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Weight of edged lenses in different eyewire shapes and sizes
Ophthal. Physiol. Opt. Vol. 1.5, No. 1, pp. 37-44, 1995 Copyright 0 1995 Elsevier Science Ltd for British College of Optometrists Printed in Great Britain. All rights reserved 0275-5408195 $10.00 + 0.00
Weight shapes
of edged lenses and sizes
Churk-Yan
Tang
Department Honcj Kong
of Optometry
and Radiography,
in different
Hong
Kong
Polytechnic,
eyewire
Hung
Horn,
Kowloon,
Summary Weights of edged lenses in different eyewire shapes and sizes were studied through mathematically generated eyewire shapes. The patterns of weight variation are often regular and predictable. The relation of lens weight changes in neighbouring sizes of a given eyewire shape and that in different boxed vertical lens dimensions of a given general eyewire appe#arance were established. These results were verified with physically edged lenses. Thro.jgh these relations, it will be possible to predict the amount of weight change in a pair of spectacles upon changing the frame size and in some cases of eyewire shape modification. Ophthal.
Physiol.
Opt.
1995,
15, 37-44
The veight of spectacles is of great concern to both practitioners and patients. The shape and size of the eyewire are two Important factors governing the weight of the finished spectacles. There have been preliminary studies on the weight of lenses of different frame shapes and sizes’X2. It was found that the weight difference between lenses of adjacent size is often small. In another report3, it was stated that a change of eyewire size by less than 1 mm could change the weight by 10%. In this investigation, weights of lenses were first considered using mathematically generated eyeaire shapes. The patterns of lens weight variation with eyeaire shape and size, under different amount of decentration, were studied. Observations from these studies were then verified by weighing some edged lenses.
custom-ground. This practice is not unrealistic, since the costs of these specially made lenses should remain less expensive than most of the aspheric stock lenses, in which reducing lens thickness is one of the primary concerns. Some positive lenses with a constant centre thickness across different finished sizes were also included for comparison. Lens weights were studied using mathematically generated eyewire shapes. Three basic eyewire shapes were generated: regular oval (with the round shape being a limiting case), pilot and square shapes (Figure la and Table 2). Each of the latter two shapes is composed of sections of a tilted nasal and an erect temporal ellipse. The shape is then completed by suitable segments of the upper and lower horizontal tangential lines about the ellipses. These shapes have an initial boxed size of 58 X 58mm. The weight of a shaped lens was calculated by summing multiple slices of the lens about the optical centre. Further details on the methods of generating eyewire shapes and calculating lens weights are available in a previous study’. Shallower shapes were also produced by modifying the apical radii of the component ellipses. With the horizontal lens dimension (HLD)5 maintained at 58mm, the vertical lens dimensions (VLD)’ were sampled from 58mm to 40mm in four equal steps for each of the three basic eyewire shapes. Eyewires of different VLD derived from a basic shape have a similar general appearance and are categorized into a shape series. The three shape series: oval, pilot and square, are shown in Figure la. To examine
Method Crown glass and CR39 lenses in four back vertex powers were studied. Lens forms were assumed according to published data from a lens manufacturer4. The principal properties of these media and characteristics of the lens forms are given in Table 1. To give the minimum weight, the centre thickness of an edged positive lens was adjusted to keep the minimum edge thickness at 1 mm. This implies that uncut sizes of all positive lenses would need to be Received: 10 May 1994 Revised form: 19 July 1994
37
Ophthal.
38
Physiol.
Opt.
1995
15: No 1
a
OVAL Series
PILOT Series
SQUARE Series BOXEDSIZE
58152
58/58
58/40
58146
b
"A" Series Al
A2
A3
A4
" B" Series
Figure 1. Eyewire shapes: (a) mathematically generated shapes. The outer shapes have a boxed horizontal dimension of 58mm and the vertical lens dimensions are 58, 52, 46 and 40mm. inner shapes are modified from the outer shapes by appropriate proportional factors. (b) shape templates cut from spectacle frames. The boxed size is inserted within each eyewire shape.
Lens Table Lens
1. Major
properties
of the lenses
material
Crown glass n = 1.523 Specific gravity
Plastic (CR39) n = II ,498 Spec.ific gravity
used
Back
= 1 .32
vertex
power
(D)
Front
surface
thickness
adjusted
to maintain
Table 2. Major features of the sections two iconic sections and two segments -.
power
(D)
Centre
Tang
thickness
+ 3.00 + 2.00 + 8.00 + 10.50
0.9 0.8 * *
- 5.00 - 9.00 + 5.00
+ 3.00 + 2.00 + 9.00
2.0 2.0 * x
the minimum
+ll.OO edge
composing the eyewire of horizontal tangential
thickness
39
shapes. lines
(mm)
at 1 mm.
Each eyewire
Boxed size (mm/mm)
R, (mm)
R, (mm)
P”
PI
‘Square’
58158 58152 58146 58140
92.54 82.97 73.40 63.82
91.71 82.22 72.73 63.25
10.00 10.00 10.00 10.00
10.00 10.00 10.00 10.00
‘Pilo-.:’
58158 58152 58146 58140
61.42 55.07 48.71 42.36
50.23 45.03 39.84 34.64
3.50 3.50 3.50 3.50
‘Ova!’
