# Optical Forums > Ophthalmic Optics >  Are plus lenses heavier than minus?

## Robert Martellaro

There is a thread over at sci.med.vision (05-18-05 Re: Farsighted v. Nearsighted Ratio) that turned into a discussion of the advantages and disadvantages of being hyperopic or myopic, with one person declaring that with all things being equal (edge and CT, power, design, material etc), plus is heavier than minus. Another person thought minus would be heavier than plus. I ran a series of powers through my dispensing software and plus was about 20% heavier in low to medium powers, with the differential increasing with higher powers. I'm *guessing* that a meniscus lens is more efficient, mass wise, with minus powers, compared to plus. Two questions...

1. Is my software describing the situation correctly?

2. If it is, why is plus heavier?

Thanks

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## chip anderson

Plus has to have added center thickness in order to have enough thickness at the farthest extremity from the eye (optical) center.  This will increase greatly with large frames and some shapes such as tear-drop.

Minus lenses increase in edge thickness for the opposite reason but do not need anything added at the center to make them strong enough for the edge mount at the thinnest point.

Chip

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## Robert Martellaro

Chip,

My example was using cr39 with a 2.0 CT for the minus and 2.0mm ET for the plus, round 60mm with no DEC, using best form BCs. In other words a level playing field. Still, the minus powers keep coming up lighter than plus powers. Any further thoughts for this optical theory challenged optician? 

Thanks

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## 35oldguy

Using the math method if you have a +2.00 D lens with a ET of 2mm you will have a CT of 3.8 CT with zero decentration.  Or with a -2.00 D lens 2.0 CT you will have a 3.8 ET. Weight difference? Have you put them on a scale to see if there is a difference? I wonder?!!!




> Chip,
> 
> My example was using cr39 with a 2.0 CT for the minus and 2.0mm ET for the plus, round 60mm with no DEC, using best form BCs. In other words a level playing field. Still, the minus powers keep coming up lighter than plus powers. Any further thoughts for this optical theory challenged optician? 
> 
> Thanks

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## DrG

Since I am the person who made this claim on sci.med.vision, here is my reasoning:

Represent a simple plus lens as a hemisphere with radius of R.

Represent a simple minus lens as a cylinder containing the same 
hemisphere as above, where the height of the cylinder is the same as R, 
and the area of the base is represented by pi times the radius squared. 
The volume of the cylinder is pi*R^2*h or pi*R^3. Subtracting the 
volume of the sphere from the volume of the cylinder gives the volume 
of the simple minus lens with a CT of zero.  Compare this to the volume 
of the simple plus lens with ET of zero. 

If I use 2 as the value of R in the example, I come up with 16.75 for 
the volume of the hemisphere = plus lens, and I come up with 
25.12-16.75 = 8.37 for the volume of the minus lens.  For those who are math challenged, use 1 for the value of R. 

Please check my math.

DrG

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## chip anderson

Frames, optical centers and P.D.s make the playing field very convoluted.

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## Robert Martellaro

> Frames, optical centers and P.D.s make the playing field very convoluted.


True. But this discussion concerns two identical lens configurations except for power.

Cr39 using best form base curves, 2mm CT for minus, 2mm ET for the plus, no bevel, identical shape. My program gives me these numbers-

60mm diameter -1.00DS weighs 8.53g and 10.57g in plus
60mm diameter -3.00DS weighs 10.13g and 15.3g in plus
60mm diameter -7.00DS weighs 13.82g and 26.1g in plus

DrG,

Your math is fine. I like the idea of determining the volume of the different shapes. Here's a link to a nice math tool...

http://grapevine.abe.msstate.edu/~fto/tools/vol/ ).

Now, is there a tool that will calculate the weight taking into consideration the complex curves of Rx meniscus lenses? 

