1. Introduction

Characterization of part shape is of fundamental importance. Because no single framework is ideally matched to the diversity of requirements for different contexts, it is helpful to have a range of options for this task. To minimize confusion, however, it is essential to have clear statement of the limitations, strengths, and goals of each option. For example, because the methods discussed here are founded upon a single-valued sag function, they are not directly applicable to hyper-hemispheres. Nevertheless, these methods apply to a wide range of surface shapes that hold technological significance. My concern is that the errors in the analysis in [1] may generate confusion where the industry needs clarity.

2. Specific examples

In the first two examples considered in [1], see Eqs. (35) and (38), a slope-orthogonal basis, so-called Q polynomials, was found to give a superior fit by orders of magnitude (see their Fig. 5). In these cases, the best-fit sphere’s curvature times the aperture radius is 0.306394. This corresponds to the sine of the angle between the transverse axis and the tangent to the best-fit sphere, say $\Theta$, of about $18°$. In place of sag, i.e. $z$-displacement, the description in terms of Q operates by reference to the displacement along the normal to the best-fit sphere (or conicoid, more generally) and this distinction becomes more significant as $\Theta$ approaches $90°$. Although it will not always be an advantage, working in terms of the normal displacement is oftentimes appropriate within the contexts of metrology and fabrication.

Two significant changes were made for the third example given by Eq. (41), which in combination with a computational error, create room for confusion. Aside from the irrelevant constant offset and sign change, there is an anisotropic factor of 2 introduced in the exponent as well as the loss of the prefactor of $1/\sqrt{2\pi }$ from Eqs. (35) and (38). That prefactor makes a significant difference because the curvature of the best-fit sphere is now 0.964400103. An incorrect curvature of 0.261082 ($\Theta \approx 15°$) is reported in [1] and this leads to invalid and misleading results in their Fig. 7. Because the ordering and membership of the terms is not explained in [1], however, it is impossible to make detailed comparisons. Nevertheless, when retaining all the terms up to order 6 as defined in Eq. (1).2) of [2], there are 14 coefficients as well as a curvature and a constant (piston) term, so perhaps between 14 and 16 terms for the purposes of Fig. 7 in [1]. The residual in this case for Q is now found to have an rms of 0.0035 and this means that the blue curve in that figure would be lowered by about an order of magnitude to join the other three curves. While it is important to prevent such errors from causing confusion, this misleading factor of ten is not the only matter of interest here.

The anistropic example creates the opportunity to make a general observation. When $\Theta <<90°$, the cosine factor that converts displacement along $z$ to displacement along the normal to the best-fit sphere has minimal impact. This means that Q then always leaves a similar residual as for an equal number of Zernike polynomials so their discrepancy in Fig. 7 led me to question those results. The correct curvature value means that $\Theta$is now $75°$, however. That is, as is evident in the isometric view presented at left in Fig. 1 below, the best-fit sphere is approaching a hemisphere and the cosine factor now plays a significant role. It turns out that at order 10 (34 to 36 terms in Fig. 7) the residual for Q has an rms of 0.0007, which is about 70 times larger than for the corresponding Zernike terms. The impact of that cosine factor can be appreciated by restoring the prefactor of $1/\sqrt{2\pi }$ mentioned above to find the surface at centre in Fig. 1 below. This case has a best-fit curvature of 0.556982999 (so $\Theta \approx 34°$) and the residual at order 10 is now $6.2\times {10}^{-6}$ which is within a factor of two of the result for the corresponding Zernike terms. That is, it is not the lack of rotational symmetry that makes this example more challenging for Q.

 

Fig. 1 An isometric view of the surface used as an example in [1] is shown at left together with its best-fit sphere. The central plot gives the same result with the original factor of 1/sqrt(2 pi) retained. A best-fit ellipsoid is shown at right with a conic constant of −0.42.

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For completeness, it is worth noting that in unusual cases such as when the best-fit sphere approaches or exceeds a hemisphere, it is possible –although discouraged more generally– to replace it with a best-fit conicoid, see Sec. 5 of Ref. 14 cited by [1]. Hopefully considerations related to fabrication or testing serve to resolve the associated degeneracy by indicating the appropriate value for the conic constant in such cases. As an example, for that original anisotropic surface, the case at right in Fig. 1 shows a best-fit ellipsoid where the conic constant is −0.42 (instead of 0 for a sphere) and the curvature is 1.14058116. “Best-fit” in this case means that the curvature is chosen to match the mean sag around the perimeter. With this ellipsoid as a base, the residual for Q at order 10 now has an rms that is about 20 times smaller than the result for the best-fit sphere discussed above. More generally, provided the maximum sag does not approach one half of the aperture diameter, the best-fit sphere avoids all degeneracy and typically offers greater readability.

3. Concluding remarks

The clear challenge in this area of research is to deliver general-purpose capabilities of practical value. I strongly agree with the authors’ introductory comments in [1] that “Ideally, the shape specification should be physically meaningful and invariant for the different stages of design, fabrication, testing or application” and that orthogonality “implies also good numerical behavior, avoids redundancy and ensures uniqueness of the representation, among other highly important properties.” While all of these properties can be achieved by using Zernike polynomials, it is then difficult to assess a shape by inspection of the associated coefficients. For example, such an option gives a cumbersome description of a spherical surface. An added goal therefore is to be able to easily assess a part’s departure from a best-fit sphere. The rate of change of that departure often dominates the difficulty of fabricating and testing an asphere. Ready access to this departure along with the criteria listed above were precisely the considerations that led to the development of gradient-orthogonal Q. In its default form with a best-fit sphere, there is no degeneracy at all so every coefficient has an independent physical meaning that delivers a unique and human-readable surface description.

In contrast, the degeneracy between the conic constant and primarily the fourth-order polynomial term within the (now disowned) Q_con characterization permitted an infinite variety of ways to describe a specific surface. That scourge is shared with the traditional surface description. For example, a sphere can be disguised unrecognizably in both these frameworks as a conic plus a polynomial and such obfuscation flies in the face of the agreed objectives stated in the previous paragraph. Once robust recurrence relations were discovered for handling the gradient-orthogonal Q basis, the original Q_con could be totally supplanted. Problematically, there is an even greater degeneracy within the framework proposed in [1]: the difference between Eq. (18) and (24) is just a constant offset of the underlying conicoid component. Normally, a constant offset to the sag corresponds to a translation of the surface; the shape is unchanged. Instead, this arbitrary offset now joins the conic constant as a second redundant parameter that also impacts all of the basis elements. It appears that the end result is a doubly bewildering diversity of options that can, for example, disguise a sphere. Given that ensuring uniqueness was among the agreed goals of this area of work, failing to achieve it appears to be a significant drawback.

It was fortuitous that Q was outstandingly effective at fitting the specific rotationally symmetric surface used for demonstration in [1]; in terms of residuals, various orthogonal fits are oftentimes closer to being on a par. Further, despite the impression generated by the concluding paragraph of [1], the corrected results for the freeform example reveal that the lack of rotational symmetry need not put Q at a disadvantage in such cases. In fact, robust algorithms for fitting with Q have been shown to allow a useful characterization (even with millions of terms) for metrology from as-built surfaces, see [2]. These algorithms have recently been implemented as a public-domain Python tool, see [3]. The end result is that this single framework provides a robust option for characterizing nominal shapes for design and production as well as a promising analytical tool for handling mid-spatial frequencies during tolerancing and part qualification. Other approaches can also play important roles, of course, and are encouraged whenever they offer a clear advantage.