Cubic Spline
The cubic spline surface is described by sag values which are the distances between the vertex tangent plane and the surface. Spline surfaces are used to describe unusual correctors, headlamps, and other non-standard optical surfaces, but rarely for imaging applications because of the fundamental properties of splines. See "Comments about spline surfaces" below for more discussion.
The Cubic Spline surface uses eight values to represent the sag at one-eighth, two-eighths, and so on to eight- eighths of the clear semi-diameter or semi-diameter of that surface. Cubic spline surfaces are rotationally symmetric. All eight points must be defined. A subset cannot be used, although the clear semi-diameter or semi-diameter may be defined to exceed the useful aperture of the surface. This is often required because of the steep curvatures occasionally introduced by spline fitting. If eight points provides an overly coarse sampling, see "Extended Cubic Spline". For a more general non-rotationally symmetric surface, see "Grid Sag".
Comments about spline surfaces
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, such as third, to be continuous across segment boundaries. For this reason, splines are of limited accuracy and usefulness in high precision optical design.
A common characteristic of tracing rays through spline surfaces is rough or noisy looking ray data, with discontinuities in some results. These ray trace discontinuities are a fundamental limitation of splines, and they are not due to a flaw in OpticStudio, or a lack of numerical precision.
Higher order splines of course exist, and one way to eliminate the discontinuities is to use a higher order spline and fewer segments. In the limit, this is essentially the same as using a single high order polynomial for the whole surface, see for example the "Even Asphere". This is why high order polynomials, and not splines or NURBS, dominate in precision optical design; they are continuously smooth and differentiable to all orders.
For an excellent discussion of spline theory, properties and algorithms, see Numerical Recipes in C, by Press et al., Cambridge University Press.
PARAMETER DEFINITIONS FOR CUBIC SPLINE SURFACES
| Parameter # | Definition |
| 1 | Sag at 1/8 |
| 2 | Sag at 2/8 |
| 3 | Sag at 3/8 |
| 4 | Sag at 4/8 |
| 5 | Sag at 5/8 |
| 6 | Sag at 6/8 |
| 7 | Sag at 7/8 |
| 8 | Sag at 8/8 |
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