= Solution
\b[<Astigmatism> is a <refractive error> in which the <eye>'s focusing power differs with direction. Detect its magnitude and axis by comparing directional focus; correct regular <astigmatism> with a suitably oriented <cylindrical lens>, adding a <spherical lens> when necessary.]
For regular <astigmatism>, the two principal <meridians of the eye> are perpendicular and have different <optical powers>, commonly because the <cornea> has unequal curvatures. The <crystalline lens> can also contribute. A point object consequently does not form one sharp point image: the idealised <paraxial optics> gives two focal lines at different distances, perpendicular to one another, with an intervening region of blur. An ordinary <spherical lens> changes the focus in both meridians together and cannot by itself remove their difference. Irregular <astigmatism>, for example with a scarred <cornea>, lacks this simple pair of uniform principal powers.<Physiology of Astigmatism by the American Academy of Ophthalmology>[Introduction and sections on regular and irregular astigmatism]{ref}
Several complementary observations reveal it:
* With a distant point or suitable letter chart, an <eye> with <astigmatism> gives blur that cannot be completely removed with different <spherical lenses>. In a chart of equally dark radial lines, some directions can appear clearer or darker than others; unequal appearance suggests unequal directional focus, but does not alone supply a full prescription.
* A narrow <stenopaic slit> admits rays through a restricted <meridian of the eye>. Rotate the slit and determine the <spherical lens> giving best distant vision in each principal direction. A difference between the best powers demonstrates regular <astigmatism>. This is an especially natural optical explanation for an 1867 answer: it isolates directions rather than simply asking whether vision improves through a small round aperture.
* Modern objective <retinoscopy> observes the moving light reflex from the <retina> while trial lenses neutralise it in each meridian. <Keratometry> measures reflected images from the anterior <cornea>; <corneal topography> maps a larger area and helps detect irregularity. These measure different things: <keratometry> does not by itself include all contributions from the <crystalline lens>. Subjective <refraction examination> refines the prescription with trial lenses, often using a <crossed-cylinder test>.<Assessment of visual function by the College of Optometrists (2025)>[Unit of Competence 7, objective retinoscopy and subjective refraction, including power and cylinder axis]{ref}
The prescription must specify both the difference in <optical power> and its orientation. If the required correcting powers in the two principal directions are $R_1$ and $R_2$, choose the cylinder axis along direction 1. In the ideal <thin lens> model:
$$
\boxed{S=R_1,\qquad C=R_2-R_1,\qquad R(\theta)=S+C\sin^2(\theta-\alpha)}.
$$
Here $S$ is the spherical correcting power, $C$ the cylindrical correcting power and $\alpha$ the cylinder axis, all powers measured in <dioptres>. A <cylindrical lens> contributes no extra power along its axis and its full power perpendicular to that axis. For example, required corrections of $-1$ and $-3$ <dioptres> are supplied by a sphere of $-1$ <dioptre> and a cylinder of $-2$ <dioptres>, with its axis in the direction requiring the smaller negative correction. These are correcting powers, not the uncorrected <eye>'s intrinsic powers; confusing the two reverses the intended correction.
Spectacles with this sphere-and-cylinder combination are the central historical remedy. Present options also include <toric contact lenses>, appropriate rigid <contact lenses> for some irregular cases, and selected <refractive surgery>. The suitability of these depends on the cause and condition of the <cornea>; a spectacle cylinder cannot generally cancel all irregularities of an irregular <cornea>.<Astigmatism by the National Eye Institute>[Treatment options: spectacles, contact lenses and surgery; modern additions are not attributed to 1867]{ref}
Back to article page
