# Improved Analytical Theory of Ophthalmic Lens Design
## 3 Results  
We continue to use \(x\) as the transverse (radial) coordinate, and we give the name \(z(x)\) to the sag of the back surface. For surfaces of revolution with low curvature, the local tangential and sagittal radii can be obtained from the simple relations \(R_{T}(x)=1/z^{\prime\prime}(x)\) and \(R_{S}=x/z^{\prime}(x)\)[22]. Henceforth, the corresponding refractive powers are  
\[F_{2T}(x)=-(n-1)z^{\prime\prime}(x), \tag{20}\]  
\[F_{2S}(x)=-(n-1)\frac{z^{\prime}(x)}{x}. \tag{21}\]  
In principle, given any aspherical back surface with revolution symmetry, \(z(x)\), we can compute its local refractive powers and then substitute in Equations (18) and (19) to obtain an approximation of the oblique powers of the lens. Then we would seek the conditions for which \(E_{T}(x)=E_{S}(x)=0\), that is, there are no oblique aberrations; the lens wearer perceives the same power for any oblique sight direction. In general, this cannot be achieved, and some classical compromises have been proposed and thoroughly studied [6]:  
* Point focal lens. Imposes the oblique astigmatism to be zero for all \(x\), \(E_{A}(x)=0\). * Percival lens. Imposes the mean power to be equal to the central power for all \(x\): \(E_{P}(x)=0\). * Zero-tangential-error lens. Imposes the tangential power to be equal to the central power for all \(x\): \(E_{T}(x)=0\). Instead of investigating each of these compromises separately, we may impose that a weighted average of the oblique (tangential and sagittal) powers must equal the central power,  
\[vF_{T}+uF_{S}=(u+v)P, \tag{22}\]  
where the weights \(u\) and \(v\) can take negative values. Any two pairs \((u,v)\) and \((u^{\prime},v^{\prime})\) lead to the same solution if they are linearly dependent, so condition (22) is fully determined by a single parameter, and a relation between \(u\) and \(v\) must be chosen. One such relation that will allow the generation of all interesting balances between tangential, sagittal, and central power while keeping a finite range of the parameters \(u\) and \(v\) is \[u^{2}+v^{2}=1, \tag{23}\]  
which normalizes the 2-tupla \((u,v)\). Condition (22), along with normalization (23), can also be written as a balanced elimination of both the tangential and sagittal errors,  
\[E_{T}\cos\alpha+E_{S}\sin\alpha=0, \tag{24}\]  
where \(v=\cos\alpha\) and \(\alpha\) runs from \(-\pi\) to \(\pi\) to generate all the interesting cases. The solutions for condition (22) includes the classical ones:  
* Point focal lens, \(u=-v=-\sqrt{2}/2\), and \(\alpha=-\pi/4\). * Percival lens, \(u=v=\sqrt{2}/2\), and \(\alpha=\pi/4\). * Lens with zero tangential error, \(u=0\), \(v=1\), and \(\alpha=0\). Although not considered a classical solution, the lens with zero sagittal error can also be obtained when \(u=1\) and \(v=0\) (\(\alpha=\pi/2\)). In the traditional third-order theory, suitable functions describing \(F_{2S}(x)\) and \(F_{2T}(x)\) are substituted in Equations (18) and (19); the resulting expressions are expanded in powers of \(x\), and the results are simplified by retaining up to second-order terms in \(x\). This procedure leads to the generic expressions \(E=x^{2}f_{sph}(B,P,Q_{1},Q_{2},n,L)\) we previously introduced. With this approach, condition (22) only adds marginal generality, as the dependence of _any_ combination of oblique error will always be quadratic with \(x\), and will vanish for all \(x\) if \(f_{sph}\) can be zeroed. We show next that analytical results around condition (22) can be easily obtained without the need to neglect higher powers of \(x\) in Equations (18) and (19). With the outgoing results we are able to explore how (22) is satisfied as a function of the transverse coordinate, and that brings a richer understanding of the full-field lens optimization problem. Finally, we will later see that among the family of oblique powers that satisfy condition (22), we can find the solution to the minimization of more interesting merit functions--for example, those related to optical blur averaged across the field of view.