Francis E. Washer and Walter R. Darling, "Factors Affecting the Accuracy of Distortion Measurements Made on the Nodal Slide Optical Bench*," J. Opt. Soc. Am. 49, 517-534 (1959)
In evaluating distortion with the visual nodal slide optical bench, the transverse displacements to right and left of center upon which the values of distortion are based are likely to be asymmetric with respect to the initial setting made at the axial focus. Several factors that contribute to this condition are (1) transverse displacement of the rear nodal point of the lens with respect to the center of rotation of the nodal slide, (2) misalignment of the collimator, and (3) prism effect on the lens. The manner in which these factors affect the observations are discussed. It is shown that the accuracy of the values of distortion are not adversely affected provided proper care is taken. A test for consistency of observations is developed and demonstrated. Results of measurements on several lenses are given that demonstrate the effect of the various factors.
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a Comparison of the magnitudes of ΔMβ as a function of β for the three factors. Column 1 shows the effect of lateral displacement of the nodal point from the center of rotation (d0 = 50 microns); column 2 shows the effect of misalignment of the collimator axis with respect to the bench ways of amount ∊c = 1 minute for f = 150 mm; column 3 shows the results for a prism effect of ∊0 = 1 minute and f = 150 mm; and column 4 gives the results for a combination of prism deviation ∊0 and lateral displacement d0 = −f∊0 for f = 150 mm. All values of ΔMβ are given in microns.
Angular separation from axis β
ΔMβ
1
2
3
4
degrees
0
0.0
0.0
0.0
0.0
5
0.2
0.2
0.4
0.6
10
0.8
0.7
1.8
2.5
15
1.7
1.6
4.2
5.8
20
3.0
2.8
7.9
10.6
25
4.7
4.6
13.1
17.2
30
6.7
6.8
20.3
26.2
35
9.0
9.7
30.3
38.3
40
11.7
13.4
44.4
54.7
45
14.6
18.2
64.4
77.3
Column 1, value of ΔMβ computed using equation ΔMβ = ΔNβ = d0 ×(cosβ − 1) for d0 = 50 microns; column 2, value of ΔMβ computed using equation ΔMβ = ΔTβ = f∊c(secβ − 1) for ∊c = 1 minute, f = 150 mm, f∊c = 44 microns; column 3, values of ΔMβ computed using equation ΔMβ = ΔPβ = f∊0[(∊/∊0 cosβ) − 1] for ∊0 = 1 minute, f = 150 mm, and f∊0 = 44 microns; column 4, values of ΔMβ computed with equation ΔMβ = ΔP′+ΔNβ′ = f∊0{[∊/∊0 cosβ) − 1] − f∊0(cosβ − 1)} for f∊0 = 44 microns.
Table II
Values of the various functions of β used in evaluating ΔNβ, ΔTβ, ΔPβ, and ΔPβ′ for a series of values of β.
β
1 − cosβ
secβ − 1
degrees
0
0.0000
0.0000
0.0000
0.0000
5
0.0038
0.0038
0.0102
0.0140
10
0.0152
0.0154
0.0418
0.0570
15
0.0341
0.0353
0.0966
0.1307
20
0.0603
0.0642
0.1800
0.2403
25
0.0937
0.1035
0.2977
0.3914
30
0.1340
0.1547
0.4618
0.5958
35
0.1808
0.2207
0.6890
0.8698
40
0.2340
0.3055
1.0089
1.2429
45
0.2929
0.4142
1.4631
1.7560
Table III
Reciprocals of the various functions of β used in evaluating d0, f∊c, and f∊0 for a series of values of β.
β
Reciprocals of the functions of β given in Table II.
degrees
0
…
…
…
…
5
263.2
263.2
98.0
71.4
10
66.2
64.9
23.9
17.5
15
29.3
28.3
10.4
7.65
20
16.6
15.6
5.56
4.16
25
10.7
9.66
3.36
2.56
30
7.46
6.46
2.16
1.68
35
5.53
4.53
1.45
1.15
40
4.27
3.27
0.991
0.804
45
3.41
2.41
0.684
0.570
Table IV
Values of the various functions of β relative to F(β) = 100 at β = 45° for a series of values of β. These values are useful in plotting the values of ΔMβ in its various forms.
β
Relative values of the functions of β given in Table II
1 − cosβ
secβ − 1
degrees
0
0.0
0.0
0.0
0.0
5
1.3
0.9
0.7
0.8
10
5.2
3.7
2.9
3.2
15
11.6
8.5
6.6
7.4
20
20.6
15.5
12.3
13.7
25
32.0
25.0
20.4
22.3
30
45.8
37.4
31.6
33.9
35
61.7
53.3
47.1
49.5
40
79.9
73.8
69.0
70.8
45
100.0
100.0
100.0
100.0
Table V
Evaluation of the distortion referred to the calibrated focal length, Dc, based on measurements Rβ and Lβ made in position I for wide angle lens No. 1 having a focal length of 150 mm. The zero point of coordinates for Rβ and Lβ is at 242 microns on the transverse scale of the viewing microscope. All values are given in microns.
β
Rβ
Lβ
Dβ
Δf tanβ
Dc
degrees
0
0
0
0
0
0
0
5
0
0
0
0
−7
−7
10
6
−4
5
5
−13
−8
15
26
−18
22
23
−20
3
20
54
−41
48
51
−28
23
25
91
−70
80
88
−35
53
30
127
−95
111
128
−44
84
35
154
−110
132
161
−53
108
40
140
−85
112
146
−64
82
45
20
66
−23
−32
−76
−108
Table VI
Evaluation of the distortion, Dc, for wide angle lens No. 1 in position II (rotated 180° about the optical axis from position I). The zero point of coordinates for Rβ and Lβ is at 232 microns on the transverse scale of the viewing microscope. All values are given in microns.
