Axial, Tangential & Refractive Corneal Curvature Maps: True Refractive Power vs Geometric Curvature

The Axial map assumes all surface normal vectors intersect the central optical reference axis:

$$r_{\text{axial}} = \frac{y}{\sin \alpha} \implies K_{\text{axial}} = \frac{n_k - 1}{r_{\text{axial}}}$$

Because the calculation is anchored to the central axis, Axial maps heavily smooth out local contour variations. They excel at presenting an intuitive global overview of corneal power and determining gross astigmatism axis, but they significantly underestimate cone peak steepness and overestimate cone diameter in keratoconus.

The Tangential map calculates the true local instantaneous radius of curvature derived from the 1st and 2nd spatial derivatives of the surface profile ($z''(y)$), without reference to a central axis:

r_{\text{tangential}} = \frac{\left[1 + (z')^2\right]^{3/2}}{|z''|} \implies \text{Pinpoints True Cone Apex & Boundary}

Tangential maps reveal sharp anatomical boundaries, uncovering the exact apex of an ectatic cone, tiny post-radial keratotomy scars, or subtle decentered excimer laser ablations.

While Axial maps use paraxial Gaussian approximations ($n=1.3375$), Refractive Power maps apply rigorous Snell's law ray tracing for incoming parallel rays. They accurately depict peripheral optical aberration and true functional focal power across the entrance pupil.