Syntheses of Spectral-Distribution Curves

W. L. Brewer; F. R. Holly

doi:10.1364/JOSA.38.000858

Journal of the Optical Society of America
Vol. 38,
Issue 10,
pp. 858-874
(1948)
•https://doi.org/10.1364/JOSA.38.000858

Syntheses of Spectral-Distribution Curves

W. L. Brewer and F. R. Holly

Not Accessible

Your library or personal account may give you access

Get PDF
Email
Share
Get Citation
Copy Citation Text
W. L. Brewer and F. R. Holly, "Syntheses of Spectral-Distribution Curves," J. Opt. Soc. Am. 38, 858-874 (1948)

Export Citation
- BibTex
- Endnote (RIS)
- HTML
- Plain Text
Citation alert
Save article

Abstract

A method is presented for obtaining spectral distributions having colorimetric characteristics which have been pre-specified in terms of a set of tristimulus values or trichromatic coefficients. The distributions may be chosen to correspond to energy radiated from light sources, or to the transmittances or reflectances of materials acting on such energy. Tables and figures are included to facilitate a rapid solution to any given problem. Several examples are given which illustrate the method and indicate the kinds of applications possible.

Full Article | PDF Article

More Like This

The Normal Probability Curve as an Approximation to the Distribution Curve of the High Intensity Searchlight

Frank Benford and John E. Bock
J. Opt. Soc. Am. 38(6) 527-531 (1948)

Counting Metameric Object Colors

W. S. Stiles and G. W. Wyszecki
J. Opt. Soc. Am. 52(3) 313-328 (1962)

The Effect of Chromatic Adaptation on Achromaticity

Harry Helson and Walter C. Michels
J. Opt. Soc. Am. 38(12) 1025-1032 (1948)

Previous Article Next Article

Cited By

You do not have subscription access to this journal. Cited by links are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access Optica Member Subscription

Figures (13)

You do not have subscription access to this journal. Figure files are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access Optica Member Subscription

Tables (4)

You do not have subscription access to this journal. Article tables are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access Optica Member Subscription

Equations (39)

You do not have subscription access to this journal. Equations are available to subscribers only. You may subscribe either as an Optica member, or as an authorized user of your institution.

Contact your librarian or system administrator
or
Login to access Optica Member Subscription