β
Rβ
Lβ
Dβ
Δf tanβ
Dc
degrees
0
0
0
0
0
0
0
5
3
−3
3
3
−10
−7
10
13
−12
12
12
−19
−7
15
34
−26
30
31
−29
2
20
66
−49
58
62
−40
22
25
106
−80
93
103
−51
52
30
150
−108
129
149
−63
86
35
179
−127
153
187
−77
110
40
164
−100
132
172
−92
80
45
57
58
0
0
−110
−110
Table VII
Evaluation of the angle of collimator misalignment ∊c from analysis of the 〈ΔMβ〉 values derived from the measurements given in Tables V and VI. A value of ∊c = 2.5 minutes accounts for most of the variation in ΔMβ; the small residual arises from small transverse displacements of the nodal point. All values are given in microns. 〈f∊c〉 = 112 microns.
β
〈ΔMβ〉
f∊c
ΔTβ
〈ΔMβ〉 − ΔTβ
degrees
0
0
0
0
0
0
0
5
0
0
0
0
0
0
10
1
0
0
0
2
−2
15
4
4
4
113
4
0
20
6
8
7
109
7
0
25
10
13
12
116
12
0
30
16
21
18
116
17
1
35
22
26
24
109
25
−1
40
28
32
30
98
34
−4
45
43
58
50
121
46
4
Table VIII
Comparison of the (Rβ − Lβ)/2 values for wide angle lens No. 1 in positions I and II. The difference can be explained by a 33 micron longitudinal displacement of the rear nodal point with respect to the center of rotation. All values are given in microns.
β
x0
x0 sinβ
I
II
0
0
0
0
5
0
3
−3
−34
−3
0
10
5
12
−7
−40
−
−1
15
22
30
−8
−31
−8
0
20
48
58
−10
−29
−11
1
25
80
93
−13
−31
−14
1
30
111
129
−18
−36
−16
−2
35
132
153
−21
−37
−19
−2
40
112
132
−20
−31
−21
1
45
−23
0
−23
−32
−23
0
Table IX
Values of (Rβ − Lβ)/2 for five separate sets of measurement made on wide angle lens No. 2 are shown in part (a) together with the average for runs 1, 2, 4, and 5. Values of the distortion, Dc are listed in part (b) for each set together with the value of PEs. All values are given in microns.
(a)
β
Values of (Rβ − Lβ)/2 for run number
Average for runs 1. 2, 4, 5
1
2
3
4
5
degrees
0
0
0
0
0
0
0
5
−4
−3
−4
−7
−1
−4
10
−7
−7
−7
−13
−2
−7
15
−11
−12
−12
−22
−6
−13
20
−18
−18
−19
−30
−10
−19
25
−27
−20
−26
−39
−16
−26
30
−30
−31
−32
−46
−19
−32
35
−23
−24
−39
−42
−10
−25
40
−14
−14
−33
−3
2
−16
45
−26
−21
−45
−43
−6
−24
(b)
Values of Dc for run number
β
1
2
3
4
5
Average
PEs
degrees
0
0
0
0
0
0
0
0
5
−1
0
1
−1
0
0
±1
10
0
−1
3
−2
−2
0
1
15
−2
−3
2
−5
−2
−2
2
20
−6
−6
0
−7
−6
−5
2
25
−12
−4
−2
−11
−10
−8
4
30
−13
−14
−4
−14
−14
−12
3
35
−1
−2
−8
−3
−3
−4
2
40
13
14
4
14
14
12
3
45
2
8
−6
7
4
3
4
Table X
Values of ΔMβ for the same measurements for which values of (Rβ − Lβ)/2 are given in Table IX. The average value 〈ΔMβ〉 is shown together with computed value ΔTβ = − 81 ×(secβ − 1) in part (a). In part (b) of the table, the departures of the measured value of ΔMβ from the computed value ΔTβ are shown. All values are given in microns.
(a)
β
Values of ΔMβ for run
〈ΔMβ〉
ΔTβ = −81(secβ − 1)
1
2
3
4
5
degrees
0
0
0
0
0
0
0
0
5
0
0
0
0
0
0
0
10
1
−1
−1
0
−1
0
−1
15
−2
−3
−6
1
−6
−3
−3
20
−5
−2
−7
−4
−7
−5
−5
25
−9
−2
−11
−7
−11
−8
−8
30
−14
−14
−16
−9
−14
−13
−13
35
−18
−19
−16
−16
−20
−18
−18
40
−24
−26
−25
−24
−25
−25
−25
45
−31
−33
−33
−31
−36
−33
−34
(b)
Departures of measured values of ΔMβ from ΔTβ for run
β
1
2
3
4
5
〈ΔMβ〉
degrees
0
0
0
0
0
0
0
5
0
0
0
0
0
0
10
2
0
0
1
0
1
15
1
0
−3
4
−3
0
20
0
3
−2
1
−2
0
25
−1
6
−3
1
−3
0
30
−1
−1
−3
4
−1
0
35
0
−1
2
2
−2
0
40
1
−1
0
1
0
0
45
3
1
1
3
−2
3
Table XI
Departures from the average value of (Rβ − Lβ)/2 for runs 1, 2, 4, and 5 for each value shown in Table IX(a) are shown in part (a) of this table as Δ[(Rβ − Lβ)/2]. In part (b) the computed values of the longitudinal displacement of the nodal point from the center of rotation, x0 = Δ[(Rβ − Lβ)/2] cscβ, are given. All values are in microns.