58158 58152 58146 58140
29.00 23.31 18.24 13.79
29.00 23.31 18.24 13.79
1 .oo 0.80 0.63 0.48
ShaFe
Churk-Yan
5.00 9.00 5.00 9.00
+ 9.00 “Centre
size and shape:
in the study
+ +
= 2.55
weight,
shape
is constructed
ROT,
to)
by four
portions:
L, (mm)
L, (mm)
8.1 8.1 8.1 8.1
42.50 44.10 45.71 47.31
35.06 37.43 39.80 42.18
3.00 3.00 3.00 3.00
33.7 33.7 33.7 33.7
30.17 33.05 35.93 38.81
5.66 11.08 16.49 21.90
1.00 0.80 0.63 0.48
0 0 0 0
0 0 0 0
0 0 0 0
R,, apical radius of the conic section in the nasal side. in the temporal side. R t, apical radius of the conic section P,, conic factor of Baker in the nasal conic section. Pt, conic factor of Baker in the temporal conic section. ROT.,, angle of rotation of the nasal conic section. L,, length of the upper tangential line connecting the two conic sections. L,, length of the lower tangential line connecting the two conic sections.
the ,:ffect of changing lens sizes on weight, congruent shapes were generated by appropriate proportional factors. The resultant eyewires have their HLD ranging from 58mm to 36mm, in 12 equal steps. Congruent lenses of different sizes are grouped into a size series. Some examples of these smaller lenses are also illustrated in Figure la. Nasal lens decentrations relative to the boxed centres of the shapes, up to lOmm, were considered. Results from these studies were verified by physically edging and weighing lenses (see Table 3). Lenses were edged with a flat-bevel using spectacle templates (see Figure lb). These shapes approximate two shape series of changing VLD. They were used to show the weight changes in shallower shapes. Four individual shapes, Al, Bl , B3 and B4 were selected for studies of weight variations on changing lens size. Weights of edged lenses were measured with an electronic pan-balance giving an accuracy of fO.1 g.
Results and analysis When only one of the parameters governing edged lens weight is varied, it is likely that weights of two lenses bear approximately a constant ratio. Semilogarithmic plots are useful in revealing the ratios between two entities. In this semilogarithmic space, weights were considered with the lens size. The difference between two points in this space gives their weight ratio. Vuriation of lens weights with boxed horizontal lens dimensions For each of the three eyewire shapes, weights of congruent lenses of different sizes were calculated. The variations of lens weight were studied with the horizontal lens dimensions (HLD). This simulated the situation of weight
40
Ophthal.
Physiol.
Opt.
1995
15: No 1
Table 3. Comparisons of estimated and measured lens weight. Lenses were edged according to a sample of templates cut from currently available spectacle frames (see Figure Ib). Equation (2). with the slope value (m) as indicated, is used to give the estimated lens weights. Estimations are based on the weight of the second lens in each of the following series Lens size
Eye wire
(mm)
Measured
weight
Cg,
Estimated
weight
Cgl
Error
(g)
Shape series A HLD = 50.0mm
“46.2 “43.2
13.4 12.3
13.6 -
0.2
- 8.00 D lens F7 = +3.35D t = 1.03 mm n = 1.700 m = 0.015
“38.8 “36.0
10.4 9.1
10.6 9.6
0.2(2) 0.5 (5)
Shape series B HLD = 57.7 + 0.3mm - 8.00 D lens Fl = +1.50D t = 1.03mm n = 1.523 m = 0.015
“56.0 “51.6 “47.1 “41.8
30.1 27.3 23.5 18.7
31.8
1.7 (6)
t60.6 1-54.4 t48.6 t42.4
25.0 21.5 17.2 13.3
26.6
1.6 (7)
17.6 14.2
0.4 0.9
f60.6 t57.7 t51.6 t43.3
25.0 20.3 13.6 6.3
24.8
-0.2
13.3 7.5
-0.3 (-2) 1.2 (19)
t59.7 1-55.4 t49.7 t41.6
23.1 16.2 11.2 5.5
22.2
-0.9
(-4)
10.6 5.9
-0.6 0.4
(-5) (7)
T56.5 t54.0 t50.7 t46.6
13.3 11.2 8.6 6.5
13.3 8.9 6.7
0.3 0.2
(4) (3)
t59.2 t55.0 f51.2 t47.1
37.0 29.2 22.3 16.5
39.0
2.0
(5)
22.5 16.9
0.2 0.4
(I) (3)
Size series based Fixed blank size + 5.00 D lens Fl = +8.00D t = 6.2mm n = 1.523 m = 0.015
on shape
Size series based e = 1.3+O.lmm + 5.00 D lens F7 = +8.00D n = 1.523 m = 0.030
on shape
Size series based -8.OOD lens F7 = +1.50D t = 1.02mm n = 1.523 m = 0.030
on shape
Size series based - 7.00 D lens Fl = +2.00D t = 2.0mm n = 1.498 m = 0.030
on shape
Size series based - 9.00 D lens F7 = +3.25D t = 0.9mm n = 1.800 m = 0.030
on shape
B4
B4
Al
B3
Bl
23.4 19.5
-0.1 0.8
(2)
(I) (4)
(2) (7)
(-
1)
0 (0)
*Vertical lens dimension. tHorizontal lens dimension. Figures in parentheses are percentages. Fl is the lens front surface power.