(i−i₀)	1	i	30−i	(i−i₀)²	$1 + sin \frac{i - i_{0}}{10} π$	$1 + sin \frac{i - i_{0}}{30} π$	${[sin \frac{i - i_{0}}{30} π]}^{2}$	$1 + {[sin \frac{i - i_{0}}{15} π]}^{3}$	$1 + {[sin \frac{i - i_{0}}{30} π]}^{3}$	$e \frac{- {(i - i_{0})}^{2}}{2 a^{2}}$ a=1	$e \frac{- {(i - i_{0})}^{2}}{2 a^{2}}$ a=2	$e \frac{- {(i - i_{0})}^{2}}{2 a^{2}}$ a=3	$e \frac{- {(i - i_{0})}^{2}}{2 a^{2}}$ a=5	(i−i₀)
−56	—	—	—	—	—	—	—	—	1.06729	—	—	—	—	−56
−55	—	—	—	—	—	1.50000	—	—	1.12500	—	—	—	—	−55
−54	—	—	—	—	—	1.58779	—	—	1.20307	—	—	—	—	−54
−53	—	—	—	—	—	1.66913	—	—	1.29959	—	—	—	—	−53
−52	—	—	—	—	—	1.74314	—	—	1.41041	—	—	—	—	−52
−51	—	—	—	—	—	1.80902	—	—	1.52951	—	—	—	—	−51
−50	—	—	—	—	—	1.86603	—	—	1.64952	—	—	—	—	−50
−49	—	—	—	—	—	1.91355	—	—	1.76241	—	—	—	—	−49
−48	—	—	—	—	—	1.95106	—	—	1.86024	—	—	—	—	−48
−47	—	—	—	—	—	1.97815	—	—	1.93586	—	—	—	—	−47
−46	—	—	—	—	—	1.99452	—	—	1.98366	—	—	—	—	−46
−45	—	—	—	—	—	2.00000	—	—	2.00000	—	—	—	—	−45
−44	—	—	—	—	—	1.99452	—	—	1.98366	—	—	—	—	−44
−43	—	—	—	—	—	1.97815	—	—	1.93586	—	—	—	—	−43
−42	—	—	—	—	—	1.95106	—	—	1.86024	—	—	—	—	−42
−41	—	—	—	—	—	1.91355	—	—	1.76241	—	—	—	—	−41
−40	—	—	—	—	—	1.86603	—	—	1.64952	—	—	—	—	−40
−39	—	—	—	—	—	1.80902	—	—	1.52951	—	—	—	—	−39
−38	—	—	—	—	—	1.74314	—	—	1.41041	—	—	—	—	−38
−37	—	—	—	—	—	1.66913	—	—	1.29959	—	—	—	—	−37
−36	—	—	—	—	—	1.58779	—	—	1.20307	—	—	—	—	−36
−35	—	—	—	—	—	1.50000	—	—	1.12500	—	—	—	—	−35
−34	—	—	—	—	—	1.40674	—	—	1.06729	—	—	—	—	−34
−33	—	—	—	—	—	1.30902	—	—	1.02951	—	—	—	—	−33
−32	—	—	—	—	—	1.00000	—	—	1.00889	—	—	—	—	−32
−31	—	—	—	—	—	1.20791	—	—	1.00114	—	—	—	—	−31
−30	—	—	—	900	—	1.10453	—	—	1.00000	0.00000	0.00000	0.00000	0.00000	−30
−29	—	—	—	841	—	0.89547	—	—	0.99886	0.00000	0.00000	0.00000	0.00000	−29
−28	—	—	—	784	—	0.79209	0.04323	1.06729	0.99101	0.00000	0.00000	0.00000	0.00000	−28
−27	—	—	—	729	—	0.69098	0.09549	1.20307	0.97049	0.00000	0.00000	0.00000	0.00000	−27
−26	—	—	—	676	—	0.59326	0.16543	1.41041	0.93271	0.00000	0.00000	0.00000	0.00000	−26
−25	—	—	—	625	—	0.50000	0.25000	1.64952	0.87500	0.00000	0.00000	0.00000	0.00000	−25
−24	—	—	—	576	—	0.41221	0.34549	1.86024	0.79693	0.00000	0.00000	0.00000	0.00001	−24
−23	—	—	—	529	—	0.33087	0.44774	1.98366	0.70041	0.00000	0.00000	0.00000	0.00003	−23
−22	—	—	—	484	—	0.25686	0.55226	1.98366	0.58959	0.00000	0.00000	0.00000	0.00006	−22
−21	—	—	—	441	—	0.19098	0.65451	1.86024	0.47049	0.00000	0.00000	0.00000	0.00015	−21
−20	—	—	—	400	—	0.13397	0.75000	1.64952	0.35048	0.00000	0.00000	0.00000	0.00034	−20
−19	—	—	—	361	—	0.08645	0.83457	1.41041	0.23759	0.00000	0.00000	0.00000	0.00073	−19
−18	—	—	—	324	1.58779	0.04894	0.90451	1.20307	0.13976	0.00000	0.00000	0.00000	0.00153	−18
−17	—	—	—	289	1.80902	0.02185	0.95677	1.06729	0.06414	0.00000	0.00000	0.00000	0.00309	−17
−16	—	—	—	256	1.95106	0.00548	0.98907	1.00899	0.01634	0.00000	0.00000	0.00000	0.00598	−16
−15	—	—	—	225	2.00000	0.00000	1.00000	1.00000	0.00000	0.00000	0.00000	0.00000	0.01111	−15
−14	—	—	—	196	1.95106	0.00548	0.98907	0.99101	0.01634	0.00000	0.00000	0.00002	0.01984	−14
−13	—	—	—	169	1.80902	0.02185	0.95677	0.93271	0.06414	0.00000	0.00000	0.00008	0.03405	−13
−12	—	—	—	144	1.58779	0.04894	0.90451	0.79693	0.13976	0.00000	0.00000	0.00034	0.05613	−12
−11	—	—	—	121	1.30902	0.08645	0.83457	0.58959	0.23759	0.00000	0.00000	0.00120	0.08892	−11
−10	—	—	—	100	1.00000	0.13397	0.75000	0.35048	0.35048	0.00000	0.00000	0.00387	0.13534	−10
−9	—	—	—	81	0.69098	0.19098	0.65451	0.13976	0.47049	0.00000	0.00004	0.01111	0.19790	−9
−8	—	—	—	64	0.41221	0.25686	0.55226	0.01634	0.58959	0.00000	0.00034	0.02856	0.27804	−8
−7	—	—	—	49	0.19098	0.33087	0.44774	0.01634	0.70041	0.00000	0.00219	0.06573	0.37531	−7
−6	—	—	—	36	0.04894	0.41221	0.34549	0.13976	0.79693	0.00000	0.01111	0.13534	0.48657	−6
−5	—	—	—	25	0.00000	0.50000	0.25000	0.35048	0.87500	0.00000	0.04394	0.24935	0.60653	−5
−4	—	—	—	16	0.04894	0.59326	0.16543	0.58959	0.93271	0.00034	0.13534	0.41111	0.72615	−4
−3	—	—	—	9	0.19098	0.69098	0.09549	0.79693	0.97049	0.01111	0.32465	0.60653	0.83527	−3
−2	—	—	—	4	0.41221	0.79209	0.04323	0.93271	0.99101	0.13534	0.60653	0.80076	0.92312	−2
−1	—	—	—	1	0.69098	0.89547	0.01093	0.99101	0.99886	0.60653	0.88250	0.94592	0.98020	−1
0	1	0	30	0	1.00000	1.00000	0.00000	1.00000	1.00000	1.00000	1.00000	1.00000	1.00000	0
1	1	1	29	1	1.30902	1.10453	0.01093	1.00899	1.00114	0.60653	0.88250	0.94592	0.98020	1
2	1	2	28	4	1.58779	1.20791	0.04323	0.06729	1.00899	0.13534	0.60653	0.80076	0.92312	2
3	1	3	27	9	1.80902	1.30902	0.09549	1.20307	1.02951	0.01111	0.32465	0.60653	0.83527	3
4	1	4	26	16	1.95106	1.40674	0.16543	1.41041	1.06729	0.00034	0.13534	0.41111	0.72615	4
5	1	5	25	25	2.00000	1.50000	0.25000	1.64952	1.12500	0.00000	0.04394	0.24935	0.60653	5
6	1	6	24	36	1.95106	1.58779	0.34549	1.86024	1.20307	0.00000	0.01111	0.13534	0.48657	6
7	1	7	23	49	1.80902	1.66913	0.44774	1.98366	1.29959	0.00000	0.00219	0.06573	0.37531	7
8	1	8	22	64	1.58779	1.74314	0.55226	1.98366	1.41041	0.00000	0.00034	0.02856	0.27804	8
9	1	9	21	81	1.30902	1.80902	0.65451	1.86024	1.52951	0.00000	0.00004	0.01111	0.19790	9
10	1	10	20	100	1.00000	1.86603	0.75000	1.64952	1.64952	0.00000	0.00000	0.00387	0.13534	10
11	1	11	19	121	0.69098	1.91355	0.83457	1.41041	1.76241	0.00000	0.00000	0.00120	0.08892	11
12	1	12	18	144	0.41221	1.95106	0.90451	1.20307	1.86024	0.00000	0.00000	0.00034	0.05613	12
13	1	13	17	169	0.19098	1.97815	0.95677	1.06729	1.93586	0.00000	0.00000	0.00008	0.03405	13
14	1	14	16	196	0.04894	1.99452	0.98907	1.00899	1.98366	0.00000	0.00000	0.00002	0.01984	14
15	1	15	15	225	0.00000	2.00000	1.00000	1.00000	2.00000	0.00000	0.00000	0.00000	0.01111	15
16	1	16	14	256	0.04894	1.99452	0.98907	0.99101	1.98366	0.00000	0.00000	0.00000	0.00598	16
17	1	17	13	289	0.19098	1.97815	0.95677	0.93271	1.93586	0.00000	0.00000	0.00000	0.00309	17
18	1	18	12	324	0.41221	1.95106	0.90451	0.79693	1.86024	0.00000	0.00000	0.00000	0.00153	18
19	1	19	11	361	0.69098	1.91355	0.83457	0.58959	1.76241	0.00000	0.00000	0.00000	0.00073	19
20	1	20	10	400	1.00000	1.86603	0.75000	0.35048	1.64952	0.00000	0.00000	0.00000	0.00034	20
21	1	21	9	441	1.30902	1.80902	0.65451	0.13976	1.52951	0.00000	0.00000	0.00000	0.00015	21
22	1	22	8	484	1.58779	1.74314	0.55226	0.01634	1.41041	0.00000	0.00000	0.00000	0.00006	22
23	1	23	7	529	1.80902	1.66913	0.44774	0.01634	1.29959	0.00000	0.00000	0.00000	0.00003	23
24	1	24	6	576	1.95106	1.58779	0.34549	0.13976	1.20307	0.00000	0.00000	0.00000	0.00001	24
25	1	25	5	625	2.00000	1.50000	0.25000	0.35048	1.12500	0.00000	0.00000	0.00000	0.00000	25
26	1	26	4	676	1.95106	1.40674	0.16543	0.58959	1.06729	0.00000	0.00000	0.00000	0.00000	26
27	1	27	3	729	1.80902	1.30902	0.09549	0.79693	1.02951	0.00000	0.00000	0.00000	0.00000	27
28	1	28	2	784	1.58779	1.20791	0.04323	0.93271	1.00899	0.00000	0.00000	0.00000	0.00000	28
29	1	29	1	841	1.30902	1.10453	0.01093	0.99101	1.00114	0.00000	0.00000	0.00000	0.00000	29
30	1	30	0	900	1.00000	1.00000	0.00000	1.00000	1.00000	0.00000	0.00000	0.00000	0.00000	30