Thanks for your help,

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## Darryl Meister

*65 mm Corrected Curve Hard Resin Blank, 1 mm Edge*

+4.00 D = 15.1 g
+2.00 D = 9.5 g
+1.00 D = 6.9 g

*65 mm Corrected Curve Hard Resin Blank, 2 mm Center*

-1.00 D = 11.4 g
-2.00 D = 13.9 g
-4.00 D = 18.8 g

*65 mm Corrected Curve Hard Resin Blank, 1 mm Center*

-1.00 D = 7.0 g
-2.00 D = 9.4 g
-4.00 D = 14.4 g

Best regards,
Darryl

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## Darryl Meister

> Now, is there a tool that will calculate the weight taking into consideration the complex curves of Rx meniscus lenses?


http://www.optiboard.com/forums/showthread.php?t=12650

Best regards,
Darryl

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## DrG

> *65 mm Corrected Curve Hard Resin Blank, 1 mm Edge*
> 
> +4.00 D = 15.1 g
> +2.00 D = 9.5 g
> +1.00 D = 6.9 g
> 
> *65 mm Corrected Curve Hard Resin Blank, 2 mm Center*
> 
> -1.00 D = 11.4 g
> ...


It's that easy now, is it?  OK, edge them all down to a 60mm eye, geometrically centered, and reweigh.

DrG

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## Darryl Meister

> OK, edge them all down to a 60mm eye, geometrically centered, and reweigh.


*60 mm Corrected Curve Hard Resin Blank, 1 mm Edge

*+4.00 D = 11.3 g
+2.00 D = 7.4 g
+1.00 D = 5.5 g

*60 mm Corrected Curve Hard Resin Blank, 1 mm Center*

-1.00 D = 5.6 g
-2.00 D = 7.4 g
-4.00 D = 10.9 g

And these are calculated values from a program I wrote, not measured values. Generally, the volume of a plus lens will be very similar to the volume of a minus lens of the same power when both have the same minimum thickness and lens form.

Best regards,
Darryl

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## DrG

It would seem that we have a conundrum between programs. Since my theoretical calculations predict that the plus lens of the same radius as the minus lens has more mass than the minus lens, I would have to side with Robert.

The only resolution would seem to be handguns at 6 paces, or actual laboratory scale measurements.

DrG

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## Darryl Meister

> It would seem that we have a conundrum between programs. Since my theoretical calculations predict that the plus lens of the same radius as the minus lens has more mass than the minus lens, I would have to side with Robert.


Below is the volume calculation of a +4.00 D plano-convex lens and a -4.00 D plano-concave lens at a 60 mm diameter with zero minimum thickness in hard resin. Feel free to double-check the math.

*Assumptions*

D = Diameter of lens blank (60 mm)
F = Power of major surface (4.00 D)
R = Radius of curvature of major surface
N = Refractive index (1.500)
S = Sagitta of major surface

R = 1000 * (n - 1) / F
R = 1000 * (1.500 - 1) / 4.00 = 125.0 mm

S = R - SQRT(R^2 - (D/2)^2)
S = 125 - SQRT(125^2 - 30^2) = 3.65 mm

*Volume of 4.00 Section of a Sphere*

Vs = PI/3 * S^2 (3R - S)
Vs = PI/3 * 3.65^2 (3 * 125 - 3.65) = 5180.81 mm^3 = 5.18 cm^3

This represents the volume of a +4.00 D plano-convex spectacle lens at a diameter of 60 mm with a knife edge thickness.

*Volume of Cylinder*

Vc = PI * (D/2)^2 * S
Vc = PI * 30^2 * 3.65 = 10320.13 mm^3 = 10.32 cm^3

*Net Volume of Cylinder - Section of a Sphere*

Vn = Vc - Vs
Vn = 10.32 - 5.18 = 5.14 cm^3

This represents the volume of a -4.00 D plano-concave spectacle lens at a diameter of 60 mm with a zero center thickness.

Note that there is virtually no difference between these two lens volumes.

Best regards,
Darryl

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## OPTIDONN

> Below is the volume calculation of a +4.00 D plano-convex lens and a -4.00 D plano-concave lens at a 60 mm diameter with zero minimum thickness in hard resin. Feel free to double-check the math.
> 
> *Assumptions*
> 
> D = Diameter of lens blank (60 mm)
> F = Power of major surface (4.00 D)
> R = Radius of curvature of major surface
> N = Refractive index (1.500)
> S = Sagitta of major surface
> ...


Here we just trust Darryl. The man works for Sola and boy can he make us feel stupid real quick (in a good way:) ). *The guy is a walking optical program!!*

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## Darryl Meister

> The guy is a walking optical program!!


Years of moderating the Ophthalmic Optics Forum will do that to you... ;)

Best regards,
Darryl

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## HarryChiling

The math is like poetry, this is not all opthalmic optics but is applied that way Bravo.