(a)
β
Departures from average value for sets 1, 2, 4, 5
1
2
3
4
5
degrees
0
0
0
0
0
0
5
0
1
0
−3
3
10
0
0
0
−6
5
15
2
1
1
−9
7
20
1
1
0
−11
9
25
−1
6
0
−13
10
30
2
1
0
−14
13
35
2
1
−14
−17
15
40
2
2
−17
−17
14
45
−2
3
−21
−19
18
(b)
β
1
2
3
4
5
0
5
0
11
0
−34
34
10
0
0
0
−34
29
15
8
4
4
−35
27
20
3
3
0
−32
26
25
−2
14
0
−31
24
30
4
2
0
−28
26
35
3
2
−24
−30
26
40
3
3
−26
−26
22
45
−3
4
−30
−27
25
x0(35°–45°)
1
3
−27
−28
24
Table XII
Evaluation of the distortion referred to the equivalent focal length, Dc, after making the indicated corrections in run number 3. The average value for the 5 runs is given together with the value of PEs. All values are in microns.
Dc
β
Value of the distortion referred to the calibrated focal length, Dc, for runs
Average
PEs
1
2
3
4
5
degrees
0
0
0
0
0
0
0
±0
5
−1
0
−1
−1
0
−1
0
10
0
−1
−1
−2
−2
−1
1
15
−2
−3
−3
−5
−2
−3
1
20
−6
−6
−7
−7
−6
−6
0
25
−12
−4
−10
−11
−10
−9
2
30
−13
−14
−12
−14
−14
−13
1
35
−1
−2
0
−3
−3
−2
1
40
13
14
12
14
14
13
1
45
2
8
8
7
4
6
2
Table XIII
Values of ΔMβ as a function of β for a series of azimuth angles θ for a lens having a focal length of 6 inches. These values of ΔMβ shown in part (a) are obtained with the lens facing the collimator. The values of ΔTβ and f∊c for each value of β are also shown in part (a). In part (b) are also values of ΔPβ′ shown together with the value of f∊0 for each value of β. All values are given in microns.
(a)
θ
Values of ΔMβ for β =
10°
15°
20°
25°
30°
degrees
0
−2
−3
−5
−9
−17
45
−10
−20
−34
−53
−84
90
−11
−23
−42
−69
−105
135
−15
−28
−48
−76
−111
180
−11
−22
−38
−59
−82
225
−8
−13
−24
−33
−44
270
−3
−4
−8
−8
−8
315
−2
−6
−5
−4
−5
ΔTβ
−8
−15
−26
−39
−57
f∊c
−520
−425
−405
−377
−368
(b)
Values of ΔPβ′ = ΔMβ − ΔTβ
0
6
12
21
30
40
45
−2
−5
−8
−14
−27
90
−3
−8
−16
−30
−48
135
−7
−13
−22
−37
−54
180
−3
−7
−12
−20
−25
225
0
2
2
6
13
270
5
11
18
31
49
315
6
9
21
35
52
f∊0
110
96
96
98
96
Table XIV
Comparison of the values of the quantity (Rβ − Lβ)/2 for β ranging from 0° to 30° and θ ranging from 0° to 315° for lie case of the lens facing the collimator. Part (a) shows the values of (Rβ − Lβ)/2 and their average; part (b) shows Δ[(Rβ − Lβ)/2], the departures from the average; and part (c) shows the computed values of x0 = Δ[(Rβ − Lβ)/2] cscβ, and the average value of x0. All values are in microns.
(a)
θ
Values of
for the following values of β
0°
5°
10°
15°
20°
25°
30°
degrees
0
0
6
18
41
77
137
241
45
0
5
17
38
74
134
237
90
0
6
20
42
78
140
242
135
0
2
12
32
65
123
222
180
0
6
16
39
75
134
238
225
0
8
20
43
80
141
246
270
0
5
17
38
74
133
233
315
0
2
10
28
60
114
214
Average
5
16
38
73
132
234
(b)
Values of
0
0
1
2
3
4
5
7
45
0
0
1
0
1
2
3
90
0
1
4
4
5
8
8
135
0
−3
−4
−6
−8
−9
−12
180
0
1
0
1
2
2
4
225
0
3
4
5
7
9
12
270
0
0
1
0
1
1
−1
315
0
−3
−6
−10
−13
−18
−20
(c)
Values of
〈x0〉
0
11
12
12
12
12
14
12
45
0
6
0
3
5
6
4
90
1
23
15
15
19
16
16
135
−34
−23
−23
−23
−21
−24
−23
180
11
0
4
6
5
8
6
225
34
23
19
20
21
24
21
270
0
6
0
3
2
−2
1
315
−34
−34
−39
−38
−43
−40
−40
Table XV
Values of the distortion referred to the equivalent focal length Dβ derived from Table XIV. Part (a) shows the values of the distortion for each value of azimuth angle θ; part (b) shows the value of Δf used to adjust the values of distortion to the calibrated focal length and values of Δf tanβ. All values are in microns.
(a)
θ
Values of Dβ for the following values of β
0°
5°
10°
15°
20°
25°
30°
degrees
0
0
6
18
42
82
151
278
45
0
5
17
39
79
148
274
90
0
6
20
43
83
154
279
135
0
2
12
33
69
136
256
180
0
6
16
40
80
148
275
225
0
8
20
44
85
156
284
270
0
5
17
39
79
147
269
315
0
2
10
29
64
126
247
Average
0
5
16
39
78
146
270
(b)
θ
Values of Δf tanβ
Δf
0
0
33
67
102
138
177
219
379
45
0
32
65
99
135
172
213
370
90
0
33
67
102
139
178
220
381
135
0
30
60
92
124
159
197
342
180
0
33
66
100
136
174
215
373
225
0
34
68
104
141
181
224
388
270
0
32
64
98
132
170
210
364
315
0
28
57
87
119
152
188
326
Average
0
32
64
98
133
171
211
366
Table XVI
Values of the distortion referred to the calibrated focal length Dc for the case of the lens facing the collimator. These values are derived from Table XV. Part (a) shows the values of the distortion Dc and the average; part (b) shows the departures from the average ΔDc. All values are in microns.
