Introduction / definition

According to Efe (2012) transposition as the process of changing a lens prescription from one form to another equivalent form without  changing the value i.e. it is also  to rewrite the expression of its power without actually changing its value. It can also be define as the act of converting the prescription of an ophthalmic lens form a sphere with minus cylinder form to a sphere with plus cylinder form or vice vise. Example – 3D sphere – 2D cylinder axis 180 transpose to – 5D sphere + 2D cylinder axis 90 prescription consists of the sphere, cylinder and the axis. The sphere correct hyperopic and myopia, the cylinder corrects the amount of astigmatism and the axis shows where the stigmatism correction is located.

Types of transposition

Flat Transposition

Toric transposition

 Flat transposition: This is the process of transposition whereby the cylinder power and spherical power is separated by 90. The flat transposition is use in the dispensing laboratory.

Toric transposition: This is the process of transposition whereby the cylinder power and spherical power is not separated by 90. The toric transposition is use in the production laboratory.

Lens prescription

Lens prescription can be written in any of this form

  1. Plano cylinder
  2. Sphero cylinder
  3. Cross cylinder

Plano cylinder: This has one surface plane and the other, either + or – cylinder eg (plano x 0.50DCx180)

Sphere cylinder:  This has one surface spherical and the other cylindrical eg +0.75DS -1.00DC x 50

Cross cylinder: This has one surface a cylinder and other another cylinder eg +1.50DCx180/0.75DCx90

 Rules of transposition

  1. Transposition of sphero-cylindrical prescription from plus to minus cylinder (or vice versa)


Step 1: New spherical power: add the power of the sphere to the power of cylinder (algebraically) of a given prescription

Step 2: New cylindrical power: change the sign of the cylinder (plus or minus to the opposite sign) with same power.

Step 3: New axis: add or subtract 90 from the given axis (depending on which value is higher) i.e. if is below 90, add 90 to it but if is above 90 subtract 90. Let us now apply this rule to a specific sphero – cylindrical prescription.

  1. Given + 1.00DS – 1.00DC x 90

From the on set, it is important to note that this prescription has been written in minus cylinder from. To convert it to plus cylinder form, we apply the rules as follows.

Rule 1: (new spherical power); (+1.00) + (1.00) = plano

Rule 2: (new cylindrical power) change the cylinder sign with same value = +1.00DC

Rule 3: (new axis) add or subtract 90 (depending on which value is greater) i.e if is below 90, add 90 to it but if it is above 90 subtract 90. 90 + 90 = 180. Therefore, the transposed plus cylindrical form of a given prescription is plano + 1.00DC x 180

  1. Given -2.00DS + 1.50DC x 45

To transpose this prescription from plus to its minus cylindrical form we apply same rules as follows:

Rule 1: (new sphere power) – 2.00 + 1.50 = – 0.50DS

Rule 2: (new cylinder power) change the cylinder sight with the same power -1.50DC

Rules 3: (New axis) i.e if is below 90, add 90 to it but if is above 90 subtract 90 45 + 90 = 135

Thus, the transpose form of this prescription into its minus cylindrical form is equivalent to :- 0.50DS – 135

 Transposition of sphero-cylinders into cross cylinders

A sphero-Cylindrical prescription can be broken down into two cross-cylinder that it was originally made of this form of. This form of transposition is also guided by its set of rules


Rule 1. (1st Cross-Cylinder): Take the spherical power of a given sphero-cylinder prescription and write this as a cross cylinder with its axis 900 away from the axis of a given sphero-cylindrical prescription.

Rule 2. (2nd cross-cylinder) add the spherical and cylindrical power of a given sphero-cylindrical prescription and write this as cross cylinder with its axis same as the axis of the sphero-cylindrical prescription.

Rules 3. (both cross-cylinder) combine the two cross cylinder obtain from rule 1 and rule 2 above as the equivalent cross cylinder for the sphero-cylindrical prescription.