changes in different spectacle sizes for a given eyewire design. Inspecting all the calculated weights, it is obvious
that lens weights in a semilogarithmic plot can be well fitted with straight lines. Furthermore, these lines are almost
Lens weight, size and shape: Churk-Yan parallel to each other and the slopes are within the range 0.032 k 0.006 units. Typical results are given in Figures 2a and b for some negative crown lenses and positive CR39 lenses respectively. Most practitioners are aware of the importance of selecting the appropriate uncut blank sizes in edging positive prescriptions. When a single blank size is being used, weight differences between lenses of adjacent sizes will often by mini:nal. Figure 3 gives examples comparing the slopes for case: of having a fixed minimum edge thickness and when fixed blank sizes were used. The rates of change of lens weig,It with size for latter cases are about 0.015 units. Lens weight in diferent eyewire shapes Eyewire shape modification was considered in two aspects: first: a fundamental change of eyewire shape, as in the case of ckanging a round to a square eyewire; second, while maintaining the general appearance of the eyewire unchanged, VLD was varied. The latter study would be useful
a 1, 36
I I 40
I I 44
HORIZONTAL
I a 46
I I 56
I I 52
LENS DIMENSION
(mm )
41
in understanding the effect of weight saving in some recent designs of shallow spectacle frame. Overall eyewire shape changes. With the generated shapes, weights of oval and square lenses were respectively used to represent the upper and lower limits of common edged ophthalmic lenses of the same boxed size. The results (see examples in Figure 2) show that square lenses are about 0.12 logarithmic units (about 30%) heavier than the round lenses for all the negative lenses. Weight differences for positive square and oval lenses vary within the range of 0.12 to 0.41 logarithmic units. Variation of lens weights with boxed vertical lens dimensions. Lens weights were studied with different values of VLD under a constant HLD. For three values of HLD, 36, 48 and 58mm, calculated lens weights are given in Figure 4. Variations of lens weight (in logarithmic units) with VLD could again be assumed linear. The rates of weight changes are however relatively smaller. For negative lenses, the
b I 60
Tang
I 40
II
II
44
40
HORIZONTAL
Figure 2. Variations of calculated lens weights with horizontal lens dimensions for the three are more or less linear and parallel to each other in a semilogarithmic space. The slopes are - 5.00 B, crown (a) negative crown lenses: . , - 9.00 D, crown glass, 58152 series; --, +, pilot; 0, oval; (b) positive CR39 plastic lenses: + 5.00D, CR39, 58/46 series; ‘, nasal decentration; 17], square; +, pilot; 0, oval.
II
II
52
LENS DIMENSION
56
(mm )
eyewire shapes. The variations little affected by decentrations: glass, 58/40 series; 0, square; zero decentration; -, IOmm
I
60
42
Ophthal. Physiol. Opt. 1995 15: No 1 others are size series for overall lens size changes. As with the above analyses, lens weights were expressed in logarithmic units on the ordinate and the abscissae are a linear scale of HLD in cases of size variations of congruent lenses, or the VLD under a constant HLD. The results agreed with previously observed linear properties. For variations of lens weight with HLD, slopes were about 0.033 units for all the negative lenses. For the + 5 .OOD crown lens, the slope was 0.034 units when the minimum edge thickness was maintained at 1 mm and 0.015 units when a single blank size was used. The shape series A and B gave slopes of 0.015 and 0.016 units respectively across different VLD for negative lenses. Discussion In the semilogarithmic space used to characterize changes in weight with changes in lens size, lens weight appeared to be approximately linearly related to its size and hence satisfies an equation of: log(weight) = mLD + c
I I
40
I I
44 HORIZONTAL
I I
I I
I I
56 48 52 LENS DIMENSION (mm )
Figure 3. Weight variations with lens when fixed uncut blank sizes are pilot; , + 9.00D, crown glass, blank size, ----, +9.00D, crown + 5.00D, CR39 e = l.Omm; p, fixed blank size; --, + 5.00D, CR39 e = l.Omm.
1
60
size in positive lenses used: q , square; +, 58/52 series, fixed glass, 58152 series, plastic, 58140 series, plastic, 58/40 series,
slopes are 0.015 k 0.005 units. The rates of change for positive lenses are from 0.005 to 0.020 units. The reason for the variation in the rates of change is that a change in the VLD can call for different centre thickness changes in positive lenses, depending on the eyewire shape. Measurements of physically edged lenses These studies found that the patterns of weight variation with changes in lens size and shape are often regular and predictable. It also appeared that these results are not specific to the lens powers and materials. To verify some of the observations, a sample of lenses of different powers and materials was edged according to some selected spectacle frames. The measured weights are given in Figure 5 and Table 3. A total of seven lens series was considered: two shape series with varying VLD and the
(1)
where m is the slope; c is a constant; LD is the HLD for the study of variation of lens weight for congruent lenses, or the VLD in considerations of weight changes within a shape series. By examining all the calculated and measured weights, it was found that a common slope (m) of 0.030 units for the former cases could be assumed for all negative lenses; and for positive lenses when the minimum edge thickness was maintained at 1 mm. In cases of positive lenses cut from a fixed blank size, a slope of 0.015 units would apply. For the consideration of weight variations with VLD, a slope of 0.015 units could be assumed for negative lenses. It appeared not practical to assign an average slope for positive lenses in the latter standing because of their eyewire shape dependence. Lens weight changes in neighbouring size Practitioners are concerned with the increase of lens weight when a frame larger than the one tried by their patient is ordered. From Equation (l), it can be shown that the change of lens weight in logarithmic units (Alog(weight)) between two lenses can be derived from the change of lens size (A LD) : A log (weight) = m A LD
(2)
It should also be mentioned that the change of lens weight in logarithmic units (Alog(weight)) is also equal to the logarithmic value of the weight ratio (log(weight ratio)). For variations of lens weight with the HLD, the slope
Lens weight, size and shape: Churk-Yan Tang 100
43
1008
L E ti w E I
1c
E T
g
c -I c
/a 1 24
II 28
II 32
II 36
VERTICAL
II 40
II 44
LENS DIMENSION
II 48
II 52
II 56
I 60
(mm )
b .-
1
I I
I I
I I
I I
I I
I I
I
I I
I
28
32
36
40
44
48
52
56
60
VERTICAL
LENS DIMENSION
Figure 4. Variations of lens weight with vertical lens dimensions (VLD) in different shape series. negative lenses (a), the slopes for the positive lenses (b) are more variable. CR39 plastic lenses: 0, oval; , HLD 58mm; - - -, HLD 48mm; -, HLD, 36mm; (a) - 9.00D; (b) + 9.00D.
