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Lens Transpose (Rx) & Spherical Equivalent Chart, Formula

All About Spherical Equivalent (SE) Chart & Rx Transposition or Lens Transposition (LT): Introduction, Importance, Types, Examples, Chart, Formula and Calculator.

How do you convert a plus cylinder to a minus cylinder and vice versa using lens transposition (transpose rx)? How to determine Spherical Equivalent Power? Are you looking for the answers to these questions? You are at the right place. Today, we will be discussing lens transposition and spherical equivalent in detail.

First, you will get information about lens transposition (transpose rx), types, and uses, and then we will be talking about spherical equivalent, and its uses and importance. So, without further ado let’s dive into the topic.

Spherical Equivalent and Lens Transposition (Transpose Rx)

Definition of Lens Transposition (or Rx Transpose)

Some optometrists and ophthalmologists prefer to prescribe eyeglasses in what called the plus cylinder, while others do so in the minus cylinder.

Lens transposition is simply converting the eyeglass prescription (Rx) from one form to another so as to maintain the same meridian values in both forms.

Plus cylinder prescriptions were popular in older days because in those days the instruments used to measure and cut lenses were only able to do so in positive increments.

Nowadays, the majority of labs throughout the world cut lenses in the negative (-) form, and require that any positive (+) cylinder prescriptions be converted into the negative (-) equivalent.

Be sure that this conversion does not alter the type or quality of the vision correction that the eye doctor prescribed to any patient. The conversion will affect (change) the spherical (SPH), cylindrical (CYL), and axis (AX)  parameters in the prescription, but will result in the exact same vision correction the patient was fit for by the eye doctor.

Transposition of lens simply means to rewrite the expressions of its power without actually changing them.

In most cases, eyeglass prescription is changed from plus (+) form to minus (-) form in lens transposition.

Rx transpose can be performed by first drawing an optical cross but in the majority of cases, the “three steps” technique (we will discuss below) is used.

Why are there minus and plus cylinders?

These days, plus cylinder lenses exist only in trial lenses, and phoropters (very few manufacturers still manufacture eyeglasses in the plus cylinder). It is easier to teach and learn retinoscopy in the plus cylinder. So, phoropters with plus cylinders are so popular. Ophthalmologists usually learn retinoscopy and refraction in the plus cylinder while optometrists typically use the minus cylinder.

Importance of lens transposition (transpose rx)

Optical and manufacturing laboratories tend to work in the minus cylinder as this is the way the lenses are ground for the final product. The lens becomes thinner and lighter when manufactured in a minus cylinder. Therefore, it is important for not only laboratories to know but vital for an optician, optometrist as well as and ophthalmologist. All the eyeglass prescriptions should be mentioned in minus cylinder form or it should be transposed into minus form if it is in plus cylinder format.

Lens transposition also helps opticians to get an equivalent (alternate form) prescription for a given prescription.

Types of Lens transposition

There are two types of lens transposition.

  • Simple transposition
  • Toric Transposition

Both types of lens transposition are described below.

Simple Rx Transposition of Lens

Simple transposition of lens applies to convert the lens into different forms. The simple transposition is used in the dispensing laboratory. There are three simple steps in eyeglass prescription lens transposition in simple method:

  1. Sum: In the first step, the cylindrical power and the spherical power of the lens are algebraically added to get the new spherical power.
  2. Sign: In the second step, the sign of the cylindrical power is changed (plus to minus or vice versa) by retaining the power of the cylinder.
  3. Axis: In the third step, the axis of the cylinder is rotated 90°. It is achieved by adding 90° if the axis of the cylinder is less than or equal to 90 degrees, and subtracting 90° if the axis of cylindrical power is more than 90 degrees.

(NOTE: Axis transposition can never be greater than 180)

Simple rx transposition examples

Example 1

Transpose a lens power of +1.00/-2.00 x 180 (in minus-cylinder form) to plus-cylinder form.

Step 1. Add the sphere and cylinder values to obtain the new sphere power: +1.00 – 2.00 = -1.00

Step 2. Change the sign of the cylinder axis: – 2.00 becomes +2.00.

Step 3. Change the cylinder axis by 90 degrees: 180 – 90 = 90 degrees

Combining the above three steps provides the lens power in the plus-cylinder form: -1.00/+2.00 x 090.

Example 2

Transpose a lens power of -1.00/+2.00 x 100 (in the plus-cylinder form) to minus-cylinder form.

Step 1. Add the sphere and cylinder values to obtain the new sphere power: -1.00 + 2.00 = +1.00

Step 2. Change the sign of the cylinder axis: + 2.00 becomes -2.00.

Step 3. Change the cylinder axis by 90 degrees: 100 – 90 = 10 degrees

Combining the above three steps provides the lens power in the minus-cylinder form: +1.00/-2.00 x 010.

Importance of Simple lens transposition

Most commonly the plus cylinder form is changed into minus cylinder form:

  • To reduce the unwanted central lens thickness
  • To make the lens lightweight
  • To minimize the peripheral aberration
  • For easy adaptation

Toric lens transposition

The main objective of the toric transposition of the lens is to select the proper tools in the cylinder lens surfacing. The toric transposition is used in the production laboratory. It helps to bring the lens into proper curvature and thickness. In the calculation of surfacing tools, different steps are followed.

