Due to the locally D-optimal criterion, we need to pre-specify the possible extremum point, b(0), first. When the full factorial designs are used in the first-order designs, we have the following theorem to show that we only need to consider the extremum point such that b(0) = (b(0)1 , b(0)2 , · · · , b(0)k ) with b(0)i ≥ 0, i = 1, 2, · · · , k. The reason is that our objective function is invariant to orthogonal transformation.
Theorem 1. Suppose the first-order design contains a 2k full factorial design and n0 center points. Let b(0) be the given extremum point and s∗1, · · · , s∗2k be the star points of the locally D-optimal design, ξ∗D, for b(0). There exist a vector fb(0) = (f
b(0)1 , f
b(0)2 , · · · , f b(0)k ) with bf(0)i ≥ 0, i = 1, 2, · · · , k, and the corresponding fsm, m = 1, 2, · · · , 2k, such that
³bf(0)
´T
= G¡ b(0)¢T
and (fsm)T = G (s∗m)T, where G = G(ϑ ), ϑ = (ϑ12, · · · , ϑ1k, ϑ23, · · · , ϑ2k, · · · , ϑ(k−1)k), is the rotation matrix and can be represented as the prod-uct of the Givens rotation matrices, Q
1≤i<j≤kGij(ϑij), with ϑij ∈ (π2)n, n ∈ Z. Then e
s1, · · · , fs2k are the additional star points of the locally D-optimal design for fb(0). The proof of this theorem is shown in Appendix.
In this subsection, we search the locally D-optimal star points when the full factorial design is used in the first-order design for k = 2, 3, · · · , 8. For k = 2, we find the close form of the objective function, and then numerical results are also shown. For k ≥ 3, only numerical results are shown.
4.1.1 22 Full Factorial Design
For the case of k = 2, the four star points are represented as (3) with radius r and angle θ12, and the given extremum point b(0) is represented as (4) with radius R and angle φ1. When k = 2, given n0 center points and the 22 full factorial design as the first-order design , the objective function d(r, θ12) is
(n0+ 8)2
4 (P1+ P2+ P3+ P4+ P5+ P6
P7 ),
where P1 = r4¡
(n0+4)r4− 16r2+4(n0+ 4)¢ , P2 = R2¡
(n0+4)r10+2(n0− 4)r8+2(5n0+ 8)r6+ 4(5n0+ 8)r4+16(n0− 4)r2+ 32(n0+4)¢ , P3 = R4¡
(3n0+16)r8+4(3n0+8)r6+24n0r4+16(3n0+8)r2+16(3n0+16)¢ , P4 = r4(cos 4θ12)¡
2(n0+ 8)R2(r2+ 2) + (n0+ 4)r4− 16r2+ 4(n0+ 4)¢ , P5 = −4(n0+ 8) (cos 4φ1) R4(r2+ 2)2,
P6 = (n0+ 8) (cos 4(θ12− φ1)) R4r4(r2+ 2)2, P7 = (1+cos 4θ12)r4(r2+2)2¡
(n0+ 4)r4−16r2+4(n0+ 4)¢ .
To identify the optimal additional points, we take the derivative of this objective function d(r, θ12) with respect to r and θ12 separately. It is easy to show that d(r, θ12) is a decreasing function for r. Thus the minimum of d(r, θ12) is obtained at r = √
2.
That is the four locally D-optimal star points are all on the boundary of the 2-ball with radius √
2. Given r = √
2, we find the minimum of d(√
2, θ12) for θ12 by taking the derivative of d(√
2, θ12) with respect to θ12. Finally the minimum of d(√
2, θ12) is attended at
θ12= −1 2arcsin
µ(n0+ 8)R√ 2sin 4φ1
H
¶
, (6)
where
H = 4(n0+ 8)2R4cos24φ1 − 8R2(n0+ 8)((3n0 + 8)R2+ 2n0) cos 4φ1
+(37n20+ 208n0+ 320)R4+ 16n0(3n0+ 8)R2+ 16n20 .
To demonstrate the difference between our optimal additional points and original star points, according to (6), we show some plots of φ1 and θ12 by fixing R = √
2,
0.75√ increases. That is when we put more center points, then optimal star points are close to the original star points, however, when extremum point b(0) is near the boundary of the experimental region, we need to move our star points from axes to obtain the better parameter estimations.
To show the performance of BAR sampler, we choose b(0) with R =√
2 and φ1 = 10π as an example. The numbers of the center points, n0, are 1, 2, 3, and 4. For k = 2, based on (5), we need to have one rotation angle, θ12, and one radius r in our objective function.