(m) i:quals 0.03. Hence, an HLD 2mm increment increases the weight by 0.06 log units (about 15%) and a 2 mn HLD reduction reduces lens weight by about 13 % Taking a 5X/52 series pilot shaped lens at an HLD 52mm for example, a -9.OOD crown lens weighs 19.9g. A change of 2mm in the HLD would change the weight by about 3 g (15%). When edged in the same shape, a +9.0QD crown lens weighs 35.9g. The same change of HLD would lead to about 5g weight change by the same percentage. For two f9.00D crown lenses cut from a single blank size, the slope (m) is 0.015 units. These two lenses will differ by about 2.5g (7%) if they have a size difference of 2mm. Lens weight changes when the shape is modijied vertically A specific kind of eyewire shape modification has been considered in this paper: a shape is changed by varying the VLD under a constant HLD. Shallow spectacle frames are becoming popular. They may include different shape
(mm )
Compared with the square; + , pilot; 0,
forms, ranging from square to oval shapes and may have a shape difference’ as much as 18 mm. The last two members of the A- and B-series in Figure lb are of this nature. Shallow eyewires often give an impression of being light weight spectacles because of the large reduction in the vertical lens dimension. Equation (2) with m = 0.015 units can be used to describe the weight changes in these frames for negative lenses. It is now clear that making the lenses shallower alone is not very effective in reducing lens weights. By the small slope value (m = 0.015), a reduction of VLD by 2mm will reduce the lens weight by about 7 % . The weight difference between a square and oval lens of the same boxed size has been shown to be about 0.12 log units. To make two lenses the same weight, a square lens needs to be 8mm shallower (i.e. 0.12/m where m = 0.015). Estimation of lens weight for different sizes in a given shape Equation (1) is useful in estimating edged lens weight. With the assumed slopes and the weight of a lens of a known size,
44
Ophthal.
Physiol.
Opt. 1995 15: No 1
weight of a sample of edged lenses by Obsteld’ . The errors of estimation were less than 0.7g.
100
Conclusions Edged lens weights in different eyewire sizes and in the variation of eyewire vertical lens dimensions can be described by a linear relation in a semilogarithmic space. For the variations of lens weight in different sizes of the same eyewire shape, a single slope can be assumed for negative lenses; and for positive lenses when the minimum edge thickness is kept at 1 mm. It is found that a change of the lens size by 2mm would change the lens weight by about 15%. A common slope for the variations of lens weight with different VLD but the same general shape appearance was also identified for negative lenses. In these cases, a lens shallower by 2mm would change the lens weight by about 7%. Although detailed studies were carried out only for crown glass and CR39 plastic lenses in four back vertex powers, the patterns of variations are so characteristic that observations from these studies should be applicable for lenses in any ophthalmic materials and for most powers. Practical measurements of edged lenses, including examples in high index materials, support the observations. I I
I I
I I
I
I
I
40
44
48
52
56
60
HORIZONTAL/VERTICAL
I
LENS DIMENSION
64
(mm )
Figure 5. Weight of some physically edged lenses. They have linear properties similar to those found for mathematically generated eyewire shapes. Variable VLD: ---0, -7.OOD, n = 1.700, series A; ---0, -8.OOD, n = 1.523, series B. Variable HLD: +, - 9.00 D, n = 1.800, shape Bl; ---+, - 7.00D, n = 1.498, shape B3; --+ , -8.OOD, n = 1.523, shape Al; ---+, +5.00D, n = 1.523, shape B4; -+, + 5.00 D, n = 1.523, shape B4, constant centre thickness.
weight estimations by Equation (1) were compared with the calculated lens weights. They are generally found to be accurate within k 1.Og and only in some cases of small positive lenses that the errors were over 2.Og and 10% of the actual lens weight. When edged lenses were tested, the accuracy of lens weight predictions was found to lie within the same limits (Table 3). With appropriate slope values (m), Equation (1) was again tested with the measured
Acknowledgments Most of the lenses used for the present study were donated by Rodenstock through its local agent, William S. T. Lee Company Limited. Their support is greatly appreciated. I also wish to thank Dr Brian Brown of Hong Kong Polytechnic for his comments and suggestions. References 1 Obstfeld, H. Weight of edged spectaclelenses. Ophthal. Physiol. Opt. 11, 248-251 (1991) 2 Tang, C. Y. Weight of edged spherical lenses in simulated eyewire shapes. Ophthal. Physiol. Opt. 13, 169-174 (1993) 3 Woodcock, F. R. High index lenses. The Optician 188, 21 September,23-28 (1984) 4 Rodenstock. Ophthalmic Lenses in Practice: Technical Information and Tables. Rodenstock, Munich (1987) 5 British Standard. BS 3521: Terms Relating to Ophthalmic Optics and Spectacle Frames. Part 1. Glossary of terms relating to Ophthalmic Lenses. p. 5. British Standards Institution, London, UK (1991)
Weight shapes
of edged lenses and sizes
Churk-Yan
Tang
Department Honcj Kong
of Optometry
and Radiography,
in different
Hong
Kong
Polytechnic,
eyewire
Hung
Horn,
Kowloon,
Summary Weights of edged lenses in different eyewire shapes and sizes were studied through mathematically generated eyewire shapes. The patterns of weight variation are often regular and predictable. The relation of lens weight changes in neighbouring sizes of a given eyewire shape and that in different boxed vertical lens dimensions of a given general eyewire appe#arance were established. These results were verified with physically edged lenses. Thro.jgh these relations, it will be possible to predict the amount of weight change in a pair of spectacles upon changing the frame size and in some cases of eyewire shape modification. Ophthal.