(a)
θ
Values of Dc for the following values of β
0°
5°
10°
15°
20°
25°
30°
Degrees
0
0
−27
−49
−60
−56
−26
59
45
0
−27
−48
−60
−56
−24
61
90
0
−27
−47
−59
−56
−24
59
135
0
−28
−48
−59
−55
−23
59
180
0
−27
−50
−60
−56
−26
60
225
0
−26
−48
−60
−56
−25
60
270
0
−27
−47
−59
−53
−23
59
315
0
−26
−47
−58
−55
−26
59
Average
0
−27
−48
−59
−55
−25
59
(b)
θ
Values of ΔDc
0
0
0
−1
−1
−1
−1
0
45
0
0
0
−1
−1
1
2
90
0
0
1
0
−1
1
0
135
0
−1
0
0
0
2
0
180
0
0
−2
−1
−1
−1
1
225
0
1
0
−1
−1
0
1
270
0
0
1
0
2
2
0
315
0
1
1
1
0
−1
0
Table XVII
Values of ΔMβ as a function of β for a series of azimuth θ for a lens having a focal length of 6 inches. These values of ΔMβ shown in part (a) are obtained with the lens facing the viewing microscope. The values of ΔTβ and f∊c for each value of β are also shown in part (a). In part (b) are also values of ΔPβ′ shown together with the values of f∊0 for each value of β. All values are given in microns.
(a)
θ
Values of ΔMβ
10
15
20
25
30
Degrees
0
−11
−24
−44
−69
−106
45
−4
−8
−15
−23
−37
90
−3
−4
−5
−5
−4
135
−2
−3
−5
−3
−9
180
−4
−7
−14
−21
−33
225
−2
−18
−31
−50
−76
270
−13
−26
−47
−75
−113
315
−12
−25
−45
−74
−111
ΔTβ
−7
−14
−26
−40
−61
f∊c
−454
−397
−405
−386
−394
(b)
θ
Values of ΔPβ′ = ΔMβ − ΔTβ
0
−4
−10
−18
−29
−45
45
3
6
11
17
24
90
4
10
21
35
57
135
5
11
21
37
52
180
3
7
12
19
28
225
−2
−4
−5
−10
−15
270
−6
−12
−21
−35
−52
315
−5
−11
−19
−34
−50
f∊0
102
100
99
103
101
Table XVIII
Comparison of the values of the quantity (Rβ − Lβ)/2 for β ranging from 0° to 30° and θ ranging from 0° to 315° for the case of the lens facing the viewing microscope. Part (a) shows the values of (Rβ − Lβ)/2 and their average; part (b) shows Δ(Rβ − Lβ)/2, the departures from the average; and part (c) shows the computed values of x0 = Δ(Rβ − Lβ)/2 cscβ, and the average value of x0. All values are in microns.
(a)
θ
Values of
for the following values of β
0°
5°
10°
15°
20°
25°
30°
Degrees
0
0
4
13
30
55
101
180
45
0
8
23
42
72
117
197
90
0
2
11
25
49
92
170
135
0
8
20
40
66
113
188
180
0
5
15
31
58
104
181
225
0
7
20
39
67
112
193
270
0
4
13
30
55
100
179
315
0
9
24
43
72
120
204
Average
0
6
17
35
62
107
186
(b)
Values of
0
0
−2
−4
−5
−7
−6
−6
45
0
2
6
7
10
10
11
90
0
−4
−6
−10
−13
−15
−16
135
0
2
3
5
4
6
2
180
0
−1
−2
−4
−4
−3
−5
225
0
1
3
4
5
5
7
270
0
−2
−4
−5
−7
−7
−7
315
0
3
7
8
10
13
18
(c)
Values of
x0
0
−23
−23
−19
−20
−14
−12
−16
45
23
35
27
20
24
22
25
90
−46
−35
−39
−38
−35
−32
−36
135
23
17
19
12
14
4
12
180
−11
−12
−15
−12
−7
−10
−11
225
11
17
15
15
12
14
14
270
−23
−23
−19
−20
−17
−14
−18
315
34
40
31
29
31
36
32
Table XIX
Values of the distortion referred to the equivalent focal length, Dβ, derived from Table XVII. Part (a) shows the values of Dβ for each value of azimuth angle θ; part (b) shows the value of Δf used to adjust the values of distortion to the calibrated focal length and values of Δf tanβ. All values are in microns.
(a)
θ
Values of Dβ for the following values of β
0°
5°
10°
15°
20°
25°
30°
Degrees
0
0
4
13
31
59
111
208
45
0
8
23
43
77
129
227
90
0
2
11
26
52
102
196
135
0
8
20
41
70
125
217
180
0
5
15
32
62
115
209
225
0
7
20
40
71
124
223
270
0
4
13
31
59
110
207
315
0
9
24
44
77
132
236
Average
0
6
17
36
66
118
215
(b)
θ
Values of Δf tanβ
Δf
0
0
25
50
76
103
131
163
282
45
0
28
56
86
116
149
185
320
90
0
23
46
70
96
123
152
263
135
0
27
54
82
111
142
176
305
180
0
25
50
76
104
133
164
285
225
0
27
55
83
113
145
179
311
270
0
25
50
76
103
131
163
282
315
0
29
58
89
120
154
191
331
Average
0
26
52
80
108
138
171
297
Table XX
Values of the distortion referred to the calibrated focal length, Dc, for the case of the lens facing the viewing microscope. These values are derived from Table XVIII. Part (a) shows the values of the distortion, Dc, and the average; part (b) shows the departures from the average, ΔDc. All values are in microns.