  The follow are some examples to illustrate the application of the rules

  • Given -4.00DS – 0.50DC x 90

Rule 1. (1st cross-cylinder) -4.00DC x 180

Rule 2. (2nd cross-cylinder) -4.50DC x 90

Rule 3. (3rd cross-cylinder) +4.00DC x 180/-4.50DC x 90

  • Given: -1.50DS + 0.75DC x 45

Rule 1. (1st cross-cylinder) -1.00DC x 135

Rule 2. (2nd cross-cylinder) -0.75DC x 45

Rule 3. (3rd cross-cylinder) +1.50DC x 135/-075DC x 45

 Transposition of 2 cross-cylinder into sphero cylinder

Rule 1: (sphere power) take either power of the 2 cross-cylinders as spherical power of the new sphero-cylinders as spherical power of the new sphero-cylinderical prescription.

Rule 2: (cylindrical power) subtract the power of the cylindrical chosen as sphere from the power of the second cross-cylinder and make this the cylinder of the new sphero-cylinder

Rule 3: (axis) take the axis of the cross-cylinder that was not chosen as sphere power as axis of the new sphero-cylinderical prescription

The following are some examples to illustrate application of the rule

  1. Given: +2.00DS x 180/+1.00DC x 90

Rule 1. (sphere power) + 2.00DS

Rule 2. (cylindrical power) +1.00 – (+2.00) = -1.00DC

Rule 3. (axis): 90

Therefore, the sphero-cylindrical prescription is: +2.00DC – 1.00DC x 90

  1. Given: -1.00DC x 135/-3.00DCx45

Rule 1. (sphere power) -3.00DS

Rule 2. (cylindrical power) -1.00 – (-3.00)

-1.00 + 3.00 = +2.00

Rule 3. (axis): 135

Therefore. The sphero-cylindrical prescription is: -3.00DC + 2.00DC x 135

Toric transposition

A toric lens is practical same as a cylindrical lens. By these design the surfaces of toric lenses are more curved (or bent) than the surfaces of cylindrical lenses. But unlike the flat cylindrical lens prescription, the toxic prescription will indicate the various curves that have been incorporated during the process of production to arrive at a particular prescription. This explain why a toric lens prescription have the following three parts

  1. The base curve (BC)
  2. The cross curve (CC)
  3. The sphere curve (SC)

To produce a particular prescription from a lens blanks these three components with their different powers will be generated in the factory. This process is called surfacing

Since most of the physical properties of toric lenses are similar to cylindrical lens and vice versa. This means that given a specific toric lens prescription, the sphero-cylindrical prescription of equivalent power can be deduced from it by applying the following rules.

Rule 1 (sphere power) = base curve + sphere curve

Rule 2 (cylinder power) = cross curve – base curve

Rule 3 (axis) same as axis of cross curve

The following example is meant to illustrate the application of those rules:

Rule 1 (sphere power): +2.00 + (-3.00) = -1.00DS

Rule 2 (cylinder power) +4.00 – 2.00 = +2.00DC

Rules 3 (axis) = 180

Sphero-cylindrical prescription: -1.00DS + 2.00DC x 180

 Importance of transposition

After an optometrist has done a refraction for a patient, the optometrist takes the prescription to the optician and the lens prescription is given to be + 1.00DS-2.00DC x 180 and the optician check the stocks of lens available and founds that (+1.00DS – 2.00DC x 180) prescription is not in stock, what the optician need to do is to transpose the prescription.

+1.00DS – 2.00DC x 180

– 1.00DS + 2.00DC x 90.

So instead of the optician finding it as a problem of getting + 1.00DS-2.00DC x 180 the optician simply go for -1.00DS + 2.00DC x90 it makes dispensing cylindrical lenses easy instead of keeping your patient  waiting or ordering for more lenses when unknowing to you it is your stock.


In conclusion, transposition is important because it helps and optician to get an equivalent (same) prescription for a given prescription. It is also important for an optician to be acquitted with the knowledge of transposition. An optician should not be ignorant of keeping a patient waiting or placing an order for a cylindrical lens prescription when there is an alternative form of that prescription in stock, all you need to do is to transpose.


Efe Odjimogho, O.D. FNCO (2012). In laboratory manual, senior lecturer optometry department faulty of life science, university of Benin.

Mr. Nsubuisi Igboekwe (2013). Ophthalmic workshop dispensing lab ii

San Francisco, American Academy of ophthalmology 2011-2012

Allied Health on the job training kit 2011

Dallas American Health Association Academy of ophthalmology 2011 (Revised as appropriate)

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