Physiol.
Opt.
1995,
15, 37-44
The veight of spectacles is of great concern to both practitioners and patients. The shape and size of the eyewire are two Important factors governing the weight of the finished spectacles. There have been preliminary studies on the weight of lenses of different frame shapes and sizes’X2. It was found that the weight difference between lenses of adjacent size is often small. In another report3, it was stated that a change of eyewire size by less than 1 mm could change the weight by 10%. In this investigation, weights of lenses were first considered using mathematically generated eyeaire shapes. The patterns of lens weight variation with eyeaire shape and size, under different amount of decentration, were studied. Observations from these studies were then verified by weighing some edged lenses.
custom-ground. This practice is not unrealistic, since the costs of these specially made lenses should remain less expensive than most of the aspheric stock lenses, in which reducing lens thickness is one of the primary concerns. Some positive lenses with a constant centre thickness across different finished sizes were also included for comparison. Lens weights were studied using mathematically generated eyewire shapes. Three basic eyewire shapes were generated: regular oval (with the round shape being a limiting case), pilot and square shapes (Figure la and Table 2). Each of the latter two shapes is composed of sections of a tilted nasal and an erect temporal ellipse. The shape is then completed by suitable segments of the upper and lower horizontal tangential lines about the ellipses. These shapes have an initial boxed size of 58 X 58mm. The weight of a shaped lens was calculated by summing multiple slices of the lens about the optical centre. Further details on the methods of generating eyewire shapes and calculating lens weights are available in a previous study’. Shallower shapes were also produced by modifying the apical radii of the component ellipses. With the horizontal lens dimension (HLD)5 maintained at 58mm, the vertical lens dimensions (VLD)’ were sampled from 58mm to 40mm in four equal steps for each of the three basic eyewire shapes. Eyewires of different VLD derived from a basic shape have a similar general appearance and are categorized into a shape series. The three shape series: oval, pilot and square, are shown in Figure la. To examine
Method Crown glass and CR39 lenses in four back vertex powers were studied. Lens forms were assumed according to published data from a lens manufacturer4. The principal properties of these media and characteristics of the lens forms are given in Table 1. To give the minimum weight, the centre thickness of an edged positive lens was adjusted to keep the minimum edge thickness at 1 mm. This implies that uncut sizes of all positive lenses would need to be Received: 10 May 1994 Revised form: 19 July 1994
37
Ophthal.
38
Physiol.
Opt.
1995
15: No 1
a
OVAL Series
PILOT Series
SQUARE Series BOXEDSIZE
58152
58/58
58/40
58146
b
"A" Series Al
A2
A3
A4
" B" Series
Figure 1. Eyewire shapes: (a) mathematically generated shapes. The outer shapes have a boxed horizontal dimension of 58mm and the vertical lens dimensions are 58, 52, 46 and 40mm. inner shapes are modified from the outer shapes by appropriate proportional factors. (b) shape templates cut from spectacle frames. The boxed size is inserted within each eyewire shape.
Lens Table Lens
1. Major
properties
of the lenses
material
Crown glass n = 1.523 Specific gravity
Plastic (CR39) n = II ,498 Spec.ific gravity
used
Back
= 1 .32
vertex
power
(D)
Front
surface
thickness
adjusted
to maintain
Table 2. Major features of the sections two iconic sections and two segments -.
power
(D)
Centre
Tang
thickness
+ 3.00 + 2.00 + 8.00 + 10.50
0.9 0.8 * *
- 5.00 - 9.00 + 5.00
+ 3.00 + 2.00 + 9.00
2.0 2.0 * x
the minimum
+ll.OO edge
composing the eyewire of horizontal tangential
thickness
39
shapes. lines
(mm)
at 1 mm.
Each eyewire
Boxed size (mm/mm)
R, (mm)
R, (mm)
P”
PI
‘Square’
58158 58152 58146 58140
92.54 82.97 73.40 63.82
91.71 82.22 72.73 63.25
10.00 10.00 10.00 10.00
10.00 10.00 10.00 10.00
‘Pilo-.:’
58158 58152 58146 58140
61.42 55.07 48.71 42.36
50.23 45.03 39.84 34.64
3.50 3.50 3.50 3.50
‘Ova!’
