Martin's Formula for Tilt...

**Pete Hanlin** · 03-20-2001, 05:13 PM

Having just completed another pair of "wrap" frames for a moderately myopic patient, I am once again pondering the effect of the tilt on the Rx.

Dragging out my copy of Optical Formulas by Stoner, I notice the note that is placed right at the end of the section on tilt that states that compensating the Rx for face form (or the pd, for that matter) is NOT recommended. Question is, "why not?" I suppose I can understand the idea of not adjusting the pd (because of the induced prism), but why not alter the Rx to compensate for the way the tilt affects it?

Illuminate me once again,
Pete
PS- I usually do adjust the p.d. by 1mm for every 4 degrees of tilt... that seems to work pretty well.

**Homer** · 03-20-2001, 10:53 PM

Pete, you have me on this one.

Why wouldn't you adjust the horizontal position of the OC for tilt (panto or retro) and the PD for the amount of wrap?

Seems that the old formula for induced plus cylinder at 180 for panto could be just turned 90 degrees.

On of the things not ever mentioned is that possibly the old formula, usually more applicable to post cataract lenses, had much more do do with the distance portion of the lens than reading..... for if the lower section of the lens was moved closer to the eye by panto than the effective plus power would be reduced.

The question for me is: In an Rx with low to moderate power, would the power changes be more perceivable than the induced prism?

Second question: What exact power changes would compensated, and why? If one is compensating for wrap or tilt one must also then ask if is the inferior or superior / nasal or temporal portions of the lens that are most important... for to improve one is to have a negative effect on the other.

Maybe it is only vertex distances that we are talking about and that would make things much more simple.

**David Wilson** · 03-20-2001, 11:04 PM

Pete,
As I see it there are two related issues here: one, adjusting the position of the OC to ensure that the principal axis passes through the centre of rotation (ie dropping it 1mm for every 2 degrees of pantoscopic tilt, or in this case decentering it in a similar fashion); and two, adjusting the prescription to allow for the effects of the tilt (Martin's formula etc). This idea has been adopted by some manufacturers in the design of their progressives, notably the Zeiss Individual and the Rodenstock Multigressiv. They refer to it as "as worn", and they require the BVD, mono PDs, heights, pantoscopic tilt and (in some cases, the dihedral angle (face wrap). However, they are doing this with a progressive. With single vision, assuming that you adopt the principal axis/centre of rotation rule, this would not be necessary. With the original refraction, using the refractor head (phoropter) or a trial frame, the lenses will be set at a pantoscopic tilt of 0 and the OC would be on pupil height with the principal axis, therefore, passing through the eye's centre of rotation. So, if the client, while looking through the refractor head looked eccentrically through a point, say, 10mm from the OC they would be viewing at a certain angle. Fitting the specs using the principal axis/centre of rotation rule would ensure that the client, wearing their specs, would be looking through the lens at the same point at the same angle (assuming the same BVD).
A related issue is whether S/V aspherics as readers should be set on distance PD, to comply with the centre of rotation rule, or near CD to eliminate unwanted prism. If you are interested, my view is to set them on distance PD (our fusional convergence reserves will sort out the prism). Anyway, the prism created will be similar to what they were experiencing in their single vision lenses prior to becoming presbyopic.
Just a thought.
Regards
David Wilson

**Darryl Meister** · 03-21-2001, 01:13 AM

Hi Pete,

Fun stuff for this forum, eh?

Martin's rule is essentially a way to compensate for the oblique astigmatism introduced by tilting a thin lens. This is similar to the oblique astigmatism you get when you look away from the center of a lens. As David and Homer have pointed out, tilting a lens shifts the optical axis away from the center of rotation, which means that the line of site is at an angle to the lens in its primary position of gaze (eye looking straight).

Tilting the lens changes the three Rx components:
1. Sphere power is increased
2. Cylinder is induced, equal in sign to the sphere power
3. The axis is oriented at the axis of rotation (if a lens is wrapped, it is tilted about a vertical axis so the cylinder axis is 90 degrees)

Anyay, at a typical vertex distance, which is about 27 mm from the center of the eye, one degree of rotation is roughly equal to 1/2 mm across the lens surface (27 mm * tan 1 = 0.5 mm). So, if you displace the optical center of the lens 1 mm for every 2 degrees of tilt, you can keep the axis passing through the center of rotation of the eye.

Now for the practical stuff... Except in relatively high powers (cataract lenses were a good example) or high amounts of lens tilt, the astigmatism induced by lens tilt is pretty small -- and can safely be ignored. For vertical frame tilts, like pantoscopic and retroscopic, it's usually better to choose a well fitting frame and play with the tilt than to worry about decentering the lenses. In any event, the prism imbalance will generally be small if you do compensate by decentering the OC's.

For wraps, compensating for astigmatism by decentering the lenses in low- to moderate-powers may produce more prism imbalance than the correction of a small amount of unwanted cylinder power is worth. (Since the OC's have to be moved out in a disjunctive fashion, horizontal prism imbalance is induced.)

Actually, that's another topic... Tilting a lens also changes the prismatic effect experienced by the wearer (even if the OC is directly in front of the eye).

This is all further complicated if the Rx has cylinder power already, since you would have to actually subtract the crossed cylinder power from the Rx to compensate for tilt. A simple program could do this for you pretty easily though.

In any event, it's good for opticians to at least pay attention to this stuff. It is certainly better to be aware of a potential problem and how to fix it, than to be dumbfounded with no explanation when the patient complains about his or her eyeglasses. ;)

Best regards,
Darryl

[This message has been edited by Darryl Meister (edited 03-20-2001).]