	x	y	m
f=1	.33335	.33381	319.99
f=i	.44460	.41519	4119.67
f=30−i	.24971	.27263	5479.89

	f=(i−i₀)²
i₀	x	y	m
0	.52005	.42553	68755
5	.56547	.41687	35558
1	.57328	.39406	26759
8	.56936	.37419	23319
9	.55689	.34700	20520
10	.53326	.31206	18360
11	.49688	.27028	16841
12	.44846	.22477	15961
13	.39191	.18045	15721
14	.33360	.14267	16122
15	.28017	.11518	17162
17	.20359	.09314	21162
20	.16140	.11153	31962
25	.17276	.16720	62762
30	.19952	.20777	109562

	$f = 1 + sin \frac{i - i_{0}}{10} π$
i₀	x	y	m
0	.31498	.17360	366.344
2	.25602	.19845	323.130
4	.21244	.29048	278.715
5	.20622	.35818	261.528
6	.21522	.43058	250.064
7	.23988	.49538	245.444
8	.27653	.54029	248.120
10	.35794	.54832	273.626
11	.39007	.51701	293.959
12	.41221	.47186	316.840
13	.42408	.42011	340.028
14	.42663	.36724	361.255
15	.42120	.31697	378.442
16	.40911	.27175	389.906
18	.36932	.20307	391.850

	$f = 1 + sin \frac{i - i_{0}}{30} π$
i₀	x	y	m
0	.34108	.36684	554.262
5	.38269	.39388	496.388
10	.43081	.41903	391.248
15	.48898	.43841	267.013
20	.54896	.42959	156.972
23	.55628	.37985	110.438
24	.54437	.34999	99.315
25	.52168	.31296	90.609
26	.48717	.27027	84.417
27	.44180	.22516	80.805
28	.38905	.18222	79.814
30	.28332	.12022	85.708
32	.20732	.10097	101.841
35	.16277	.12614	143.582
40	.18003	.19976	248.722
45	.22192	.25893	372.957
50	.26327	.30269	482.998
55	.30228	.33725	549.361

	$f = {[sin \frac{i - i_{0}}{30} π]}^{2}$
i₀	x	y	m
0	.36021	.46729	190.520
2	.41762	.48731	182.654
4	.46798	.48478	170.870
6	.50281	.45528	157.205
8	.51158	.39694	144.022
10	.48499	.31522	133.600
12	.42217	.22712	127.742
14	.33705	.15792	127.460
16	.25440	.12801	132.804
18	.19594	.14117	112.848
20	.17052	.18658	155.857
22	.17567	.24885	169.581
24	.20410	.31522	181.647
26	.24830	.37717	189.969
28	.30206	.42924	193.107

	$f = 1 + {[sin \frac{i - i_{0}}{15} π]}^{3}$
i₀	x	y	m
0	.21693	.28087	338.626
2	.24817	.33856	345.737
4	.28168	.38402	361.068
6	.31433	.41219	376.361
8	.35555	.43377	373.601
10	.40923	.45314	349.371
12	.45809	.45561	320.826
14	.47225	.42208	304.808
16	.44999	.36095	298.480
18	.41677	.29720	287.566
20	.38227	.24353	270.057
22	.33346	.20490	261.948
24	.27032	.18875	276.547
26	.22084	.19803	305.722
28	.20358	.23003	329.058