58158 58152 58146 58140
29.00 23.31 18.24 13.79
29.00 23.31 18.24 13.79
1 .oo 0.80 0.63 0.48
ShaFe
Churk-Yan
5.00 9.00 5.00 9.00
+ 9.00 “Centre
size and shape:
in the study
+ +
= 2.55
weight,
shape
is constructed
ROT,
to)
by four
portions:
L, (mm)
L, (mm)
8.1 8.1 8.1 8.1
42.50 44.10 45.71 47.31
35.06 37.43 39.80 42.18
3.00 3.00 3.00 3.00
33.7 33.7 33.7 33.7
30.17 33.05 35.93 38.81
5.66 11.08 16.49 21.90
1.00 0.80 0.63 0.48
0 0 0 0
0 0 0 0
0 0 0 0
R,, apical radius of the conic section in the nasal side. in the temporal side. R t, apical radius of the conic section P,, conic factor of Baker in the nasal conic section. Pt, conic factor of Baker in the temporal conic section. ROT.,, angle of rotation of the nasal conic section. L,, length of the upper tangential line connecting the two conic sections. L,, length of the lower tangential line connecting the two conic sections.
the ,:ffect of changing lens sizes on weight, congruent shapes were generated by appropriate proportional factors. The resultant eyewires have their HLD ranging from 58mm to 36mm, in 12 equal steps. Congruent lenses of different sizes are grouped into a size series. Some examples of these smaller lenses are also illustrated in Figure la. Nasal lens decentrations relative to the boxed centres of the shapes, up to lOmm, were considered. Results from these studies were verified by physically edging and weighing lenses (see Table 3). Lenses were edged with a flat-bevel using spectacle templates (see Figure lb). These shapes approximate two shape series of changing VLD. They were used to show the weight changes in shallower shapes. Four individual shapes, Al, Bl , B3 and B4 were selected for studies of weight variations on changing lens size. Weights of edged lenses were measured with an electronic pan-balance giving an accuracy of fO.1 g.
Results and analysis When only one of the parameters governing edged lens weight is varied, it is likely that weights of two lenses bear approximately a constant ratio. Semilogarithmic plots are useful in revealing the ratios between two entities. In this semilogarithmic space, weights were considered with the lens size. The difference between two points in this space gives their weight ratio. Vuriation of lens weights with boxed horizontal lens dimensions For each of the three eyewire shapes, weights of congruent lenses of different sizes were calculated. The variations of lens weight were studied with the horizontal lens dimensions (HLD). This simulated the situation of weight
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Ophthal.
Physiol.
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1995
15: No 1
Table 3. Comparisons of estimated and measured lens weight. Lenses were edged according to a sample of templates cut from currently available spectacle frames (see Figure Ib). Equation (2). with the slope value (m) as indicated, is used to give the estimated lens weights. Estimations are based on the weight of the second lens in each of the following series Lens size
Eye wire
(mm)
Measured
weight
Cg,
Estimated
weight
Cgl
Error
(g)
Shape series A HLD = 50.0mm
“46.2 “43.2
13.4 12.3
13.6 -
0.2
- 8.00 D lens F7 = +3.35D t = 1.03 mm n = 1.700 m = 0.015
“38.8 “36.0
10.4 9.1
10.6 9.6
0.2(2) 0.5 (5)
Shape series B HLD = 57.7 + 0.3mm - 8.00 D lens Fl = +1.50D t = 1.03mm n = 1.523 m = 0.015
“56.0 “51.6 “47.1 “41.8
30.1 27.3 23.5 18.7
31.8
1.7 (6)
t60.6 1-54.4 t48.6 t42.4
25.0 21.5 17.2 13.3
26.6
1.6 (7)
17.6 14.2
0.4 0.9
f60.6 t57.7 t51.6 t43.3
25.0 20.3 13.6 6.3
24.8
-0.2
13.3 7.5
-0.3 (-2) 1.2 (19)
t59.7 1-55.4 t49.7 t41.6
23.1 16.2 11.2 5.5
22.2
-0.9
(-4)
10.6 5.9
-0.6 0.4
(-5) (7)
T56.5 t54.0 t50.7 t46.6
13.3 11.2 8.6 6.5
13.3 8.9 6.7
0.3 0.2
(4) (3)
t59.2 t55.0 f51.2 t47.1
37.0 29.2 22.3 16.5
39.0
2.0
(5)
22.5 16.9
0.2 0.4
(I) (3)
Size series based Fixed blank size + 5.00 D lens Fl = +8.00D t = 6.2mm n = 1.523 m = 0.015
on shape
Size series based e = 1.3+O.lmm + 5.00 D lens F7 = +8.00D n = 1.523 m = 0.030
on shape
Size series based -8.OOD lens F7 = +1.50D t = 1.02mm n = 1.523 m = 0.030
on shape
Size series based - 7.00 D lens Fl = +2.00D t = 2.0mm n = 1.498 m = 0.030
on shape
Size series based - 9.00 D lens F7 = +3.25D t = 0.9mm n = 1.800 m = 0.030
on shape
B4
B4
Al
B3
Bl
23.4 19.5
-0.1 0.8
(2)
(I) (4)
(2) (7)
(-
1)
0 (0)
*Vertical lens dimension. tHorizontal lens dimension. Figures in parentheses are percentages. Fl is the lens front surface power.