(a)
θ
Values of Dc for the following values of β
0°
5°
10°
15°
20°
25°
30°
Degrees
0
0
−21
−37
−45
−44
−20
45
45
0
−20
−33
−43
−39
−20
42
90
0
−21
−35
−44
−44
−21
44
135
0
−19
−34
−41
−41
−17
41
180
0
−20
−35
−44
−42
−18
45
225
0
−20
−35
−43
−42
−21
44
270
0
−21
−37
−45
−44
−21
44
315
0
−20
−34
−45
−43
−22
45
Average
0
−20
−35
−44
−42
−20
44
(b)
θ
Values of ΔDc
0
0
−1
−2
−1
−2
0
1
45
0
0
2
1
3
0
−2
90
0
−1
0
0
−2
−1
0
135
0
1
1
3
1
3
−3
180
0
0
0
0
0
2
1
225
0
0
0
1
0
−1
0
270
0
−1
−2
−1
−2
−1
0
315
0
0
1
−1
−1
−2
1
Tables (20)
Table I
a Comparison of the magnitudes of ΔMβ as a function of β for the three factors. Column 1 shows the effect of lateral displacement of the nodal point from the center of rotation (d0 = 50 microns); column 2 shows the effect of misalignment of the collimator axis with respect to the bench ways of amount ∊c = 1 minute for f = 150 mm; column 3 shows the results for a prism effect of ∊0 = 1 minute and f = 150 mm; and column 4 gives the results for a combination of prism deviation ∊0 and lateral displacement d0 = −f∊0 for f = 150 mm. All values of ΔMβ are given in microns.
Angular separation from axis β
ΔMβ
1
2
3
4
degrees
0
0.0
0.0
0.0
0.0
5
0.2
0.2
0.4
0.6
10
0.8
0.7
1.8
2.5
15
1.7
1.6
4.2
5.8
20
3.0
2.8
7.9
10.6
25
4.7
4.6
13.1
17.2
30
6.7
6.8
20.3
26.2
35
9.0
9.7
30.3
38.3
40
11.7
13.4
44.4
54.7
45
14.6
18.2
64.4
77.3
Column 1, value of ΔMβ computed using equation ΔMβ = ΔNβ = d0 ×(cosβ − 1) for d0 = 50 microns; column 2, value of ΔMβ computed using equation ΔMβ = ΔTβ = f∊c(secβ − 1) for ∊c = 1 minute, f = 150 mm, f∊c = 44 microns; column 3, values of ΔMβ computed using equation ΔMβ = ΔPβ = f∊0[(∊/∊0 cosβ) − 1] for ∊0 = 1 minute, f = 150 mm, and f∊0 = 44 microns; column 4, values of ΔMβ computed with equation ΔMβ = ΔP′+ΔNβ′ = f∊0{[∊/∊0 cosβ) − 1] − f∊0(cosβ − 1)} for f∊0 = 44 microns.
Table II
Values of the various functions of β used in evaluating ΔNβ, ΔTβ, ΔPβ, and ΔPβ′ for a series of values of β.
β
1 − cosβ
secβ − 1
degrees
0
0.0000
0.0000
0.0000
0.0000
5
0.0038
0.0038
0.0102
0.0140
10
0.0152
0.0154
0.0418
0.0570
15
0.0341
0.0353
0.0966
0.1307
20
0.0603
0.0642
0.1800
0.2403
25
0.0937
0.1035
0.2977
0.3914
30
0.1340
0.1547
0.4618
0.5958
35
0.1808
0.2207
0.6890
0.8698
40
0.2340
0.3055
1.0089
1.2429
45
0.2929
0.4142
1.4631
1.7560
Table III
Reciprocals of the various functions of β used in evaluating d0, f∊c, and f∊0 for a series of values of β.
β
Reciprocals of the functions of β given in Table II.
degrees
0
…
…
…
…
5
263.2
263.2
98.0
71.4
10
66.2
64.9
23.9
17.5
15
29.3
28.3
10.4
7.65
20
16.6
15.6
5.56
4.16
25
10.7
9.66
3.36
2.56
30
7.46
6.46
2.16
1.68
35
5.53
4.53
1.45
1.15
40
4.27
3.27
0.991
0.804
45
3.41
2.41
0.684
0.570
Table IV
Values of the various functions of β relative to F(β) = 100 at β = 45° for a series of values of β. These values are useful in plotting the values of ΔMβ in its various forms.
β
Relative values of the functions of β given in Table II
1 − cosβ
secβ − 1
degrees
0
0.0
0.0
0.0
0.0
5
1.3
0.9
0.7
0.8
10
5.2
3.7
2.9
3.2
15
11.6
8.5
6.6
7.4
20
20.6
15.5
12.3
13.7
25
32.0
25.0
20.4
22.3
30
45.8
37.4
31.6
33.9
35
61.7
53.3
47.1
49.5
40
79.9
73.8
69.0
70.8
45
100.0
100.0
100.0
100.0
Table V
Evaluation of the distortion referred to the calibrated focal length, Dc, based on measurements Rβ and Lβ made in position I for wide angle lens No. 1 having a focal length of 150 mm. The zero point of coordinates for Rβ and Lβ is at 242 microns on the transverse scale of the viewing microscope. All values are given in microns.
β
Rβ
Lβ
Dβ
Δf tanβ
Dc
degrees
0
0
0
0
0
0
0
5
0
0
0
0
−7
−7
10
6
−4
5
5
−13
−8
15
26
−18
22
23
−20
3
20
54
−41
48
51
−28
23
25
91
−70
80
88
−35
53
30
127
−95
111
128
−44
84
35
154
−110
132
161
−53
108
40
140
−85
112
146
−64
82
45
20
66
−23
−32
−76
−108
Table VI
Evaluation of the distortion, Dc, for wide angle lens No. 1 in position II (rotated 180° about the optical axis from position I). The zero point of coordinates for Rβ and Lβ is at 232 microns on the transverse scale of the viewing microscope. All values are given in microns.