	$f = 1 + {[sin \frac{i - i_{0}}{30} π]}^{3}$
i₀	x	y	m
0	.34347	.39248	484.103
4	.38399	.38455	474.660
8	.40104	.36186	425.998
12	.40566	.37304	337.283
16	.42850	.43783	244.627
20	.47876	.47322	186.135
22	.49172	.43442	172.103
24	.47984	.35944	164.800
26	.43985	.27117	160.789
28	.37680	.19516	157.823
30	.30192	.15158	155.867
32	.23235	.15108	157.164
34	.18792	.18813	165.310
36	.17926	.23926	183.706
40	.22777	.29329	254.948
44	.26816	.27904	351.447
48	.27492	.26757	430.032
52	.27509	.29681	467.867
56	.29761	.35483	479.181

	$e \frac{- {(i - i_{0})}^{2}}{2 a^{2}}, a = 1$
i₀	x	y	m
0	.17220	.00501	3.605
4	.16210	.01300	48.321
6	.14303	.03439	47.316
8	.09854	.13435	28.169
9	.06115	.28292	20.411
10	.03235	.51115	17.217
11	.04506	.70795	18.177
12	.09829	.78960	21.840
13	.16775	.78428	26.636
14	.23909	.74048	31.434
15	.30938	.68213	35.779
16	.37872	.61776	39.517
18	.51068	.48836	43.896
20	.61723	.38230	41.289
22	.68190	.31791	30.380
26	.72697	.27302	6.390
30	.73421	.26579	0.442

	$e \frac{- {(i - i_{0})}^{2}}{2 a^{2}}, a = 2$
i₀	x	y	m
0	.16843	.00771	17.316
4	.15626	.02113	79.741
6	.14007	.05099	82.351
8	.10949	.15734	61.117
9	.09172	.27663	50.770
10	.08503	.44042	44.756
11	.10325	.59693	43.975
12	.14684	.69030	47.679
13	.20449	.71487	54.267
14	.26676	.69503	62.057
15	.32943	.65281	69.696
16	.39066	.60110	76.191
18	.50328	.49477	82.710
20	.59364	.50569	76.833
22	.65604	.34365	58.524
26	.71488	.28505	16.560
30	.73109	.26891	1.773

	$e \frac{- {(i - i_{0})}^{2}}{2 a^{2}}, a = 3$
i₀	x	y	m
0	.16285	.01406	37.885
4	.14947	.04035	98.939
6	.13698	.08527	105.725
8	.12345	.19347	93.790
9	.12223	.28315	86.315
10	.13147	.39084	80.993
11	.15569	.49664	79.204
12	.19477	.57660	81.235
13	.24378	.61833	86.441
14	.29677	.62511	93.576
15	.34963	.60784	101.165
16	.40025	.57710	107.772
18	.49119	.50245	113.469
20	.56558	.43249	105.036
22	.62235	.38298	83.524
26	.69059	.30924	31.423
30	.71982	.28014	5.639

	$e \frac{- {(i - i_{0})}^{2}}{2 a^{2}}, a = 5$
i₀	x	y	m
0	.15189	.05231	75.815
4	.14765	.11674	127.356
6	.15178	.17644	142.228
8	.16739	.25876	148.958
9	.18182	.30626	150.470
10	.20156	.35484	151.571
11	.22666	.40108	152.764
12	.25646	.44145	154.338
13	.28970	.47315	156.309
14	.32481	.49472	158.428
15	.36023	.50618	160.227
16	.39472	.50869	161.114
18	.45794	.49420	157.822
20	.51175	.46561	145.263
22	.55650	.43302	123.519
26	.62352	.37419	67.810
30	.66771	.33170	24.979

	x	y	m
f=1	.31004	.31646	3 1600
f=i	.42154	.40953	38.4113
f=30−i	.23408	.25306	56.3876

	f=(i−i₀)²
i₀	x	y	m
0	.50276	.42948	616.83
5	.55401	.42541	311.72
7	.56192	.40052	233.91
8	.55627	.37765	204.49
9	.54040	.34602	181.38
10	.51185	.30560	164.60
11	.46998	.25847	154.14
12	.41728	.20920	149.99
13	.35946	.16374	152.17
14	.30353	.12731	160.67
15	.25526	.10254	175.48
17	.19068	.08511	224.08
20	.15775	.10363	344.36
25	.16810	.15401	671.24
30	.19058	.19122	1156.12

	$f = 1 + sin \frac{i - i_{0}}{10} π$
i₀	x	y	m
0	.28353	.15574	3.73776
2	.23074	.18007	3.37098
4	.19599	.26748	2.92360
5	.19313	.33284	2.72370
6	.20397	.40532	2.56650
7	.22933	.47439	2.46740
8	.26611	.52672	2.43609
10	.34841	.54910	2.58217
11	.38020	.51955	2.74526
12	.40068	.47237	2.94895
13	.40957	.41636	3.17329
14	.40821	.35862	3.39633
15	.39858	.30405	3.59623
16	.38256	.25570	3.75343
18	.33759	.18457	3.88384

	$f = 1 + sin \frac{i - i_{0}}{30} π$
i₀	x	y	m
0	.31846	.35020	5.44303
5	.35917	.38130	4.77163
10	.40824	.41249	3.66838
15	.47039	.44028	2.42890
20	.53697	.43819	1.38531
23	.54258	.38295	.97414
24	.52723	.34840	.83321
25	.49972	.30586	.81724
26	.46001	.25798	.77693
27	.41064	.20933	.76273
28	.35665	.16533	.77479
30	.25773	.10705	.87689
32	.19338	.09200	1.07878
35	.15860	.11663	1.54830
40	.17418	.18361	2.65155
45	.20994	.23917	3.89102
50	.24633	.28229	4.93462
55	.28187	.31803	5.50269

	$f = {[sin \frac{i - i_{0}}{30} π]}^{2}$
i₀	x	y	m
0	.34180	.45659	1.83452
2	.40056	.48588	1.70705
4	.45390	.49066	1.55762
6	.49055	.46274	1.41205
8	.49654	.39831	1.29551
10	.46195	.30604	1.22817
12	.39129	.21052	1.22165
14	.30596	.14132	1.27710
16	.23149	.11482	1.38491
18	.18325	.12875	1.52646
20	.16406	.17082	1.67726
22	.16991	.22824	1.81124
24	.19515	.29144	1.90523
26	.23484	.35388	1.94298
23	.28497	.41057	1.91797