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## Darryl Meister

Okay, I was feeling a bit inspired, so I decided to prove mathematically exactly _why_ a plano-convex lens is roughly equal in volume to a plano-concave lens of the same diameter and power.

Recall that the volume of a section of a sphere is given by,

Vs = PI/3 * S^2 (3R - S)

which represents a plano-convex lens with a zero edge thickness.

Since S (the sag) will usually be small in relation to the radius of curvature R, this formula can be simplified using,

Vs = PI/3 * S^2 * 3R

Now recall that to calculate the volume of a plano-concave lens, we must first determine the volume of a cylinder of equivalent thickness using,

Vc = PI * (D/2)^2 * S

We then subtract the volume of the section of a sphere from the volume of this cylinder in order to determine the volume of a plano-concave lens with a zero center thickness,

Vn = Vc - Vs

Moreover, if we want to determine the _difference_ in volume between a plano-concave lens and a plano-convex lens, you would subtract the volume of the plano-convex lens from the volume of the plano-concave lens,

Dif = Vn - Vs = (Vc - Vs) - Vs = Vc - 2 * Vs

Now, we substitute for Vc and Vs,

Dif = Vc - 2 * Vs
Dif = PI * (D/2)^2 * S - 2 * PI/3 * S^2 * 3R

Now, again since the sag S is usually small relative to the radius or curvature R, we can substitute this _approximate sag formula_ for S,

S = (D/2)^2 / (2R)

Finally, we substitute and then simplify,

Dif = PI * (D/2)^2 * S - 2 * PI/3 * S^2 * 3R
Dif = PI * (D/2)^2 * (D/2)^2 / (2R) - 2 * PI/3 * [(D/2)^2 / (2R)]^2 * 3R
Dif = PI * (D/2)^4 / (2R) - 2 * PI * R * (D/2)^4 / (4R^2)
Dif = PI * (D/2)^4 / (2R) - PI * (D/2)^4 / (2R)
Dif = 0

This proves mathematically that when the radius of curvature R is relatively long compared to the sagitta S of the surface, a plano-convex lens produces no difference in volume from a plano-concave lens of the same power and diameter.

Best regards,
Darryl

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## DrG

According to your "approximate" formula, your answer is "approximately" true.  :hammer: 

DrG

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## OPTIDONN

DrG do you have a version of this equation? If you do please share. I like to copy them down and try to learn them. The more the merrier!:cheers:

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## DrG

I don't have my own version, but I can certainly oblige. How many versions would you like?
:drop: 

DrG

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## OPTIDONN

I'll take anything and everything!

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## Darryl Meister

> According to your "approximate" formula, your answer is "approximately" true.


But ultimately true enough to prove that there is no real advantage to plus versus minus power when it comes to lens volume -- at least until you start factoring in things like minimum thickness differences. For our +/-4.00 lens at 60 mm, even the exact difference is less than 0.8%.

Best regards,
Darryl

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## DrG

> But ultimately true enough to prove that there is no real advantage to plus versus minus power when it comes to lens volume -- at least until you start factoring in things like minimum thickness differences. For our +/-4.00 lens at 60 mm, even the exact difference is less than 0.8%.
> 
> Best regards,
> Darryl


Theoretically, there is a difference, even if it is less than one percent for your example.  In practice, there are thickness differences, as the thicker parts of the minus lens are discarded during edging.  But, as you have shown, there are ways of minimizing those differences.

DrG

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## Darryl Meister

> In practice, there are thickness differences, as the thicker parts of the minus lens are discarded during edging.


Very true, especially as the frame difference increases. But also remember that minus lenses generally have more minimum thickness than plus lenses (e.g., a 2.0 CT versus a 1.0 ET), which will offset some of this. Still for a _shaped_ lens you could probably argue that the plus lens would be heavier, though the actual magnitude of the difference would depend upon several factors.

Best regards,
Darryl

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## Darryl Meister

For a typical frame (52 x 35 mm) with 3 mm of decentration, the volumes of the _shaped_ lenses would be.

*-4.00 D: 4.39 cm^3*


*+4.00 D: 5.07 cm^3*


Best regards,
Darryl

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## Darryl Meister

> Still for a _shaped_ lens you could probably argue that the plus lens would be heavier, though the actual magnitude of the difference would depend upon several factors.