• n0 = 1
When n0 = 1, θ12 is -0.2152 from (6), and then the four locally D-optimal star points are
±
1.3816 −0.3020 0.3020 1.3816
, (7)
which are different from the original star points of the spherical CCD clearly.
Now we employ the BAR sampler with Nt = 10, T (t) ∝ t−2, and 400 times iterations, and then the four locally D-optimal star points are
±
1.3814 −0.3031 0.3031 1.3814
,
which are very close to the points shown in (7). Figure 3 presents the tendency of det¡
Mb−1(ξ)¢ .
0 1000 2000 3000 4000 5000 6000 7000 8000 100
150 200 250 300 350 400 450 500 550
Figure 3: For k = 2 and n0 = 1, the tendency of det¡
Mb−1(ξ)¢
for 2 × 10 × 400 = 8000 steps. (Nt = 10 and 400 iterations.)
• n0 = 2
When n0 = 2, θ12 is -0.1753 from (6), and then the four locally D-optimal star points are
±
1.3925 −0.2467 0.2467 1.3925
, (8)
which are different from the original star points of the spherical CCD clearly.
Now we employ the BAR sampler with Nt = 10, T (t) ∝ t−2, and 400 times iterations, and then the four locally D-optimal star points are
±
1.3921 −0.2490 0.2490 1.3921
,
which are very close to the points shown in (8). Figure 4 presents the tendency of det¡
Mb−1(ξ)¢ .
0 1000 2000 3000 4000 5000 6000 7000 8000 100
150 200 250 300 350 400 450 500
Figure 4: For k = 2 and n0 = 2, the tendency of det¡
Mb−1(ξ)¢
for 2 × 10 × 400 = 8000 steps. (Nt = 10 and 400 iterations.)
• n0 = 3
When n0 = 3, θ12 is -0.1519 from (6), and then the four locally D-optimal star points are
±
1.3979 −0.2140 0.2140 1.3979
, (9)
which are different from the original star points of the spherical CCD clearly.
Now we employ the BAR sampler with Nt = 10, T (t) ∝ t−2, and 400 times iterations, and then the four locally D-optimal star points are
±
1.3978 −0.2148 0.2148 1.3978
,
which are very close to the points shown in (9). Figure 5 presents the tendency of det¡
Mb−1(ξ)¢ .
0 1000 2000 3000 4000 5000 6000 7000 8000 100
200 300 400 500 600 700 800 900
Figure 5: For k = 2 and n0 = 3, the tendency of det¡
Mb−1(ξ)¢
for 2 × 10 × 400 = 8000 steps. (Nt = 10 and 400 iterations.)
• n0 = 4
When n0 = 4, θ12 is -0.1365 from (6), and then the four locally D-optimal star points are
±
1.4011 −0.1925 0.1925 1.4011
, (10)
which are different from the original star points of the spherical CCD clearly.
Now we employ the BAR sampler with Nt = 10, T (t) ∝ t−2, and 400 times iterations, and then the four locally D-optimal star points are
±
1.4013 −0.1905 0.1905 1.4013
,
which are very close to the points shown in (10). Figure 6 presents the tendency of det¡
Mb−1(ξ)¢ .
0 1000 2000 3000 4000 5000 6000 7000 8000 100
200 300 400 500 600 700 800
Figure 6: For k = 2 and n0 = 4, the tendency of det¡
Mb−1(ξ)¢
for 2 × 10 × 400 = 8000 steps. (Nt = 10 and 400 iterations.)
4.1.2 23 Full Factorial Design
Since a locally D-optimal design depends on b(0), we choose b(0) by fixing the values of R and φj, j = 1, 2, as R = 0.25√
3, 0.5√
3, 0.75√
3, and √
3, and φj = 0, 10π, 2π10, 3π10, and 4π10. Thus totally there are 4 × 5 × 5 = 100 cases here.
To save the space, we only demonstarte one case by setting R =√
3, φ1 = 0, and φ2 = 10π. The numbers of the center points, n0, are 1, 2, 3, and 4. For k = 3, based on (5), we need to have three rotation angles, θ1, θ2 and θ3, and one radius r in our objective function, and then we employ the BAR sampler with Nt = 10, T (t) ∝ t−2, and 400 times iterations to find the six locally D-optimal star points numerically.
