Refractive errors and astigmatism: myopia, hyperopia and Sturm's conoid
Quick answer
A refractive error means light from a distant object does not focus on the retina when accommodation is relaxed. In myopia the focus falls in front of the retina and is corrected with minus (concave) lenses; in hyperopia it falls behind the retina and is corrected with plus (convex) lenses. In astigmatism the eye has different power in two meridians, so a point of light forms two focal lines separated by the interval of Sturm (Sturm's conoid); a cylindrical lens at the right axis brings both lines together on the retina.
Open the simulatorEmmetropia, myopia, hyperopia and astigmatism
| Where distant light focuses | Cause | Correction | |
|---|---|---|---|
| Emmetropia | On the retina | Optical power matches eye length | None |
| Myopia (short-sightedness) | In front of the retina | Eye too long (axial) or optics too strong (refractive) | Minus (concave) lens |
| Hyperopia (long-sightedness) | Behind the retina | Eye too short (axial) or optics too weak (refractive) | Plus (convex) lens |
| Astigmatism | Two focal lines at different distances | Cornea or lens curved differently in two meridians | Cylindrical (toric) lens |
As a rule of thumb, each millimeter of change in axial length changes the refraction by about 2.5 to 3 dioptres.
Astigmatism and Sturm's conoid
- A regularly astigmatic eye has two principal meridians, of greatest and least power, 90 degrees apart.
- Each meridian focuses light into a line, so a point of light forms two focal lines separated by the interval of Sturm. The whole three-dimensional bundle of rays is Sturm's conoid.
- Between the two lines lies the circle of least confusion, where the blur patch is smallest. Correcting the spherical equivalent places it on the retina.
- With-the-rule astigmatism: the vertical meridian is steeper; it is corrected with minus cylinder near axis 180 or plus cylinder near axis 90.
- Against-the-rule astigmatism: the horizontal meridian is steeper; it is corrected with minus cylinder near axis 90.
- Oblique astigmatism: the principal meridians lie roughly between 30° and 60°, or 120° and 150°.
Classifying astigmatism by where the focal lines fall
| Type | Position of the two focal lines |
|---|---|
| Simple myopic | One on the retina, one in front of it |
| Simple hyperopic | One on the retina, one behind it |
| Compound myopic | Both in front of the retina |
| Compound hyperopic | Both behind the retina |
| Mixed | One in front of the retina and one behind it |
Reading a spherocylindrical prescription
A prescription is written sphere / cylinder × axis, for example −2.00 / −1.00 × 180. The sphere is the power along the axis meridian; the sphere plus the cylinder is the power in the meridian 90 degrees away. The axis (1 to 180 degrees) gives the orientation of the cylinder axis, which itself has no cylinder power.
To transpose between minus and plus cylinder: add the cylinder to the sphere, change the sign of the cylinder, and rotate the axis by 90 degrees. −2.00 / −1.00 × 180 becomes −3.00 / +1.00 × 090.
The spherical equivalent is the sphere plus half the cylinder — here −2.50 D.
Vertex distance
A lens acts where it sits. The power needed at the cornea (for a contact lens) is Fc = F ÷ (1 − d × F), where F is the spectacle power in dioptres and d is the vertex distance in meters. A −10.00 D spectacle lens at 12 mm corresponds to about −8.93 D at the cornea, and a +10.00 D lens to about +11.36 D. The difference becomes clinically important above about ±4.00 D.
From refractive error to subjective refraction
- Start from an objective result: retinoscopy or autorefraction.
- Find the most plus (least minus) lens that gives the best visual acuity, to avoid over-minusing.
- Refine astigmatism with the Jackson cross-cylinder: axis first, then power, keeping the spherical equivalent constant.
- Check the sphere with the duochrome test: letters clearer on red → add minus; clearer on green → add plus.
- Balance the two eyes, then check near vision and add a reading addition if needed.
Practice it on the virtual patient
- Open the refraction simulator and switch between emmetropia, myopia and hyperopia.
- Change the cause between axial (eye length) and refractive (corneal curvature) and watch where the focus falls.
- Add astigmatism and view both meridians to see the two focal lines of Sturm's conoid.
- Apply sphere, cylinder and axis trial lenses until the focal lines collapse onto the retina, then change the vertex distance and watch the effective power change.
Frequently asked questions
Is myopia corrected with plus or minus lenses?
Myopia is corrected with minus (concave, diverging) lenses, which move the focus back onto the retina. Hyperopia is corrected with plus (convex, converging) lenses.
What is the circle of least confusion?
The circle of least confusion is the point between the two focal lines of an astigmatic eye where the blur patch is smallest. Correcting the spherical equivalent places it on the retina.
How do you calculate the spherical equivalent?
Add half of the cylinder to the sphere. For −1.00 / −2.00 × 090 the spherical equivalent is −1.00 + (−1.00) = −2.00 D.
What is with-the-rule astigmatism?
In with-the-rule astigmatism the vertical corneal meridian is steeper than the horizontal one. It is common in younger people and is corrected with minus cylinder near axis 180 or plus cylinder near axis 90.
Related guides
- Streak retinoscopy: with motion, against motion and neutralization Read with and against motion, neutralize both meridians and correct for working distance.
- Extraocular muscles: anatomy, actions and nerve supply Origins, insertions, actions and nerve supply of the six eye muscles — and why actions change with gaze.
- Visual pathway lesions and the visual field defects they cause Localize a lesion from its field defect, from the optic nerve to the occipital cortex.