A cylindrical lens has power in one meridian only
A sphero-cylindrical (toric) lens has different powers in two principal meridians at right angles. A toric surface is curved like the side of a rugby ball or a ring doughnut

## Cylindrical lenses

The axis is the meridian with no power. All the power is in the meridian at 90° to the axis
A point object forms a focal line parallel to the axis
Prescriptions use standard (TABO) axis notation: 0–180°, measured anticlockwise from the patient's left as the examiner faces them, the same for both eyes

## Power crosses and notation

Worked example: −2.00 / −1.50 × 180 has −2.00D in the 180° meridian and −3.50D in the 90° meridian
Spherical equivalent = sphere + half the cylinder

## Simple transposition

Worked example: +2.00 / −1.00 × 90 = +1.00 / +1.00 × 180
Check with power crosses: both have +2.00D in the 90° meridian and +1.00D in the 180° meridian

## Toric transposition

Used to specify a lens with one spherical surface and one toric surface of a chosen base curve (the flatter meridian of the toric surface)

Sphere curve: +2.00 − (−6.00) = +8.00D
Base curve: −6.00D, axis 180
Cross curve: −6.00 + (−1.00) = −7.00D, axis 90
Written +8.00 DS / −6.00 DC × 180 / −7.00 DC × 90
Check: 90° meridian 8 − 6 = +2.00D; 180° meridian 8 − 7 = +1.00D, as in the original prescription

## The conoid of Sturm

A sphero-cylinder does not form a point image. The pencil of light (the conoid of Sturm) has

A first focal line from the stronger meridian and a second focal line from the weaker meridian
Each focal line is at right angles to the meridian that formed it
The interval of Sturm between them, which grows with the size of the cylinder

The circle of least confusion is the smallest, circular, cross-section of the beam

It lies dioptrically midway between the focal lines, at the focus of the spherical equivalent
This is why placing the circle of least confusion on the retina (prescribing the spherical equivalent) gives the best uncorrected vision for a given amount of astigmatism

Types of astigmatic eye (simple, compound, mixed) are named by where the focal lines fall relative to the retina (see astigmatism)

## The cross-cylinder

A cross-cylinder has equal and opposite powers in its two principal meridians

±0.25 = +0.25 DS / −0.50 DC. ±0.50 = +0.50 DS / −1.00 DC
Its spherical equivalent is zero, so it does not move the circle of least confusion. It only changes the interval of Sturm

The handle is at 45° to the principal meridians. Twirling the handle swaps the plus and minus meridians
This is the basis of the Jackson cross-cylinder technique for refining cylinder axis and power (see subjective refraction)

Obliquely crossed cylinders combine into a new sphero-cylinder at an intermediate axis. If a correcting cylinder is rotated off axis by θ, a residual cylinder of 2C sin θ appears
For a toric IOL, each degree of misrotation loses about 3.5% of its effect, and at 30° the residual astigmatism equals the original (2 sin 30° = 1)
Hand identification: rotating a cylindrical lens in front of a cross makes the lines appear to scissor; a spherical lens does not do this