Usually in ZEMAX we work with parametric models of surfaces and objects, so that (for example) a radius of curvature defines the surface of a lens at any point on its surface. This is both compact and easy to optimize.

A 'freeform' surface by distinction is often defined by points on the surface. A curve-fitting routine fits these points to some function, and then the function is used to define the surface height at all points on the surface. So rather than define the coefficients of a function that yield a surface, we define the points on a surface that yield the coefficients of the polynomial.

The functions used to fit the input data are usually chosen to be smooth functions, such as cubic splines, Bezier curves, or NURBS. The Freeform-Z non-sequential object is formed by drawing a cubic spline curve through a series of points in the YZ plane, and then sweeping this curve around the Z axis to form either a solid volume or shell. Cubic splines are formed by a piece-wise concatenation of curved segments. Within the bounds of each segment, the curve is defined by a third order polynomial. The polynomial coefficients describing each segment are determined from the sag values of the defined segment boundaries. The determination of the coefficients is driven by the boundary requirements that the curve goes through the defined points, and both the first and second derivatives be continuous across segment boundaries.

For a third order spline, it is not possible to require higher order derivatives to be continuous across segment boundaries. For this reason, splines are of limited accuracy and usefulness in high precision (imaging) optical design. Note that fundamental optical properties, such as surface power, are determined  by second order derivatives, and basic aberrations such as coma and spherical are controlled by third and fourth order derivatives.

This is, of course, why high order polynomials are commonly used for aberration correction in imaging system design.

However spline shapes can be very useful in non-imaging work, as 'optical power' can be added wherever needed. In the following pages we will design a collimator optic for an Osram LED. The Freeform-z object is used, along with its associated FREZ optimization operand. The two should be used together, as the FREZ operand allows the freeform object to be controlled during optimization at any point on its surface, and not just at the points where the defining data exists.