# Improved Analytical Theory of Ophthalmic Lens Design
## 5 Conclusions  
This work shows that it is possible to use high-order polynomial terms to describe the optical surface of an ophthalmic lens while at the same time maintaining the second-order approximation of the incident and refraction angles typically used in the so-called third-order theory of spectacle lenses. There is previous work that used high-order aspherical terms to correct oblique aberrations in ophthalmic lenses [1,17], but the calculation has always been based on multiparametric optimization. Here, we presented analytical expressions for the surface coefficients \(c_{2}\) up to \(c_{8}\) and a general recursive relation for higher-order terms. The need for higher-order polynomial terms in the surface description may seem to contradict the second-order approximation in angles shown in Equation (9), but in fact, both are independent: high-order asphericity may bring richness to the central region of the lens where the incident angles are still small, as demonstrated by the numerical results and corroborated by the analytical approach.  
We verified that this formulation provides a more accurate estimation of the oblique powers than previous third-order approximations because they closely match the equivalent ray-traced solutions.  
A generalization of the three classical solutions (point-focal lens, Percival lens, and lens with zero tangential error) was proposed, depending on a parameter \(u\). We have shown that these generalized solutions are well-defined for all base curves and powers for a wide range of values of the parameter \(u\). Moreover, we have shown that the oblique powers of the solutions obtained with this parametrization tend to be parabolic and smooth. Moreover, it seems that the surfaces that minimize a broad set of nonlinear merit functions  
Figure 9: Tangential (red line) and sagittal (blue line) powers up to the term \(c_{8}\) for a lens with \(P=-4\) D, \(B=0.5\) D, \(n=1.5\), and \(L_{2}^{\prime}=37\) D. The continuous lines represent the analytical solution obtained with the approximation described in the text for \(u=0.316\), while the dashed lines correspond to the oblique power of the lens computed by multiparametric minimization of the merit function (39) with weights \((1,1,0,0)\).  
belong to the set of functions that meets the general compromise (22). This hypothesis allows the analytical solving of minimization problems relevant in ophthalmic lenses.  
Though modern ophthalmic lens design will rely on numerical computation to optimize general lenses with tilts, prescription astigmatism, even progressive lenses, the tools presented in this paper offer new insight on the behavior of the oblique powers and their relationship with the surface geometry and allow fast testing of new design hypotheses.  
Conceptualization, E.P.; methodology, E.P., J.A.G.-P. and J.A.; software, E.P.; validation, E.P., J.A.G.-P. and J.A.; formal analysis, E.P.; investigation, E.P., J.A.G.-P. and J.A.; resources, J.A.G.-P. and J.A.; data curation, J.A.G.-P. and J.A.; writing--original draft preparation, E.P.; writing--review and editing, J.A.G.-P. and J.A.; visualization, E.P., J.A.G.-P. and J.A.; supervision, J.A.G.-P. and J.A.; project administration, E.P., J.A.G.-P. and J.A.; funding acquisition, J.A.G.-P. and J.A. All authors have read and agreed to the published version of the manuscript.  
This work was partially financed by the European Fund for Regional Development (EFRD-FEDER) and the Spanish Agencia Estatal de Investigacion, through the grant DPI2016-75272-R.  
Data sharing not applicable.  
The authors declare no conflict of interest.