β
Rβ
Lβ
Dβ
Δf tanβ
Dc
degrees
0
0
0
0
0
0
0
5
3
−3
3
3
−10
−7
10
13
−12
12
12
−19
−7
15
34
−26
30
31
−29
2
20
66
−49
58
62
−40
22
25
106
−80
93
103
−51
52
30
150
−108
129
149
−63
86
35
179
−127
153
187
−77
110
40
164
−100
132
172
−92
80
45
57
58
0
0
−110
−110
Table VII
Evaluation of the angle of collimator misalignment ∊c from analysis of the 〈ΔMβ〉 values derived from the measurements given in Tables V and VI. A value of ∊c = 2.5 minutes accounts for most of the variation in ΔMβ; the small residual arises from small transverse displacements of the nodal point. All values are given in microns. 〈f∊c〉 = 112 microns.
β
〈ΔMβ〉
f∊c
ΔTβ
〈ΔMβ〉 − ΔTβ
degrees
0
0
0
0
0
0
0
5
0
0
0
0
0
0
10
1
0
0
0
2
−2
15
4
4
4
113
4
0
20
6
8
7
109
7
0
25
10
13
12
116
12
0
30
16
21
18
116
17
1
35
22
26
24
109
25
−1
40
28
32
30
98
34
−4
45
43
58
50
121
46
4
Table VIII
Comparison of the (Rβ − Lβ)/2 values for wide angle lens No. 1 in positions I and II. The difference can be explained by a 33 micron longitudinal displacement of the rear nodal point with respect to the center of rotation. All values are given in microns.
β
x0
x0 sinβ
I
II
0
0
0
0
5
0
3
−3
−34
−3
0
10
5
12
−7
−40
−
−1
15
22
30
−8
−31
−8
0
20
48
58
−10
−29
−11
1
25
80
93
−13
−31
−14
1
30
111
129
−18
−36
−16
−2
35
132
153
−21
−37
−19
−2
40
112
132
−20
−31
−21
1
45
−23
0
−23
−32
−23
0
Table IX
Values of (Rβ − Lβ)/2 for five separate sets of measurement made on wide angle lens No. 2 are shown in part (a) together with the average for runs 1, 2, 4, and 5. Values of the distortion, Dc are listed in part (b) for each set together with the value of PEs. All values are given in microns.
(a)
β
Values of (Rβ − Lβ)/2 for run number
Average for runs 1. 2, 4, 5
1
2
3
4
5
degrees
0
0
0
0
0
0
0
5
−4
−3
−4
−7
−1
−4
10
−7
−7
−7
−13
−2
−7
15
−11
−12
−12
−22
−6
−13
20
−18
−18
−19
−30
−10
−19
25
−27
−20
−26
−39
−16
−26
30
−30
−31
−32
−46
−19
−32
35
−23
−24
−39
−42
−10
−25
40
−14
−14
−33
−3
2
−16
45
−26
−21
−45
−43
−6
−24
(b)
Values of Dc for run number
β
1
2
3
4
5
Average
PEs
degrees
0
0
0
0
0
0
0
0
5
−1
0
1
−1
0
0
±1
10
0
−1
3
−2
−2
0
1
15
−2
−3
2
−5
−2
−2
2
20
−6
−6
0
−7
−6
−5
2
25
−12
−4
−2
−11
−10
−8
4
30
−13
−14
−4
−14
−14
−12
3
35
−1
−2
−8
−3
−3
−4
2
40
13
14
4
14
14
12
3
45
2
8
−6
7
4
3
4
Table X
Values of ΔMβ for the same measurements for which values of (Rβ − Lβ)/2 are given in Table IX. The average value 〈ΔMβ〉 is shown together with computed value ΔTβ = − 81 ×(secβ − 1) in part (a). In part (b) of the table, the departures of the measured value of ΔMβ from the computed value ΔTβ are shown. All values are given in microns.
(a)
β
Values of ΔMβ for run
〈ΔMβ〉
ΔTβ = −81(secβ − 1)
1
2
3
4
5
degrees
0
0
0
0
0
0
0
0
5
0
0
0
0
0
0
0
10
1
−1
−1
0
−1
0
−1
15
−2
−3
−6
1
−6
−3
−3
20
−5
−2
−7
−4
−7
−5
−5
25
−9
−2
−11
−7
−11
−8
−8
30
−14
−14
−16
−9
−14
−13
−13
35
−18
−19
−16
−16
−20
−18
−18
40
−24
−26
−25
−24
−25
−25
−25
45
−31
−33
−33
−31
−36
−33
−34
(b)
Departures of measured values of ΔMβ from ΔTβ for run
β
1
2
3
4
5
〈ΔMβ〉
degrees
0
0
0
0
0
0
0
5
0
0
0
0
0
0
10
2
0
0
1
0
1
15
1
0
−3
4
−3
0
20
0
3
−2
1
−2
0
25
−1
6
−3
1
−3
0
30
−1
−1
−3
4
−1
0
35
0
−1
2
2
−2
0
40
1
−1
0
1
0
0
45
3
1
1
3
−2
3
Table XI
Departures from the average value of (Rβ − Lβ)/2 for runs 1, 2, 4, and 5 for each value shown in Table IX(a) are shown in part (a) of this table as Δ[(Rβ − Lβ)/2]. In part (b) the computed values of the longitudinal displacement of the nodal point from the center of rotation, x0 = Δ[(Rβ − Lβ)/2] cscβ, are given. All values are in microns.
