Tscherning's Ellipses: Wollaston vs Ostwalt Branches & The Neutralization of Marginal Oblique Astigmatism

When an eye fixates through an off-axis angle $\theta$, paraxial symmetry is destroyed. The single spherical wavefront splits into orthogonal tangential ($F_t'$) and sagittal ($F_s'$) refracted wave-fronts, creating Marginal Oblique Astigmatism ($MOA$):

$$MOA = F_t' - F_s' \propto \theta^2 \cdot \left[ A \cdot F_1^2 + B \cdot F_1 F_2 + C \cdot F_2^2 + \dots \right]$$

Setting $MOA = 0$ yields a quadratic equation in base curve $F_1$, plotting an ellipse on Cartesian axes of Base Curve vs Back Vertex Power.

Because the quadratic equation has two real roots, Tscherning's ellipse features two distinct solutions for every power:

1. Wollaston Branch (Steep Meniscus): Employs extremely steep base curves ($F_1 > +12.00\text{ to }+18.00\ \text{D}$). Completely eliminates both oblique astigmatism and distortion, but is cosmetically hideous, bulges forward, and hits the eyelashes.
2. Ostwalt Branch (Flatter Meniscus): Employs gentle base curves ($F_1 \approx +3.00\text{ to }+8.00\ \text{D}$). Flattens the cosmetic profile while neutralizing oblique astigmatism within clinical tolerances ($<0.12\ \text{D}$). All modern 'best form' spherical lenses follow the Ostwalt branch.

For an ophthalmic polymer of index $n = 1.50$, the quadratic discriminant ($\Delta = b^2 - 4ac$) remains positive only between $+7.25\ \text{D}$ and $-23.00\ \text{D}$. Above $+7.25\ \text{D}$, no spherical lens can eliminate oblique astigmatism! This physical boundary forced the invention of aspheric surfacing to correct high hyperopia and aphakia.