changes in different spectacle sizes for a given eyewire design. Inspecting all the calculated weights, it is obvious
that lens weights in a semilogarithmic plot can be well fitted with straight lines. Furthermore, these lines are almost
Lens weight, size and shape: Churk-Yan parallel to each other and the slopes are within the range 0.032 k 0.006 units. Typical results are given in Figures 2a and b for some negative crown lenses and positive CR39 lenses respectively. Most practitioners are aware of the importance of selecting the appropriate uncut blank sizes in edging positive prescriptions. When a single blank size is being used, weight differences between lenses of adjacent sizes will often by mini:nal. Figure 3 gives examples comparing the slopes for case: of having a fixed minimum edge thickness and when fixed blank sizes were used. The rates of change of lens weig,It with size for latter cases are about 0.015 units. Lens weight in diferent eyewire shapes Eyewire shape modification was considered in two aspects: first: a fundamental change of eyewire shape, as in the case of ckanging a round to a square eyewire; second, while maintaining the general appearance of the eyewire unchanged, VLD was varied. The latter study would be useful
a 1, 36
I I 40
I I 44
HORIZONTAL
I a 46
I I 56
I I 52
LENS DIMENSION
(mm )
41
in understanding the effect of weight saving in some recent designs of shallow spectacle frame. Overall eyewire shape changes. With the generated shapes, weights of oval and square lenses were respectively used to represent the upper and lower limits of common edged ophthalmic lenses of the same boxed size. The results (see examples in Figure 2) show that square lenses are about 0.12 logarithmic units (about 30%) heavier than the round lenses for all the negative lenses. Weight differences for positive square and oval lenses vary within the range of 0.12 to 0.41 logarithmic units. Variation of lens weights with boxed vertical lens dimensions. Lens weights were studied with different values of VLD under a constant HLD. For three values of HLD, 36, 48 and 58mm, calculated lens weights are given in Figure 4. Variations of lens weight (in logarithmic units) with VLD could again be assumed linear. The rates of weight changes are however relatively smaller. For negative lenses, the
b I 60
Tang
I 40
II
II
44
40
HORIZONTAL
Figure 2. Variations of calculated lens weights with horizontal lens dimensions for the three are more or less linear and parallel to each other in a semilogarithmic space. The slopes are - 5.00 B, crown (a) negative crown lenses: . , - 9.00 D, crown glass, 58152 series; --, +, pilot; 0, oval; (b) positive CR39 plastic lenses: + 5.00D, CR39, 58/46 series; ‘, nasal decentration; 17], square; +, pilot; 0, oval.
II
II
52
LENS DIMENSION
56
(mm )
eyewire shapes. The variations little affected by decentrations: glass, 58/40 series; 0, square; zero decentration; -, IOmm
I
60
42
Ophthal. Physiol. Opt. 1995 15: No 1 others are size series for overall lens size changes. As with the above analyses, lens weights were expressed in logarithmic units on the ordinate and the abscissae are a linear scale of HLD in cases of size variations of congruent lenses, or the VLD under a constant HLD. The results agreed with previously observed linear properties. For variations of lens weight with HLD, slopes were about 0.033 units for all the negative lenses. For the + 5 .OOD crown lens, the slope was 0.034 units when the minimum edge thickness was maintained at 1 mm and 0.015 units when a single blank size was used. The shape series A and B gave slopes of 0.015 and 0.016 units respectively across different VLD for negative lenses. Discussion In the semilogarithmic space used to characterize changes in weight with changes in lens size, lens weight appeared to be approximately linearly related to its size and hence satisfies an equation of: log(weight) = mLD + c
I I
40
I I
44 HORIZONTAL
I I
I I
I I
56 48 52 LENS DIMENSION (mm )
Figure 3. Weight variations with lens when fixed uncut blank sizes are pilot; , + 9.00D, crown glass, blank size, ----, +9.00D, crown + 5.00D, CR39 e = l.Omm; p, fixed blank size; --, + 5.00D, CR39 e = l.Omm.
1
60
size in positive lenses used: q , square; +, 58/52 series, fixed glass, 58152 series, plastic, 58140 series, plastic, 58/40 series,
slopes are 0.015 k 0.005 units. The rates of change for positive lenses are from 0.005 to 0.020 units. The reason for the variation in the rates of change is that a change in the VLD can call for different centre thickness changes in positive lenses, depending on the eyewire shape. Measurements of physically edged lenses These studies found that the patterns of weight variation with changes in lens size and shape are often regular and predictable. It also appeared that these results are not specific to the lens powers and materials. To verify some of the observations, a sample of lenses of different powers and materials was edged according to some selected spectacle frames. The measured weights are given in Figure 5 and Table 3. A total of seven lens series was considered: two shape series with varying VLD and the
(1)
where m is the slope; c is a constant; LD is the HLD for the study of variation of lens weight for congruent lenses, or the VLD in considerations of weight changes within a shape series. By examining all the calculated and measured weights, it was found that a common slope (m) of 0.030 units for the former cases could be assumed for all negative lenses; and for positive lenses when the minimum edge thickness was maintained at 1 mm. In cases of positive lenses cut from a fixed blank size, a slope of 0.015 units would apply. For the consideration of weight variations with VLD, a slope of 0.015 units could be assumed for negative lenses. It appeared not practical to assign an average slope for positive lenses in the latter standing because of their eyewire shape dependence. Lens weight changes in neighbouring size Practitioners are concerned with the increase of lens weight when a frame larger than the one tried by their patient is ordered. From Equation (l), it can be shown that the change of lens weight in logarithmic units (Alog(weight)) between two lenses can be derived from the change of lens size (A LD) : A log (weight) = m A LD
(2)
It should also be mentioned that the change of lens weight in logarithmic units (Alog(weight)) is also equal to the logarithmic value of the weight ratio (log(weight ratio)). For variations of lens weight with the HLD, the slope
Lens weight, size and shape: Churk-Yan Tang 100
43
1008
L E ti w E I
1c
E T
g
c -I c
/a 1 24
II 28
II 32
II 36
VERTICAL
II 40
II 44
LENS DIMENSION
II 48
II 52
II 56
I 60
(mm )
b .-
1
I I
I I
I I
I I
I I
I I
I
I I
I
28
32
36
40
44
48
52
56
60
VERTICAL
LENS DIMENSION
Figure 4. Variations of lens weight with vertical lens dimensions (VLD) in different shape series. negative lenses (a), the slopes for the positive lenses (b) are more variable. CR39 plastic lenses: 0, oval; , HLD 58mm; - - -, HLD 48mm; -, HLD, 36mm; (a) - 9.00D; (b) + 9.00D.