(a)
β
Departures from average value for sets 1, 2, 4, 5
1
2
3
4
5
degrees
0
0
0
0
0
0
5
0
1
0
−3
3
10
0
0
0
−6
5
15
2
1
1
−9
7
20
1
1
0
−11
9
25
−1
6
0
−13
10
30
2
1
0
−14
13
35
2
1
−14
−17
15
40
2
2
−17
−17
14
45
−2
3
−21
−19
18
(b)
β
1
2
3
4
5
0
5
0
11
0
−34
34
10
0
0
0
−34
29
15
8
4
4
−35
27
20
3
3
0
−32
26
25
−2
14
0
−31
24
30
4
2
0
−28
26
35
3
2
−24
−30
26
40
3
3
−26
−26
22
45
−3
4
−30
−27
25
x0(35°–45°)
1
3
−27
−28
24
Table XII
Evaluation of the distortion referred to the equivalent focal length, Dc, after making the indicated corrections in run number 3. The average value for the 5 runs is given together with the value of PEs. All values are in microns.
Dc
β
Value of the distortion referred to the calibrated focal length, Dc, for runs
Average
PEs
1
2
3
4
5
degrees
0
0
0
0
0
0
0
±0
5
−1
0
−1
−1
0
−1
0
10
0
−1
−1
−2
−2
−1
1
15
−2
−3
−3
−5
−2
−3
1
20
−6
−6
−7
−7
−6
−6
0
25
−12
−4
−10
−11
−10
−9
2
30
−13
−14
−12
−14
−14
−13
1
35
−1
−2
0
−3
−3
−2
1
40
13
14
12
14
14
13
1
45
2
8
8
7
4
6
2
Table XIII
Values of ΔMβ as a function of β for a series of azimuth angles θ for a lens having a focal length of 6 inches. These values of ΔMβ shown in part (a) are obtained with the lens facing the collimator. The values of ΔTβ and f∊c for each value of β are also shown in part (a). In part (b) are also values of ΔPβ′ shown together with the value of f∊0 for each value of β. All values are given in microns.
(a)
θ
Values of ΔMβ for β =
10°
15°
20°
25°
30°
degrees
0
−2
−3
−5
−9
−17
45
−10
−20
−34
−53
−84
90
−11
−23
−42
−69
−105
135
−15
−28
−48
−76
−111
180
−11
−22
−38
−59
−82
225
−8
−13
−24
−33
−44
270
−3
−4
−8
−8
−8
315
−2
−6
−5
−4
−5
ΔTβ
−8
−15
−26
−39
−57
f∊c
−520
−425
−405
−377
−368
(b)
Values of ΔPβ′ = ΔMβ − ΔTβ
0
6
12
21
30
40
45
−2
−5
−8
−14
−27
90
−3
−8
−16
−30
−48
135
−7
−13
−22
−37
−54
180
−3
−7
−12
−20
−25
225
0
2
2
6
13
270
5
11
18
31
49
315
6
9
21
35
52
f∊0
110
96
96
98
96
Table XIV
Comparison of the values of the quantity (Rβ − Lβ)/2 for β ranging from 0° to 30° and θ ranging from 0° to 315° for lie case of the lens facing the collimator. Part (a) shows the values of (Rβ − Lβ)/2 and their average; part (b) shows Δ[(Rβ − Lβ)/2], the departures from the average; and part (c) shows the computed values of x0 = Δ[(Rβ − Lβ)/2] cscβ, and the average value of x0. All values are in microns.
(a)
θ
Values of
for the following values of β
0°
5°
10°
15°
20°
25°
30°
degrees
0
0
6
18
41
77
137
241
45
0
5
17
38
74
134
237
90
0
6
20
42
78
140
242
135
0
2
12
32
65
123
222
180
0
6
16
39
75
134
238
225
0
8
20
43
80
141
246
270
0
5
17
38
74
133
233
315
0
2
10
28
60
114
214
Average
5
16
38
73
132
234
(b)
Values of
0
0
1
2
3
4
5
7
45
0
0
1
0
1
2
3
90
0
1
4
4
5
8
8
135
0
−3
−4
−6
−8
−9
−12
180
0
1
0
1
2
2
4
225
0
3
4
5
7
9
12
270
0
0
1
0
1
1
−1
315
0
−3
−6
−10
−13
−18
−20
(c)
Values of
〈x0〉
0
11
12
12
12
12
14
12
45
0
6
0
3
5
6
4
90
1
23
15
15
19
16
16
135
−34
−23
−23
−23
−21
−24
−23
180
11
0
4
6
5
8
6
225
34
23
19
20
21
24
21
270
0
6
0
3
2
−2
1
315
−34
−34
−39
−38
−43
−40
−40
Table XV
Values of the distortion referred to the equivalent focal length Dβ derived from Table XIV. Part (a) shows the values of the distortion for each value of azimuth angle θ; part (b) shows the value of Δf used to adjust the values of distortion to the calibrated focal length and values of Δf tanβ. All values are in microns.
(a)
θ
Values of Dβ for the following values of β
0°
5°
10°
15°
20°
25°
30°
degrees
0
0
6
18
42
82
151
278
45
0
5
17
39
79
148
274
90
0
6
20
43
83
154
279
135
0
2
12
33
69
136
256
180
0
6
16
40
80
148
275
225
0
8
20
44
85
156
284
270
0
5
17
39
79
147
269
315
0
2
10
29
64
126
247
Average
0
5
16
39
78
146
270
(b)
θ
Values of Δf tanβ
Δf
0
0
33
67
102
138
177
219
379
45
0
32
65
99
135
172
213
370
90
0
33
67
102
139
178
220
381
135
0
30
60
92
124
159
197
342
180
0
33
66
100
136
174
215
373
225
0
34
68
104
141
181
224
388
270
0
32
64
98
132
170
210
364
315
0
28
57
87
119
152
188
326
Average
0
32
64
98
133
171
211
366
Table XVI
Values of the distortion referred to the calibrated focal length Dc for the case of the lens facing the collimator. These values are derived from Table XV. Part (a) shows the values of the distortion Dc and the average; part (b) shows the departures from the average ΔDc. All values are in microns.