	$f = 1 + {[sin \frac{i - i_{0}}{15} π]}^{3}$
i₀	x	y	m
0	.20529	.25920	3.56352
2	.23281	.31683	3.53984
4	.26420	.36442	3.60852
6	.29588	.39450	3.71500
8	.33581	.41941	3.63878
10	.38943	.44652	3.31079
12	.44138	.45816	3.93903
14	.45597	.42390	2.76350
16	.42802	.35292	2.77190
18	.38926	.28230	2.76066
20	.35211	.22690	2.64949
22	.30411	.18808	2.60888
24	.24663	.17185	2.82161
26	.20556	.18069	3.20820
28	.19316	.21079	3.49972

	$f = 1 + {[sin \frac{i - i_{0}}{30} π]}^{3}$
i₀	x	y	m
0	.32255	.37736	4.72782
4	.36126	.37202	4.56031
8	.37545	.34875	4.07922
12	.37885	.36059	3.22039
16	.40572	.43316	2.28732
20	.46471	.48069	1.68453
22	.47792	.43899	1.55399
24	.46037	.35414	1.51149
26	.41223	.25657	1.52105
28	.34472	.17737	1.55205
30	.27287	.13563	1.59211
32	.21244	.13693	1.65183
34	.17728	.17247	1.75962
36	.17272	.22050	1.94860
40	.21687	.27318	2.63420
44	.25126	.25986	3.58536
48	.25529	.24814	4.39152
52	.25530	.27651	4.76593
56	.27765	.33544	4.79887

	$e \frac{- {(i - i_{0})}^{2}}{2 a^{2}}, a = 1$
i₀	x	y	m
0	.17192	.00491	0.02832
4	.16197	.01313	0.54481
6	.14303	.03437	0.54844
8	.09868	.13299	0.32525
9	.06247	.27466	0.22933
10	.03351	.49436	0.18102
11	.04337	.69712	0.17674
12	.09743	.78795	0.20387
13	.16921	.78260	0.24956
14	.24090	.73913	0.30165
15	.31004	.68161	0.34965
16	.37775	.61869	0.38539
18	.50801	.49096	0.40280
20	.61593	.38359	0.35220
22	.68175	.31808	0.25206
26	.72694	.27306	0.05208
30	.73540	.20460	0.00332

	$e \frac{- {(i - i_{0})}^{2}}{2 a^{2}}, a = 2$
i₀	x	y	m
0	.16784	.00810	0.16996
4	.15577	.02158	0.89183
6	.14002	.05021	0.94322
8	.11067	.14729	0.69409
9	.09319	.25583	0.56172
10	.08522	.41160	0.47355
11	.10173	.57261	0.44263
12	.14575	.67736	0.46365
13	.20494	.70960	0.52134
14	.26746	.69330	0.59558
15	.32860	.65352	0.66828
16	.38746	.60424	0.72486
18	.49710	.50085	0.75622
20	.58900	.41029	0.66837
22	.65420	.34550	0.49242
26	.71480	.28517	0.13654
30	.73126	.26874	0.01408

	$e \frac{- {(i - i_{0})}^{2}}{2 a^{2}}, a = 3$
i₀	x	y	m
0	.16201	.01481	0.39863
4	.14897	.03978	1.10263
6	.13702	.08033	1.19080
8	.12378	.17654	1.04340
9	.12188	.25829	0.94214
10	.12961	.36106	0.85999
11	.15213	.46843	0.81468
12	.19025	.55594	0.81145
13	.23909	.60683	0.84424
14	.29193	.62106	0.89938
15	.34413	.60891	0.96009
16	.39367	.58154	1.01035
18	.48298	.51013	1.03381
20	.55810	.43981	0.92683
22	.61729	.38195	0.71664
26	.68942	.31043	0.26112
30	.71961	.28036	0.04595

	$e \frac{- {(i - i_{0})}^{2}}{2 a^{2}}, a = 5$
i₀	x	y	m
0	.15102	.05060	0.82581
4	.14645	.10788	1.39868
6	.14944	.16118	1.55264
8	.16248	.23660	1.59991
9	.17503	.28162	1.59621
10	.19265	.32918	1.58386
11	.21563	.37631	1.56927
12	.24361	.41950	1.55673
13	.27554	.45556	1.54776
14	.30993	.48231	1.54102
15	.34519	.49899	1.53287
16	.37993	.50619	1.51826
18	.44435	.49841	1.44929
20	.49973	.47303	1.30584
22	.54619	.44121	1.09040
26	.61678	.38053	0.58134
30	.66401	.33534	0.20963