• n0 = 1
When n0 = 1, the six locally D-optimal star points are
±
3. These points are different from the original star points of the spherical CCD clearly.
• n0 = 2
When n0 = 2, the six locally D-optimal star points are
±
3. These points are different from the original star points of the spherical CCD clearly.
• n0 = 3
When n0 = 3, the six locally D-optimal star points are
±
3. These points are different from the original star points of the spherical CCD clearly.
• n0 = 4
When n0 = 4, the six locally D-optimal star points are
±
3. These points are different from the original star points of the spherical CCD clearly.
We try to summarize some finds in our all 500 cases. For r, from our numerical results, we find that r =√
3. Then for the rotation angles, θ1, θ2, and θ3, basically we want to see that at least one of θ1, θ2, and θ3 are not equal to zero, because it means that our optimal additional points are not on axes anymore. The summarizations are as follows.
(I) φ2 = 0
In this case, the possible extremum point b(0) is on the third axis, i.e. b(0) = (0, 0, R). The following two figures, Figures 7 and 8, are the plots of | θi| with n0
and R separately. From these figures, we conclude that when φ2 = 0, the original star points are also our optimal star points.
(a) (b) (c)
Figure 7: For k = 3, R = √
3, and φ2 = 0, (a) | θ1| v.s. n0, (b) | θ2| v.s. n0, and (c) | θ3| v.s. n0 .
(a) (b) (c)
Figure 8: For k = 3, n0 = 1, and φ2 = 0, (a) | θ1| v.s. R, (b) | θ2| v.s. R, and (c)
| θ3| v.s. R . The unit of R is√ 3.
(II) φ2 6= 0
When φ2 6= 0, all or some of the rotation angles θ1, θ2, and θ3 are not equal to zeros. That is our optimal star points should not on axes. We also have the following two figures, Figures 9 and 10, to show the relations of | θi| and n0, R.
Basically more center points we put in the first-order design, | θi| are smaller.
That is our optimal additional points are more close to axes. But for large value of R, | θi| is large. Then at this time, our additional points are far from axes.
These two summarizations are similar as what we find in the case of k = 2.
(a) (b) (c)
Figure 9: For k = 3, R = 0.75√
3, φ1 = 10π, and φ2 = 2π10, (a) | θ1| v.s. n0, (b) | θ2| v.s. n0, and (c) | θ3| v.s. n0 .
(a) (b) (c)
Figure 10: For k = 3, n0 = 2, φ1 = 3π10, and φ2 = 4π10, (a) | θ1| v.s. R, (b) | θ2| v.s.
R, and (c) | θ3| v.s. R . The unit of R is √ 3.
4.1.3 24 Full Factorial Design
Since a locally D-optimal design depends on b(0), we choose b(0) by fixing the values of R and φj, j = 1, 2, 3, as R = 0.25√
4, 0.5√
4, 0.75√
4, and √
4, and φj = 0, 10π, 2π10,
3π
10, and 4π10. Thus totally there are 4 × 5 × 5 × 5 = 500 cases here.
We show one case with R = √
4, φ1 = 0, φ2 = 0, and φ3 = 10π as an example here. The numbers of the center points, n0, are 1, 2, 3, and 4. For k = 4, based on (5), we need to have six rotation angles, θ1, θ2, · · · , θ6, and one radius r in our objective function, and then we employ the BAR sampler with Nt = 10, T (t) ∝ t−2, and 400 times iterations to find the eight locally D-optimal star points numerically.
• n0 = 1
When n0 = 1, the eight locally D-optimal star points are
±
−1.0786 −1.6842 0.0047 −0.0047 1.6843 −1.0785 0.0021 −0.0047 0.0011 0.0036 1.9524 0.4339 0.0013 −0.0075 −0.4339 1.9524
4. These points are different from the original star points of the spherical CCD clearly.
• n0 = 2
When n0 = 2, the eight locally D-optimal star points are
±
1.9251 −0.5420 −0.0064 −0.0028 0.5420 1.9252 −0.0057 0.0004 0.0081 0.0034 1.9627 0.3842 0.0011 −0.0018 −0.3843 1.9627
4. These points are different from the original star points of the spherical CCD clearly.
• n0 = 3
When n0 = 3, the eight locally D-optimal star points are
±
0.7257 −1.8637 0.0007 0.0072 1.8637 0.7257 0.0073 −0.0054
−0.0066 −0.0006 1.9742 0.3202 0.0035 0.0089 −0.3202 1.9742
4. These points are different from the original star points of the spherical CCD clearly.
• n0 = 4
When n0 = 4, the eight locally D-optimal star points are
±
1.9612 0.3921 0.0064 0.0031
−0.3921 1.9612 −0.0075 −0.0051
−0.0083 0.0067 1.9755 0.3120
−0.0028 0.0034 −0.3120 1.9755
4. These points are different from the original star points of the spherical CCD clearly.
4.1.4 25 Full Factorial Design
Since a locally D-optimal design depends on b(0), we choose b(0) by fixing the values of R and φj, j = 1, 2, · · · , 4, as R = 0.25√ times iterations to find the ten locally D-optimal star points numerically.
