied is: scaling –> translation –> rotation –> elliptical transform.

The translation, rotation, and elliptical arguments are optional.

ε = r₂/r₁ where r₂ is the new smaller radius, r₁ the original

In particular the radial variable corresponding to the rescaled exit pupil is normalized such that: ρ = r/r₂; {0 ≤ ρ ≤ 1} r: radial pupil position, r₂: max. radius ΔW₂(ρ₂, θ) = ΔW₁(ερ₂, θ)

For translation the shift must be within the bounds of the scaling applied such that: 0.0 ≤ ε + |δ| ≤ 1.0.

For elliptical pupils (usually the result of measuring the wavefront off-axis), the major radius is defined such that it equals the radius of the circle and so ω[1] is the fraction of the circular pupil covered by the minor radius (this is approximated well by a cosine projection factor for angles up to 40 degrees); ω[2] is then the direction of the stretching applied under transformation in converting the ellipse to a circle before fitting the expansion coefficients.

The transformed expansion coefficients are computed using a fast and accurate algorithm suitable for high orders; it is based on a formulation presented by Lundström & Unsbo (2007).