I should clarify that this statement holds true for lenses of moderate to high power, or in frames requiring a great deal of decentration or a relatively large minimum blank size. As the powers drop below +/-3.00 D, or the decentration becomes negligible, the advantage in thickness can actually reverse in favor of plus lenses.

Best regards,
Darryl

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## Robert Martellaro

> Still for a _shaped_ lens you could probably argue that the plus lens would be heavier, though the actual magnitude of the difference would depend upon several factors.


And that's exactly why my program was misreporting the weight of plus lenses. The program I use allows me to choose from ten different shapes, and plots the edge thickness at eight equally spaced points around the clock. This seems to be working properly for minus powers but when I choose round for the shape in plus powers the lens weight is higher than if I choose a square shape, which is just the opposite of what one would expect with plus powers. I guess I'll have to trial and error each shape and re-map the keyboard. 

Darryl,

Thanks for spending so much time and effort putting this matter to rest. Thanks also for pointing me towards the OpticsLite program. Many useful tools.

Respectfully

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## Darryl Meister

> Thanks for spending so much time and effort putting this matter to rest


No prob! It was an interesting exercise.

Best regards,
Darryl

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## DrG

> I should clarify that this statement holds true for lenses of moderate to high power, or in frames requiring a great deal of decentration or a relatively large minimum blank size. As the powers drop below +/-3.00 D, or the decentration becomes negligible, the advantage in thickness can actually reverse in favor of plus lenses.
> 
> Best regards,
> Darryl


I'm sorry to have taken so long to respond, but I think you are agreeing that plano/concave lenses have less mass than plano/convex lenses, all things being equal including the minimum thickness. And that the difference in mass increases with increasing sag, and approaches zero as the sag approaches zero....or the base curve approaches a cone -- on a theoretical basis, of course.

I do appreciate the different perspectives from a practical point of view, though.

Best regards,
DrG

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## Darryl Meister

Here is a summary of my conclusions:

1. For _uncut_, knife-thickness plano-concave and plano-convex lenses, there is no practical difference in volume. As the diameter or power of the lens blank increases, the plano-convex lens would normally increase in volume slightly faster than the plano-concave lens of the same curvature (refer to the mathematics above). However, plus lenses must be reduced in curvature somewhat in order to compensate for the gain in vertex power caused by thickness. As it turns out, the reduction in volume as a consequence of this reduction in curvature very nearly equals the gain in volume produced as the mathematical approximation breaks down. I had a look at a +10.00 D plano-convex lens and a -10.00 D plano-concave lens, both at a 70 mm diameter, and the difference in volume was only 0.02 cm^3. This is even less than it was for +/-4.00 D lenses at 60 mm.

2. For _shaped_ lenses of typical thickness and base curvature, minus lenses generally produce less volume in moderate to high powers than plus lenses. For powers below +/-3.00 D, plus lenses will generally have similar -- and often less -- volume (factoring in the smaller minimum thickness) than minus lenses, unless the frame requires significant decentration or is of an unusual shape.

Best regards,
Darryl

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## drk

Great thread. I always assumed that a plus lens weighed more than a minus, but I've been incorrect.

How about this twist: are aspheric plus lenses flattened greater than aspheric minus lenses are steepened? In other words, on the whole, has the industry produced higher eccentricity plus lenses than minus?  It would seem so.

My preconceived notion is that aspheric design benefits plus much more than minus.

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## Steve Machol

Is it possible that because the center of gravity in a plus lens is farther from the face it would appear to be heavier (i.e., more pressure on the nose) than a similarly powered minus lens?  Just a thought.

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## Darryl Meister

> My preconceived notion is that aspheric design benefits plus much more than minus


I think the biggest issue with plus versus minus aspheric lens designs is the fact that base curves for minus lenses are already relatively flat, and don't require a great deal of modification from "best form" shapes. Further, asphericity on the _back_ surface of a minus lens will result in maximum thickness reduction (and vice versa for aspheric plus lenses), though most semi-finished aspheric lens blanks are aspheric on the front.

Nevertheless, the asphericity required to improve the optical performance of a minus lens results in nearly as much thickness reduction as it does for a comparable plus lens -- even when the asphericity is on the front surface. However, the reduction in volume (and weight) is not as significant as with plus lenses. (At least according to the simple aspherics I tested in OpticsLite.)