(m) i:quals 0.03. Hence, an HLD 2mm increment increases the weight by 0.06 log units (about 15%) and a 2 mn HLD reduction reduces lens weight by about 13 % Taking a 5X/52 series pilot shaped lens at an HLD 52mm for example, a -9.OOD crown lens weighs 19.9g. A change of 2mm in the HLD would change the weight by about 3 g (15%). When edged in the same shape, a +9.0QD crown lens weighs 35.9g. The same change of HLD would lead to about 5g weight change by the same percentage. For two f9.00D crown lenses cut from a single blank size, the slope (m) is 0.015 units. These two lenses will differ by about 2.5g (7%) if they have a size difference of 2mm. Lens weight changes when the shape is modijied vertically A specific kind of eyewire shape modification has been considered in this paper: a shape is changed by varying the VLD under a constant HLD. Shallow spectacle frames are becoming popular. They may include different shape
(mm )
Compared with the square; + , pilot; 0,
forms, ranging from square to oval shapes and may have a shape difference’ as much as 18 mm. The last two members of the A- and B-series in Figure lb are of this nature. Shallow eyewires often give an impression of being light weight spectacles because of the large reduction in the vertical lens dimension. Equation (2) with m = 0.015 units can be used to describe the weight changes in these frames for negative lenses. It is now clear that making the lenses shallower alone is not very effective in reducing lens weights. By the small slope value (m = 0.015), a reduction of VLD by 2mm will reduce the lens weight by about 7 % . The weight difference between a square and oval lens of the same boxed size has been shown to be about 0.12 log units. To make two lenses the same weight, a square lens needs to be 8mm shallower (i.e. 0.12/m where m = 0.015). Estimation of lens weight for different sizes in a given shape Equation (1) is useful in estimating edged lens weight. With the assumed slopes and the weight of a lens of a known size,
44
Ophthal.
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Opt. 1995 15: No 1
weight of a sample of edged lenses by Obsteld’ . The errors of estimation were less than 0.7g.
100
Conclusions Edged lens weights in different eyewire sizes and in the variation of eyewire vertical lens dimensions can be described by a linear relation in a semilogarithmic space. For the variations of lens weight in different sizes of the same eyewire shape, a single slope can be assumed for negative lenses; and for positive lenses when the minimum edge thickness is kept at 1 mm. It is found that a change of the lens size by 2mm would change the lens weight by about 15%. A common slope for the variations of lens weight with different VLD but the same general shape appearance was also identified for negative lenses. In these cases, a lens shallower by 2mm would change the lens weight by about 7%. Although detailed studies were carried out only for crown glass and CR39 plastic lenses in four back vertex powers, the patterns of variations are so characteristic that observations from these studies should be applicable for lenses in any ophthalmic materials and for most powers. Practical measurements of edged lenses, including examples in high index materials, support the observations. I I
I I
I I
I
I
I
40
44
48
52
56
60
HORIZONTAL/VERTICAL
I
LENS DIMENSION
64
(mm )
Figure 5. Weight of some physically edged lenses. They have linear properties similar to those found for mathematically generated eyewire shapes. Variable VLD: ---0, -7.OOD, n = 1.700, series A; ---0, -8.OOD, n = 1.523, series B. Variable HLD: +, - 9.00 D, n = 1.800, shape Bl; ---+, - 7.00D, n = 1.498, shape B3; --+ , -8.OOD, n = 1.523, shape Al; ---+, +5.00D, n = 1.523, shape B4; -+, + 5.00 D, n = 1.523, shape B4, constant centre thickness.
weight estimations by Equation (1) were compared with the calculated lens weights. They are generally found to be accurate within k 1.Og and only in some cases of small positive lenses that the errors were over 2.Og and 10% of the actual lens weight. When edged lenses were tested, the accuracy of lens weight predictions was found to lie within the same limits (Table 3). With appropriate slope values (m), Equation (1) was again tested with the measured
Acknowledgments Most of the lenses used for the present study were donated by Rodenstock through its local agent, William S. T. Lee Company Limited. Their support is greatly appreciated. I also wish to thank Dr Brian Brown of Hong Kong Polytechnic for his comments and suggestions. References 1 Obstfeld, H. Weight of edged spectaclelenses. Ophthal. Physiol. Opt. 11, 248-251 (1991) 2 Tang, C. Y. Weight of edged spherical lenses in simulated eyewire shapes. Ophthal. Physiol. Opt. 13, 169-174 (1993) 3 Woodcock, F. R. High index lenses. The Optician 188, 21 September,23-28 (1984) 4 Rodenstock. Ophthalmic Lenses in Practice: Technical Information and Tables. Rodenstock, Munich (1987) 5 British Standard. BS 3521: Terms Relating to Ophthalmic Optics and Spectacle Frames. Part 1. Glossary of terms relating to Ophthalmic Lenses. p. 5. British Standards Institution, London, UK (1991)