(i−i₀)	1	i	30−i	(i−i₀)²	$1 + sin \frac{i - i_{0}}{10} π$	$1 + sin \frac{i - i_{0}}{30} π$	${[sin \frac{i - i_{0}}{30} π]}^{2}$	$1 + {[sin \frac{i - i_{0}}{15} π]}^{3}$	$1 + {[sin \frac{i - i_{0}}{30} π]}^{3}$	$e \frac{- {(i - i_{0})}^{2}}{2 a^{2}}$ a=1	$e \frac{- {(i - i_{0})}^{2}}{2 a^{2}}$ a=2	$e \frac{- {(i - i_{0})}^{2}}{2 a^{2}}$ a=3	$e \frac{- {(i - i_{0})}^{2}}{2 a^{2}}$ a=5	(i−i₀)
−56	—	—	—	—	—	—	—	—	1.06729	—	—	—	—	−56
−55	—	—	—	—	—	1.50000	—	—	1.12500	—	—	—	—	−55
−54	—	—	—	—	—	1.58779	—	—	1.20307	—	—	—	—	−54
−53	—	—	—	—	—	1.66913	—	—	1.29959	—	—	—	—	−53
−52	—	—	—	—	—	1.74314	—	—	1.41041	—	—	—	—	−52
−51	—	—	—	—	—	1.80902	—	—	1.52951	—	—	—	—	−51
−50	—	—	—	—	—	1.86603	—	—	1.64952	—	—	—	—	−50
−49	—	—	—	—	—	1.91355	—	—	1.76241	—	—	—	—	−49
−48	—	—	—	—	—	1.95106	—	—	1.86024	—	—	—	—	−48
−47	—	—	—	—	—	1.97815	—	—	1.93586	—	—	—	—	−47
−46	—	—	—	—	—	1.99452	—	—	1.98366	—	—	—	—	−46
−45	—	—	—	—	—	2.00000	—	—	2.00000	—	—	—	—	−45
−44	—	—	—	—	—	1.99452	—	—	1.98366	—	—	—	—	−44
−43	—	—	—	—	—	1.97815	—	—	1.93586	—	—	—	—	−43
−42	—	—	—	—	—	1.95106	—	—	1.86024	—	—	—	—	−42
−41	—	—	—	—	—	1.91355	—	—	1.76241	—	—	—	—	−41
−40	—	—	—	—	—	1.86603	—	—	1.64952	—	—	—	—	−40
−39	—	—	—	—	—	1.80902	—	—	1.52951	—	—	—	—	−39
−38	—	—	—	—	—	1.74314	—	—	1.41041	—	—	—	—	−38
−37	—	—	—	—	—	1.66913	—	—	1.29959	—	—	—	—	−37
−36	—	—	—	—	—	1.58779	—	—	1.20307	—	—	—	—	−36
−35	—	—	—	—	—	1.50000	—	—	1.12500	—	—	—	—	−35
−34	—	—	—	—	—	1.40674	—	—	1.06729	—	—	—	—	−34
−33	—	—	—	—	—	1.30902	—	—	1.02951	—	—	—	—	−33
−32	—	—	—	—	—	1.00000	—	—	1.00889	—	—	—	—	−32
−31	—	—	—	—	—	1.20791	—	—	1.00114	—	—	—	—	−31
−30	—	—	—	900	—	1.10453	—	—	1.00000	0.00000	0.00000	0.00000	0.00000	−30
−29	—	—	—	841	—	0.89547	—	—	0.99886	0.00000	0.00000	0.00000	0.00000	−29
−28	—	—	—	784	—	0.79209	0.04323	1.06729	0.99101	0.00000	0.00000	0.00000	0.00000	−28
−27	—	—	—	729	—	0.69098	0.09549	1.20307	0.97049	0.00000	0.00000	0.00000	0.00000	−27
−26	—	—	—	676	—	0.59326	0.16543	1.41041	0.93271	0.00000	0.00000	0.00000	0.00000	−26
−25	—	—	—	625	—	0.50000	0.25000	1.64952	0.87500	0.00000	0.00000	0.00000	0.00000	−25
−24	—	—	—	576	—	0.41221	0.34549	1.86024	0.79693	0.00000	0.00000	0.00000	0.00001	−24
−23	—	—	—	529	—	0.33087	0.44774	1.98366	0.70041	0.00000	0.00000	0.00000	0.00003	−23
−22	—	—	—	484	—	0.25686	0.55226	1.98366	0.58959	0.00000	0.00000	0.00000	0.00006	−22
−21	—	—	—	441	—	0.19098	0.65451	1.86024	0.47049	0.00000	0.00000	0.00000	0.00015	−21
−20	—	—	—	400	—	0.13397	0.75000	1.64952	0.35048	0.00000	0.00000	0.00000	0.00034	−20
−19	—	—	—	361	—	0.08645	0.83457	1.41041	0.23759	0.00000	0.00000	0.00000	0.00073	−19
−18	—	—	—	324	1.58779	0.04894	0.90451	1.20307	0.13976	0.00000	0.00000	0.00000	0.00153	−18
−17	—	—	—	289	1.80902	0.02185	0.95677	1.06729	0.06414	0.00000	0.00000	0.00000	0.00309	−17
−16	—	—	—	256	1.95106	0.00548	0.98907	1.00899	0.01634	0.00000	0.00000	0.00000	0.00598	−16
−15	—	—	—	225	2.00000	0.00000	1.00000	1.00000	0.00000	0.00000	0.00000	0.00000	0.01111	−15
−14	—	—	—	196	1.95106	0.00548	0.98907	0.99101	0.01634	0.00000	0.00000	0.00002	0.01984	−14
−13	—	—	—	169	1.80902	0.02185	0.95677	0.93271	0.06414	0.00000	0.00000	0.00008	0.03405	−13
−12	—	—	—	144	1.58779	0.04894	0.90451	0.79693	0.13976	0.00000	0.00000	0.00034	0.05613	−12
−11	—	—	—	121	1.30902	0.08645	0.83457	0.58959	0.23759	0.00000	0.00000	0.00120	0.08892	−11
−10	—	—	—	100	1.00000	0.13397	0.75000	0.35048	0.35048	0.00000	0.00000	0.00387	0.13534	−10
−9	—	—	—	81	0.69098	0.19098	0.65451	0.13976	0.47049	0.00000	0.00004	0.01111	0.19790	−9
−8	—	—	—	64	0.41221	0.25686	0.55226	0.01634	0.58959	0.00000	0.00034	0.02856	0.27804	−8
−7	—	—	—	49	0.19098	0.33087	0.44774	0.01634	0.70041	0.00000	0.00219	0.06573	0.37531	−7
−6	—	—	—	36	0.04894	0.41221	0.34549	0.13976	0.79693	0.00000	0.01111	0.13534	0.48657	−6
−5	—	—	—	25	0.00000	0.50000	0.25000	0.35048	0.87500	0.00000	0.04394	0.24935	0.60653	−5
−4	—	—	—	16	0.04894	0.59326	0.16543	0.58959	0.93271	0.00034	0.13534	0.41111	0.72615	−4
−3	—	—	—	9	0.19098	0.69098	0.09549	0.79693	0.97049	0.01111	0.32465	0.60653	0.83527	−3
−2	—	—	—	4	0.41221	0.79209	0.04323	0.93271	0.99101	0.13534	0.60653	0.80076	0.92312	−2
−1	—	—	—	1	0.69098	0.89547	0.01093	0.99101	0.99886	0.60653	0.88250	0.94592	0.98020	−1
0	1	0	30	0	1.00000	1.00000	0.00000	1.00000	1.00000	1.00000	1.00000	1.00000	1.00000	0
1	1	1	29	1	1.30902	1.10453	0.01093	1.00899	1.00114	0.60653	0.88250	0.94592	0.98020	1
2	1	2	28	4	1.58779	1.20791	0.04323	0.06729	1.00899	0.13534	0.60653	0.80076	0.92312	2
3	1	3	27	9	1.80902	1.30902	0.09549	1.20307	1.02951	0.01111	0.32465	0.60653	0.83527	3
4	1	4	26	16	1.95106	1.40674	0.16543	1.41041	1.06729	0.00034	0.13534	0.41111	0.72615	4
5	1	5	25	25	2.00000	1.50000	0.25000	1.64952	1.12500	0.00000	0.04394	0.24935	0.60653	5
6	1	6	24	36	1.95106	1.58779	0.34549	1.86024	1.20307	0.00000	0.01111	0.13534	0.48657	6
7	1	7	23	49	1.80902	1.66913	0.44774	1.98366	1.29959	0.00000	0.00219	0.06573	0.37531	7
8	1	8	22	64	1.58779	1.74314	0.55226	1.98366	1.41041	0.00000	0.00034	0.02856	0.27804	8
9	1	9	21	81	1.30902	1.80902	0.65451	1.86024	1.52951	0.00000	0.00004	0.01111	0.19790	9
10	1	10	20	100	1.00000	1.86603	0.75000	1.64952	1.64952	0.00000	0.00000	0.00387	0.13534	10
11	1	11	19	121	0.69098	1.91355	0.83457	1.41041	1.76241	0.00000	0.00000	0.00120	0.08892	11
12	1	12	18	144	0.41221	1.95106	0.90451	1.20307	1.86024	0.00000	0.00000	0.00034	0.05613	12
13	1	13	17	169	0.19098	1.97815	0.95677	1.06729	1.93586	0.00000	0.00000	0.00008	0.03405	13
14	1	14	16	196	0.04894	1.99452	0.98907	1.00899	1.98366	0.00000	0.00000	0.00002	0.01984	14
15	1	15	15	225	0.00000	2.00000	1.00000	1.00000	2.00000	0.00000	0.00000	0.00000	0.01111	15
16	1	16	14	256	0.04894	1.99452	0.98907	0.99101	1.98366	0.00000	0.00000	0.00000	0.00598	16
17	1	17	13	289	0.19098	1.97815	0.95677	0.93271	1.93586	0.00000	0.00000	0.00000	0.00309	17
18	1	18	12	324	0.41221	1.95106	0.90451	0.79693	1.86024	0.00000	0.00000	0.00000	0.00153	18
19	1	19	11	361	0.69098	1.91355	0.83457	0.58959	1.76241	0.00000	0.00000	0.00000	0.00073	19
20	1	20	10	400	1.00000	1.86603	0.75000	0.35048	1.64952	0.00000	0.00000	0.00000	0.00034	20
21	1	21	9	441	1.30902	1.80902	0.65451	0.13976	1.52951	0.00000	0.00000	0.00000	0.00015	21
22	1	22	8	484	1.58779	1.74314	0.55226	0.01634	1.41041	0.00000	0.00000	0.00000	0.00006	22
23	1	23	7	529	1.80902	1.66913	0.44774	0.01634	1.29959	0.00000	0.00000	0.00000	0.00003	23
24	1	24	6	576	1.95106	1.58779	0.34549	0.13976	1.20307	0.00000	0.00000	0.00000	0.00001	24
25	1	25	5	625	2.00000	1.50000	0.25000	0.35048	1.12500	0.00000	0.00000	0.00000	0.00000	25
26	1	26	4	676	1.95106	1.40674	0.16543	0.58959	1.06729	0.00000	0.00000	0.00000	0.00000	26
27	1	27	3	729	1.80902	1.30902	0.09549	0.79693	1.02951	0.00000	0.00000	0.00000	0.00000	27
28	1	28	2	784	1.58779	1.20791	0.04323	0.93271	1.00899	0.00000	0.00000	0.00000	0.00000	28
29	1	29	1	841	1.30902	1.10453	0.01093	0.99101	1.00114	0.00000	0.00000	0.00000	0.00000	29
30	1	30	0	900	1.00000	1.00000	0.00000	1.00000	1.00000	0.00000	0.00000	0.00000	0.00000	30