Although I'd have to sit down and think about it, this could be due to the fact that the bulk of the volume and thickness of a plus lens is fixed by its center thickness, which is in turn fixed by the sagittal depth of the front surface for a given back curve and edge thickness. A shallower aspheric curve reduces the center thickness and volume by an amount equal to the difference in height between a spherical surface and an aspherical surface at the same diameter. However, asphericity on the surface of a minus lens rolls off the thickness toward the edge. Consequently, the reduction in thickness (and volume) as a result of the increasing asphericity of the surface only becomes pronounced as you get farther and farther out.

Think of it like this: The smaller you edge an aspheric minus lens down, the less the edge thickness will vary from the edge thickness of a comparable spherical design. However, the reduction in center thickness for an aspheric plus lens is fixed, regardless of how much you edge the lens down.




> Is it possible that because the center of gravity in a plus lens is farther from the face it would appear to be heavier (i.e., more pressure on the nose) than a similarly powered minus lens?


Interesting thought. This might make the lenses more likely to "tip off," since moving the center of gravity forward would make the weight of the lenses work against the temples by using the nosepads act as a fulcrum. The downward force on the nose would be similar, though there might be a new _torsion_ component against the bridge of the nose as a result of the weight of the lenses applying torque to the nose pads (which are acting as that fixed fulcrum).

Best regards,
Darryl

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## Tom Hubin

Hello,

This is my second attempt to post to this thread. Let's see if I can do this right this time.

I am new to this list so may not be strictly proper.

In the 1970's I was an Army optician in Denver and a civilian optician in Maryland with a Virginia license. I also taught opticianry with Ken Wagner and Jim Matthews at Essex Community College in Baltimore. These days I am an optical engineer but still poke my nose into old interests.

I was referred to this list when I asked at sci.med.vision for info and manuals for the Essilor MBX edger. I still need that info but probably need to find the right place to ask.

I just happened onto this thread regarding the weight of lenses. I too did some math in the 1970's to figure out how much a pair of glasses might weigh. At first this was just to help teach but later I cleaned it up and published the article referenced below in a Washington DC based trade magazine. 

"Weight of Lenses", T. Hubin, THE DISPENSING OPTICIAN, Jul 1978

I have a copy here someplace but no idea where at the moment. Darryl just told me how to post a file so I will try to find it, scan it, and post it as a JPG file or something like that. Wish me luck.

BTW, my math was similar to that posted in this thread for sections of spheres but I doubt that I actually put that much math detail into the published article. 

Tom Hubin
thubin@earthlink.net

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## HarryChiling

The equations that I have go as follows:

Rf=front curve radius
Rb=back curve radius
p=eccentricity of lens (asphericity)
Et=edge thickness
Vf=volume of front
Vb=volume of back
Sb=sag of back
Sf=sag of front
d=diameter of lens
den=density of lens
min=minimum edge thickness

  First I start with the getting the sag for both surfaces

       Sf = Rf/p - √[ (Rf/p)2 - (d/2)2 / p ]
Sb = Rb - √[ Rb2  (d/2)2 ]

     Then I determine the Edge Thickness

 (+) Et = min
(-) Et = Sb  Sf + min    Then I will find the volume (convert all variables into centimeters before equations)

       Vf = π*Sf2 *( 3*Rf  p*Sf )/3
Vb = π*Sb2 *( 3*Rb  Sb )/3

     Then we find the weight (diameter and Et is in mm)

     Weight = den * ( Vf  Vb + π*d2*Et/4000 )

  For reference I used Mo Jalies article "How to ensure the thinnest lensses" Optics Today April 22 2005 and various internet resources, as well as Darryls Optics Lite.  I checked it to make sure everything was correct with Darryls software Optics Lite.

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## QDO1

depends how clever the surfacer is... generally there is no difference between plus and minus, if looked at theoretically, but the crux comes when a plus lens is surfaced.  surfaced poorly, it will be thicker, surfaced to a "on the edge knife edge", and optimised it can be thinner... why? because the minus lens has a set CT that cant be made thinner, because the lens crashes out in surfacing/polishing/glazing



if the lens is aspheric, then the answer is in the math, and it really depends what aspheric curve was used, but all things being equal, the plus lens is generally thinner if well surfaced

explained another way - the smaller the lens, the more the CT of the minus lens effects overall thickness of the minus lens

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