Math model + optical elements

Core types

PolyTerm<T, N>

A single monomial:

• coefficient T coefficient
• exponent vector echar exponents[N]

TruncPoly<T, N>

A truncated multivariate polynomial as a list of terms.

Main operations:

• print()
• evaluate(x)
• differentiate(i)
• truncate(degree)
• symbolic substitution/composition (via plug_into)

Transform<T, N>

A vector of N polynomials: a multivariate map.

• A Transform4f maps 4D inputs to 4D outputs: out = T(in).

Composition is function composition:

• T3 = T2.plug_into(T1) corresponds to T3(x) = T2(T1(x)).

Optical elements

Optical elements are implemented as functions that return a Transform4f for a particular action, e.g.:

• propagate_5(d)
• refract_spherical_5(R, n1, n2, degree)
• reflect_spherical_5(R, degree)
• refract_cylindrical_x_5(R, n1, n2) / refract_cylindrical_y_5(...)

Typical pattern:

1. Create a TwoPlane5 parametrization system
2. Append element transforms via plug_into
3. Evaluate the ray transform many times (per sample)

Performance and numerical considerations

• Term growth can be large for higher degrees; use truncation (% degree) aggressively.
• Evaluation uses repeated integer powers; for very hot loops, consider caching or specialized evaluation strategies.
• Exponents are stored as unsigned char (echar); exponent overflow is possible if you exceed intended degrees.
• Composition (plug_into) is the most expensive symbolic operation; it is intended for building systems once, then evaluating many times.