	x	y	m
f=1	.33335	.33381	319.99
f=i	.44460	.41519	4119.67
f=30−i	.24971	.27263	5479.89

	f=(i−i₀)²
i₀	x	y	m
0	.52005	.42553	68755
5	.56547	.41687	35558
1	.57328	.39406	26759
8	.56936	.37419	23319
9	.55689	.34700	20520
10	.53326	.31206	18360
11	.49688	.27028	16841
12	.44846	.22477	15961
13	.39191	.18045	15721
14	.33360	.14267	16122
15	.28017	.11518	17162
17	.20359	.09314	21162
20	.16140	.11153	31962
25	.17276	.16720	62762
30	.19952	.20777	109562

	$f = 1 + sin \frac{i - i_{0}}{10} π$
i₀	x	y	m
0	.31498	.17360	366.344
2	.25602	.19845	323.130
4	.21244	.29048	278.715
5	.20622	.35818	261.528
6	.21522	.43058	250.064
7	.23988	.49538	245.444
8	.27653	.54029	248.120
10	.35794	.54832	273.626
11	.39007	.51701	293.959
12	.41221	.47186	316.840
13	.42408	.42011	340.028
14	.42663	.36724	361.255
15	.42120	.31697	378.442
16	.40911	.27175	389.906
18	.36932	.20307	391.850