**Pete Hanlin** · 03-21-2001, 11:05 AM

Thanks for the replies. I understand the formula and the theory as far as panto and retroscopic tilt goes (I spend some time discussing the need to have the optical axis run through the center of rotation in a course I teach on dispensing to myopes). I was just wondering why we wouldn't adjust a script for face form- if we choose not to move the optical center (because, as I understand it, its an either/or proposition- doing one negates the need to do the other).

If I understand Darryl correctly, he is saying that the correction of the induced cylinder isn't worth the prism that results from moving the oc. Okay, what about just compensating for the induced cylinder and leaving the p.d. alone? I'm sure a program could be written to crank out the new Rxs- adjusting for sph and cyl independant of each other.

It just seems that- in this age of "Retina Forward Design" and all- that we should be able to come up with a solution for -4.00 sph patients that want to wear a wrap with 14 degrees of face wrap per eye...

Thanks for the input!
Pete

**Darryl Meister** · 03-21-2001, 05:44 PM

Hi Pete,

After reviewing my post, I can see where I wasn't entirely clear...

You can compensate for the astigmatism created by a tilted lens by either decentering the OC according to Martin's rule or by changing the Rx by subtracting out the induced power. Decentering the OC will induce prism, which could induce prism imbalance for horizontal displacements. If you change the Rx though, no prism will be induced (although the tilt of the lens itself will induce some prism).

Changing the Rx would be the ideal solution if and when this is necessary, if you know the exact amount of tilt, but it is also the most complicated. Also, for the majority of the lenses that you put in a wrap frame the difference won't really be perceptible.

For instance, consider a +2.00 DS lens in a wrap frame that tilts the lens 10 degrees. This would require decentering each lens by roughly 5 mm out. This is going to induce about 2 prism diopters of base out prism, which will put a considerable strain on the negative fusional vergence system of the eyes. So, what did all of this prism save us? A +2.00 DS lens tilted by 10 degrees introduces less than 0.07 D of unwanted astigmatism. But, if you really wanted to, you could subtract this unwanted cylinder power from the original Rx for the ordered lenses and get the power dead on.

Anyway, I hope that makes more sense.

Best regards,
Darryl

**Pete Hanlin** · 03-22-2001, 11:14 AM

Thanks Darryl,

I just wanted to differentiate between the methods by which face form can be dealt with. Before I pass on anything as I understand it, I like to see how you understand it (that way, I know if I'm understanding it correctly ;) ).

I rarely get hyperopes wanting the wrap frames (after all, it'd be too easy if someone whose lenses naturally come with a steeper base curve would want a frame that requires such a curve...). It seems like the people attracted to wrap frames are- without fail -4.00 to -6.00 myopes wanting to look cool on the court, beach, etc...

Thanks for the info!
Pete

**EyeCon** · 03-22-2001, 06:58 PM

Pete Hanlin - " Martin's Formula for Tilt "
This topic is very timely in coming! I too am very concerned about fitting larger, wrapped frames; the trend toward larger-sized frames and the impact on lens design and optical clarity...It seems to me, if I understand this correctly, we have two otions for optical clarity in the case of larger-sized frames: 1) Adjusting the OC (dropping the OC 1mm/2 degrees panto tilt. 2)Adjusting the Rx to allow for the effects of tilt...I tend to think the vertex distance at refraction would come into play here...and the solution when fitting the really huge sunglasses this summer...let us put the case that an OD uses a standard 12mm vertex distance at exam time! Now if the customer/patient's frame ( larger-sized frame, wrapped-frame ) selection is fitted 6mm closer. That would mean an Rx of say -4.25 -1.50 x 085 fitted 6mm closer, the resultant compensated Rx should be something like: -4.12 -1.37 x 085 or if something crazy like fitting a +15.25 +1.25 x 175 fitted 4mm closer would be compensated with +15.50 -1.12 x 085!... But compensating an Rx for good optics is one thing; but good optics become more difficult when lens designs are limited. The question I have to my optical brothers and sisters is: Are the lens manufacturers getting in shape to handle the larger frame sizes...I mean the 64 eyesize, 9-base front curve behemoths that are Chloe Lunettes and Killer Loops etc.
I followed the progress from thinner and lighter lenses ( polycarbonate ), but is poly the best material for a larger sunglass frame?

**harry a saake** · 04-07-2001, 10:38 PM

Darryl, once upon a time we use to actually fabricate lenses with crossed cylinders on somewhat of a regular basis,1964 etc, glass and all that. What would be the efect of a major face wrap like pete described on a pair of actual crossed cylinders that were of a significant power and maybe on an oblique axis

**Darryl Meister** · 04-09-2001, 02:10 PM

Hi Harry,

For all intents and purposes, the effect would be the same. I am assuming that by lenses made in "cross cyl" you are referring to lenses with cylinder power on both the front and back surfaces. A lens made with a toric (cylinder) surface on the front and a toric (cylinder) surface on the back will yield a single, net spherocylindrical power. If we assume that this lens is reasonably thin, tilting the lens will have the same effect as tilting the single, net spherocylindrical power.

The math just becomes much more complex, since you now have to worry about tilting a spherocylindrical lens (particularly if you plan to compensate for it). For example, wrapping a +2.00 DS -3.00 DC x 045 lens 12 degrees will induce an Rx error of roughly +0.07 DS -0.13 DC x 041.

Best regards,
Darryl

[This message has been edited by Darryl Meister (edited 04-09-2001).]

**VVizard** · 12-06-2005, 08:46 AM

Hello, Darryl and everyone.

I wonder if there any chance to revive this thread ...

I want to ask you what calculations you did in you last message. What is Martin's formula in case of sphero-cylindrical lens... I just can't find it but only spherical case. Also, maybe you can refer me to some source for derivation of Martin's formula.

Thanks in advance, hoping that somebody will read this :)