Cross cylinder lens transposition calculator or rx transpose calculator

There are few online eyeglass transposition calculators. Two popular online calculators are:

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Spherical Equivalent Definition

The spherical equivalent is defined as a spherical power whose focal point coincides with the circle of least confusion of a spherocylindrical lens. It is calculated from the toric eyeglass lens prescription.

A spherical equivalent is just an estimate of the eye’s refractive error which is calculated by merging the spherical (myopia or hyperopia) and cylindrical (astigmatism) components of the refractive error. Usually, the spherical equivalent is not accurate enough to provide the sharpest vision.

Spherical equivalent lens places any astigmatic eye in a condition of meridional balance i.e. to change astigmatism into equally-mixed astigmatism and produce retinal images made up of disuse circles. In short, the spherical equivalent is the average of the spherocylindrical prescription.

Uses and Importance of spherical equivalent

The spherical equivalent is the appropriate method in determining the correcting lens in astigmatic eyes under the following circumstances:

  • In contact lens fitting

If a toric soft contact lens user wants a contact lens immediately for an urgent meeting but she couldn’t find the lens at her place then the optometrist can prescribe her a spherical equivalent number with slightly compromised vision.

  • To choose the intraocular lens power
  • To determine if the refractive outcome after cataract surgery was on target
  • To calculate a trial lens for visual field (automated and manual) testing and electrophysiological (VEP and ERG) testing
  • To reduce astigmatism in an eyeglass and contact lens prescription for those who have trouble adapting
  • To compare overall changes in prescriptions

Spherical Equivalent Formula

The only formula for calculating spherical equivalent from the toric lens prescription is to algebraically add ½ of the cylindrical power to the spherical power and deleting the cylinder power and the axis from the equation. In other words, the Spherical Equivalent Formula is:

Spherical Equivalent = Sphere + Cylinder/2

  • The axis of the cylinder is removed

The axis is removed because the spherocylindrical surface is converted into a sphere by a spherical equivalent. It means half of the surface is no longer steeper (or flatter) than the other. Therefore, the axis completely disappears.

  • The cylindrical power is divided by 2

The cylindrical power is only present at the steeper half of the lens. So, only half of the cylinder is taken for the calculation of the spherical equivalent.

If the end result doesn’t yield the multiple of 0.25 Diopters when the cylinder is divided by 2, the eye doctor will decide how to choose the closest multiple of 0.25 that will be the most appropriate for the patient based on various factors.

  • The spherical power and ½ cylindrical power are combined

The final step is to add together the spherical and the ½ cylindrical powers. Be cautious to account for the plus (+) or (-) sign.

Examples of Spherical Equivalent Refraction in Myopia and Hyperopia

Look at the following examples. These calculations are useful for both eyeglass and contact lens prescriptions.

Example 1

A patient has an Eyeglass prescription of -4.00/-1.00 × 90 degree

The Spherical Equivalent (SE)= -4.00 + (-1.00/2)

= -4.00 + (-0.50)

= -4.50D

Example 2

A patient has a Toric Soft Contact Lens prescription of -2.00/-1.25 × 135 degree

SE = -2.00 + (-1.25/2)

= -4.00 + (-0.625)

= -4.625D (The Eye Doctor will choose either -4.50D or -4.75D)

Example 3

A patient has an Eyeglass prescription of +1.00/-2.00 × 180 degree

SE = +1.00 + (-2.00/2)

= +1.00 + (-1.00)

= 0.00D

Example 4

A patient has an Eyeglass prescription of +3.00/-1.00 × 145 degree

SE = +3.00 + (-1.00/2)

= +3.00 + (-0.50)

= +2.50D

Example 5

A patient has an Eyeglass prescription of -2.00/+1.00 × 90 degree

SE = -2.00 + (+1.00/2)

= -2.00 + (+0.50)

= -1.50D

Spherical Equivalent Chart

Based on the SE Formula (SE= Sphere+Cylinder/2), we have presented the Spherical Equivalent of every Sphere and Cylinder combination between Sphere values of 0.00 to -7.75D and Cylinder values of 0.00 to -3.25D.

Spherical Equivalent Chart of spherical power 0.00 to -0.75D and cylindrical power 0.00 to -3.25D


Spherical Equivalent Chart of spherical power -1.00 to -1.75D and cylindrical power 0.00 to -3.25D


Spherical Equivalent Chart of spherical power -2.00 to -2.75D and cylindrical power 0.00 to -3.25D


Spherical Equivalent Chart of spherical power -3.00 to -3.75D and cylindrical power 0.00 to -3.25D


SE Chart of spherical power -4.00 to -4.75D and cylindrical power 0.00 to -3.25D


SE Chart of spherical power -5.00 to -5.75D and cylindrical power 0.00 to -3.25D


Spherical Equivalent Chart of spherical power -6.00 to -6.75D and cylindrical power 0.00 to -3.25D


SE Chart of spherical power -7.00 to -7.75D and cylindrical power 0.00 to -3.25D


Spherical equivalent calculator or converter

There is no specific online spherical equivalent calculator or spherical equivalent converter. But you can use the simple formula and the spherical equivalent chart (mentioned above) to easily calculate the spherical equivalent amount of any toric or spherocylindrical prescription either for eyeglasses or for contact lenses.

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