(a)
θ
Values of Dc for the following values of β
0°
5°
10°
15°
20°
25°
30°
Degrees
0
0
−27
−49
−60
−56
−26
59
45
0
−27
−48
−60
−56
−24
61
90
0
−27
−47
−59
−56
−24
59
135
0
−28
−48
−59
−55
−23
59
180
0
−27
−50
−60
−56
−26
60
225
0
−26
−48
−60
−56
−25
60
270
0
−27
−47
−59
−53
−23
59
315
0
−26
−47
−58
−55
−26
59
Average
0
−27
−48
−59
−55
−25
59
(b)
θ
Values of ΔDc
0
0
0
−1
−1
−1
−1
0
45
0
0
0
−1
−1
1
2
90
0
0
1
0
−1
1
0
135
0
−1
0
0
0
2
0
180
0
0
−2
−1
−1
−1
1
225
0
1
0
−1
−1
0
1
270
0
0
1
0
2
2
0
315
0
1
1
1
0
−1
0
Table XVII
Values of ΔMβ as a function of β for a series of azimuth θ for a lens having a focal length of 6 inches. These values of ΔMβ shown in part (a) are obtained with the lens facing the viewing microscope. The values of ΔTβ and f∊c for each value of β are also shown in part (a). In part (b) are also values of ΔPβ′ shown together with the values of f∊0 for each value of β. All values are given in microns.
(a)
θ
Values of ΔMβ
10
15
20
25
30
Degrees
0
−11
−24
−44
−69
−106
45
−4
−8
−15
−23
−37
90
−3
−4
−5
−5
−4
135
−2
−3
−5
−3
−9
180
−4
−7
−14
−21
−33
225
−2
−18
−31
−50
−76
270
−13
−26
−47
−75
−113
315
−12
−25
−45
−74
−111
ΔTβ
−7
−14
−26
−40
−61
f∊c
−454
−397
−405
−386
−394
(b)
θ
Values of ΔPβ′ = ΔMβ − ΔTβ
0
−4
−10
−18
−29
−45
45
3
6
11
17
24
90
4
10
21
35
57
135
5
11
21
37
52
180
3
7
12
19
28
225
−2
−4
−5
−10
−15
270
−6
−12
−21
−35
−52
315
−5
−11
−19
−34
−50
f∊0
102
100
99
103
101
Table XVIII
Comparison of the values of the quantity (Rβ − Lβ)/2 for β ranging from 0° to 30° and θ ranging from 0° to 315° for the case of the lens facing the viewing microscope. Part (a) shows the values of (Rβ − Lβ)/2 and their average; part (b) shows Δ(Rβ − Lβ)/2, the departures from the average; and part (c) shows the computed values of x0 = Δ(Rβ − Lβ)/2 cscβ, and the average value of x0. All values are in microns.
(a)
θ
Values of
for the following values of β
0°
5°
10°
15°
20°
25°
30°
Degrees
0
0
4
13
30
55
101
180
45
0
8
23
42
72
117
197
90
0
2
11
25
49
92
170
135
0
8
20
40
66
113
188
180
0
5
15
31
58
104
181
225
0
7
20
39
67
112
193
270
0
4
13
30
55
100
179
315
0
9
24
43
72
120
204
Average
0
6
17
35
62
107
186
(b)
Values of
0
0
−2
−4
−5
−7
−6
−6
45
0
2
6
7
10
10
11
90
0
−4
−6
−10
−13
−15
−16
135
0
2
3
5
4
6
2
180
0
−1
−2
−4
−4
−3
−5
225
0
1
3
4
5
5
7
270
0
−2
−4
−5
−7
−7
−7
315
0
3
7
8
10
13
18
(c)
Values of
x0
0
−23
−23
−19
−20
−14
−12
−16
45
23
35
27
20
24
22
25
90
−46
−35
−39
−38
−35
−32
−36
135
23
17
19
12
14
4
12
180
−11
−12
−15
−12
−7
−10
−11
225
11
17
15
15
12
14
14
270
−23
−23
−19
−20
−17
−14
−18
315
34
40
31
29
31
36
32
Table XIX
Values of the distortion referred to the equivalent focal length, Dβ, derived from Table XVII. Part (a) shows the values of Dβ for each value of azimuth angle θ; part (b) shows the value of Δf used to adjust the values of distortion to the calibrated focal length and values of Δf tanβ. All values are in microns.
(a)
θ
Values of Dβ for the following values of β
0°
5°
10°
15°
20°
25°
30°
Degrees
0
0
4
13
31
59
111
208
45
0
8
23
43
77
129
227
90
0
2
11
26
52
102
196
135
0
8
20
41
70
125
217
180
0
5
15
32
62
115
209
225
0
7
20
40
71
124
223
270
0
4
13
31
59
110
207
315
0
9
24
44
77
132
236
Average
0
6
17
36
66
118
215
(b)
θ
Values of Δf tanβ
Δf
0
0
25
50
76
103
131
163
282
45
0
28
56
86
116
149
185
320
90
0
23
46
70
96
123
152
263
135
0
27
54
82
111
142
176
305
180
0
25
50
76
104
133
164
285
225
0
27
55
83
113
145
179
311
270
0
25
50
76
103
131
163
282
315
0
29
58
89
120
154
191
331
Average
0
26
52
80
108
138
171
297
Table XX
Values of the distortion referred to the calibrated focal length, Dc, for the case of the lens facing the viewing microscope. These values are derived from Table XVIII. Part (a) shows the values of the distortion, Dc, and the average; part (b) shows the departures from the average, ΔDc. All values are in microns.