The concept that we will discuss here is in  Systems for Ophthalmic Dispensing on pages 62 - 68, and in Optical Formulas Tutorial, pages 161-163.  Start this module by reading.

There are two circumstances under which we normally 'tilt' a lens.  We rotate the lens horizontally, or on the 180 meridian, for pantoscopic tilt.  And we rotate the lens vertically, on the 90 meridian, for face form.  If you look back at the lesson on aberrations you will see that a narrow beam of light entering a spherical lens obliquely, or not parallel to the axis of the lens, will show marginal astigmatism, also called oblique astigmatism.  When we tilt the lens either horizontally or vertically we are potentially inducing astigmatism in the lens, regardless of whether or not it is a spherical lens or a compound lens.

Look at the diagram in Systems on the far right (F - J) on  63, and on page 64.  This diagram is showing you why we use face form when we have to decenter the optical centers (or major reference point) of the lenses from the geometrical center of the lens.  If this is done correctly then we do not create marginal astigmatism for the wearer when looking in the primary position (or straight ahead).  Likewise, in the middle (A - E) of page 63, and on page 65, Dr. Brooks talks about pantoscopic tilt with respect to the position of the optical center (or MRP) below the pupil.  Again, this is done to keep from inducing marginal astigmatism when the wearer is looking in the primary position.

If we do not adjust the glasses correctly with pantoscopic tilt and face form based on the position of the MRP, then the rotation of the lens with respect to the pupil when the eye is looking straight ahead induces extra (unwanted) sphere power, and it induces cylinder that was not intended.  If we look just at pantoscopic tilt, which is a rotation around the 180 meridian, we induce cylinder on the 180.  If we look just at face form, which is rotation around the 90 meridian, we induce cylinder on the 90th.

Suppose you are handed an Rx of -15.00 sph.  You are going to make the glasses with an aspheric or polycarbonate lens, where you know that you are to measure the height of the OC, and you place the OC directly in front of the wearer's pupil.  You give the glasses a few degrees of pantoscopic tilt to decrease surface reflections.  You know that you want a frame with as little decentration as possible to decrease edge thickness, and you put a little face form in the glasses for cosmetic purposes.  What happens to the effective power of the lens?

The OC at pupil center combined with pantoscopic tilt induces some extra minus power (since the sphere power was minus to start with) and it induces some minus cylinder on the 180 axis.  How much of each can be computed using Martin's formula for lens tilt.

Likewise, the minimal (or no) decentration combined with face form will induce some extra minus power and it will induce some minus cylinder on the 90 axis, since that is the meridian of rotation for face form.

Now, suppose a person with an Rx of +10.00 -3.00 x 180 comes in and wants you to put prescription lenses in a pair of goggles with a lot of face form.  What happens?  You induce extra + sphere power (since the power of the lens is +) and you induce extra + cylinder on the 90th.  The end result is that the person will not see correctly through the lenses even though the power of the lens may be perfect.

The concepts to "get" here are:
 

For a plus lens, incorrect pantoscopic tilt will induce extra + sphere and + cylinder on the 180 axis.	To avoid, make sure that the MRP is 1 mm below the pupil center for every 2 degrees of pantoscopic tilt.  (Do not use retroscopic tilt for an MRP that is above the pupil.  Find another way to fix this problem.)
For a minus lens, incorrect pantoscopic tilt will induce extra - sphere and - cylinder on the 180 axis.

For a plus lens, incorrect face form will induce extra + sphere and + cylinder on the 90 axis.	To avoid, use some face form if the MRP is decentered in, and use no face form if there is no decentration.  If there is a lot of decentration then there should be more face form.  (Do not use negative face form if the MRP is decentered out.)
For a minus lens, incorrect face form will induce extra - sphere and - cylinder on the 90 axis.

Find out from your instructor if you will be required to use the formulas.  If not, it would still be a good idea to read through the following examples to better understand what is happening.

In using the problems below, I give you permission to use 3 instead of 2n.  If the lens is CR39 or crown glass, 2n is close to 3.  For polycarbonate or high-index plastic, the amount of error is minimal, particularly since this is just an exercise to see what the effect is of an improper OC height/pantoscopic tilt combination.

EXAMPLE 1:
A +15.00 lens has 10 degrees of pantoscopic tilt, and the OC is placed directly in front of the pupil.  What is the effective power of this lens?

New sphere power = sphere[1 + (sin a)/3]
        Do the sine 10 first, then square it.
        Divide by 3.
        Add 1.
        Multiply by the sphere power.

sin 10 = 0.173648...
(sin 10) = 0.03015....
/3 = 0.01005....
+1 = 1.01005....
x 15.00 = +15.15 diopters.  So the lens has a little stronger plus power.

Cylinder power, induced on the 180 meridian, since that is the meridian that you are rotating around, is (new sphere)(tan a)
        Do the tangent 10 first, then square it.
        Multiply by the new sphere.

tan 10 = 0.176326....
(tan 10) = 0.031091....
x 15.15 = 0.47 diopters.

The lens will 'feel' like a +15.15 +0.47 x 180.  On the horizontal meridian this person will have an Rx that is 1/8 diopter too strong, and on the vertical meridian the Rx will be 5/8 diopter too strong.
 

EXAMPLE 2:
A -15.00 lens has 10 degrees of pantoscopic tilt, and the OC is placed directly in front of the pupil.  What is the effective power of this lens?

New sphere power = sphere[1 + (sin a)/3]

sin 10 = 0.173648...
(sin 10) = 0.03015....
/3 = 0.01005....
+1 = 1.01005....
x -15.00 = -15.15 diopters.  So the lens has a little stronger minus power.

Cylinder power induced on the 180 meridian is (new sphere)(tan a)

tan 10 = 0.176326....
(tan 10) = 0.031091....
x -15.15 = -0.47 diopters.

The lens will 'feel' like a -15.15 -0.47 x 180.  On the horizontal meridian this person will have an Rx that is 1/8 diopter too strong, and on the vertical meridian the Rx will be 5/8 diopter too strong.
 

EXAMPLE 3:
A -15.00-4.00 x 090 lens has 12 degrees of pantoscopic tilt, and the OC is placed directly in front of the pupil.  What is the effective power of this lens?

First, you need the lens power on the 180 meridian, because that is what you are rotating around.  We will be inducing cylinder on the 180 meridian, so you need the Rx in a form where the axis is 180.  (If the axis were other than 90 / 180,  then the problem is more complex and we will not do any of those.)

Rx with the axis at 180 is -19.00 +4.00 x 180

New sphere power = sphere[1 + (sin a)/3]

sin 12 = 0.207911...
(sin 12) = 0.043227....
/3 = 0.014409....
+1 = 1.014409....
x -19.00 = -19.27 diopters.  So the lens has a little stronger minus on the 180.

Cylinder power induced on the 180 meridian is (new sphere)(tan a)

 tan 12 = 0.212556....
(tan 12) = 0.045180....
x -19.27 = -0.87 diopters.

The tile will induce -0.87 cylinder on the 180.  The lens already had +4.00 cylinder on the 180, so the net effect will be to reduce the + cylinder to +3.13.   By inducing extra - cylinder on the 180 the amount f cylinder that the wearer will 'feel' will be reduced.  The effective Rx is -19.27 +3.13 x 180, or -16.14 -3.13 x 090.

REFERENCES:
Brooks & Borish, Systems for Ophthalmic Dispensing, 2nd edition pages 62 - 68.
Fannin & Grosvenor, Clinical Optics 2nd edition, Butterworth-Heinemann, pages 48-50.
Stoner, & Perkins, Optical Formulas Tutorial pages 161-163.