	$f = 1 + sin \frac{i - i_{0}}{30} π$
i₀	x	y	m
0	.34108	.36684	554.262
5	.38269	.39388	496.388
10	.43081	.41903	391.248
15	.48898	.43841	267.013
20	.54896	.42959	156.972
23	.55628	.37985	110.438
24	.54437	.34999	99.315
25	.52168	.31296	90.609
26	.48717	.27027	84.417
27	.44180	.22516	80.805
28	.38905	.18222	79.814
30	.28332	.12022	85.708
32	.20732	.10097	101.841
35	.16277	.12614	143.582
40	.18003	.19976	248.722
45	.22192	.25893	372.957
50	.26327	.30269	482.998
55	.30228	.33725	549.361

	$f = {[sin \frac{i - i_{0}}{30} π]}^{2}$
i₀	x	y	m
0	.36021	.46729	190.520
2	.41762	.48731	182.654
4	.46798	.48478	170.870
6	.50281	.45528	157.205
8	.51158	.39694	144.022
10	.48499	.31522	133.600
12	.42217	.22712	127.742
14	.33705	.15792	127.460
16	.25440	.12801	132.804
18	.19594	.14117	112.848
20	.17052	.18658	155.857
22	.17567	.24885	169.581
24	.20410	.31522	181.647
26	.24830	.37717	189.969
28	.30206	.42924	193.107

	$f = 1 + {[sin \frac{i - i_{0}}{15} π]}^{3}$
i₀	x	y	m
0	.21693	.28087	338.626
2	.24817	.33856	345.737
4	.28168	.38402	361.068
6	.31433	.41219	376.361
8	.35555	.43377	373.601
10	.40923	.45314	349.371
12	.45809	.45561	320.826
14	.47225	.42208	304.808
16	.44999	.36095	298.480
18	.41677	.29720	287.566
20	.38227	.24353	270.057
22	.33346	.20490	261.948
24	.27032	.18875	276.547
26	.22084	.19803	305.722
28	.20358	.23003	329.058

	$f = 1 + {[sin \frac{i - i_{0}}{30} π]}^{3}$
i₀	x	y	m
0	.34347	.39248	484.103
4	.38399	.38455	474.660
8	.40104	.36186	425.998
12	.40566	.37304	337.283
16	.42850	.43783	244.627
20	.47876	.47322	186.135
22	.49172	.43442	172.103
24	.47984	.35944	164.800
26	.43985	.27117	160.789
28	.37680	.19516	157.823
30	.30192	.15158	155.867
32	.23235	.15108	157.164
34	.18792	.18813	165.310
36	.17926	.23926	183.706
40	.22777	.29329	254.948
44	.26816	.27904	351.447
48	.27492	.26757	430.032
52	.27509	.29681	467.867
56	.29761	.35483	479.181

	$e \frac{- {(i - i_{0})}^{2}}{2 a^{2}}, a = 1$
i₀	x	y	m
0	.17220	.00501	3.605
4	.16210	.01300	48.321
6	.14303	.03439	47.316
8	.09854	.13435	28.169
9	.06115	.28292	20.411
10	.03235	.51115	17.217
11	.04506	.70795	18.177
12	.09829	.78960	21.840
13	.16775	.78428	26.636
14	.23909	.74048	31.434
15	.30938	.68213	35.779
16	.37872	.61776	39.517
18	.51068	.48836	43.896
20	.61723	.38230	41.289
22	.68190	.31791	30.380
26	.72697	.27302	6.390
30	.73421	.26579	0.442

	$e \frac{- {(i - i_{0})}^{2}}{2 a^{2}}, a = 2$
i₀	x	y	m
0	.16843	.00771	17.316
4	.15626	.02113	79.741
6	.14007	.05099	82.351
8	.10949	.15734	61.117
9	.09172	.27663	50.770
10	.08503	.44042	44.756
11	.10325	.59693	43.975
12	.14684	.69030	47.679
13	.20449	.71487	54.267
14	.26676	.69503	62.057
15	.32943	.65281	69.696
16	.39066	.60110	76.191
18	.50328	.49477	82.710
20	.59364	.50569	76.833
22	.65604	.34365	58.524
26	.71488	.28505	16.560
30	.73109	.26891	1.773

	$e \frac{- {(i - i_{0})}^{2}}{2 a^{2}}, a = 3$
i₀	x	y	m
0	.16285	.01406	37.885
4	.14947	.04035	98.939
6	.13698	.08527	105.725
8	.12345	.19347	93.790
9	.12223	.28315	86.315
10	.13147	.39084	80.993
11	.15569	.49664	79.204
12	.19477	.57660	81.235
13	.24378	.61833	86.441
14	.29677	.62511	93.576
15	.34963	.60784	101.165
16	.40025	.57710	107.772
18	.49119	.50245	113.469
20	.56558	.43249	105.036
22	.62235	.38298	83.524
26	.69059	.30924	31.423
30	.71982	.28014	5.639

	$e \frac{- {(i - i_{0})}^{2}}{2 a^{2}}, a = 5$
i₀	x	y	m
0	.15189	.05231	75.815
4	.14765	.11674	127.356
6	.15178	.17644	142.228
8	.16739	.25876	148.958
9	.18182	.30626	150.470
10	.20156	.35484	151.571
11	.22666	.40108	152.764
12	.25646	.44145	154.338
13	.28970	.47315	156.309
14	.32481	.49472	158.428
15	.36023	.50618	160.227
16	.39472	.50869	161.114
18	.45794	.49420	157.822
20	.51175	.46561	145.263
22	.55650	.43302	123.519
26	.62352	.37419	67.810
30	.66771	.33170	24.979

Abstract

Cited By

Figures (13)

Tables (4)

Equations (39)

Journal of the Optical Society of America