• n0 = 1
When n0 = 1, the ten locally D-optimal star points are
±
0.1418 −0.2804 −2.2138 0.0110 −0.0108
−2.1485 0.5821 −0.2115 −0.0098 0.0110 0.6029 2.1407 −0.2326 −0.0026 0.0065
−0.0072 0.0041 0.0076 2.1871 0.4652 0.0112 −0.0115 −0.0108 −0.4651 2.1871
5. These points are different from the original star points of the spherical CCD clearly.
• n0 = 2
When n0 = 2, the ten locally D-optimal star points are
±
−0.8091 2.0117 0.5462 0.0009 0.0069
−1.0203 0.1288 −1.9855 −0.0003 −0.0012
−1.8178 −0.9677 0.8714 −0.0067 0.0030
−0.0044 −0.0045 0.0013 2.1979 0.4114 0.0053 −0.0041 −0.0042 −0.4114 2.1979
5. These points are different from the original star points of the spherical CCD clearly.
• n0 = 3
When n0 = 3, the ten locally D-optimal star points are
±
−0.5068 −2.1221 −0.4895 0.0110 −0.0108
−1.1637 0.6886 −1.7809 −0.0098 0.0110 1.8409 −0.1489 −1.2605 −0.0026 0.0065
−0.0008 0.0111 −0.0052 2.2082 0.3516
−0.0020 −0.0151 0.0110 −0.3515 2.2082
5. These points are different from the original star points of the spherical CCD clearly.
• n0 = 4
When n0 = 4, the ten locally D-optimal star points are
±
0.5452 −0.9011 −1.9725 −0.0075 −0.0098 1.6356 −1.1647 0.9840 0.0063 0.0072
−1.4240 −1.6827 0.3751 0.0078 −0.0102 0.0008 0.0049 −0.0121 2.2115 0.3301
−0.0096 −0.0087 −0.0084 −0.3302 2.2115
5. These points are different from the original star points of the spherical CCD clearly.
4.1.5 26 Full Factorial Design
Since a locally D-optimal design depends on b(0), we choose b(0) by fixing the values of R and φj, j = 1, 2, · · · , 5, as R = 0.25√ objective function, and then we employ the BAR sampler with Nt = 10, T (t) ∝ t−2, and 400 times iterations to find the twelve locally D-optimal star points numerically.
• n0 = 1
When n0 = 1, the twelve locally D-optimal star points are
±
0.3624 0.4399 −0.5656 2.3141 0.0049 −0.0005 0.1582 −2.3643 0.3522 0.5108 0.0009 0.0052
−1.0777 0.3692 2.0817 0.6074 −0.0091 0.0050 2.1639 0.2831 1.1057 −0.1224 −0.0058 0.0094
−0.0008 0.0026 0.0094 −0.0031 2.4057 0.4608
−0.0063 0.0028 −0.0113 −0.0008 −0.4608 2.4057
6. These points are different from the original star points of the spherical CCD clearly.
• n0 = 2
When n0 = 2, the twelve locally D-optimal star points are
±
0.1516 −0.1873 −2.2484 −0.9415 0.0014 −0.0088
−1.7754 0.1122 −0.7585 1.5033 0.0085 0.0042
−0.3587 −2.4087 0.2300 −0.1278 −0.0015 −0.0095
−1.6421 0.3876 0.5623 −1.6844 −0.0016 0.0023 0.0053 −0.0032 0.0034 −0.0065 2.4189 0.3861 0.0029 −0.0102 −0.0070 −0.0039 −0.3861 2.4188
6. These points are different from the original star points of the spherical CCD clearly.
• n0 = 3
When n0 = 3, the twelve locally D-optimal star points are
±
0.6273 2.2099 −0.8171 −0.2344 0.0073 0.0091 2.2321 −0.5024 0.5498 −0.6805 −0.0035 −0.0085 0.7589 −0.0802 −0.2965 2.3086 −0.0068 0.0079
−0.2202 0.9259 2.2231 0.3901 −0.0015 −0.0009 0.0037 −0.0083 0.0047 0.0050 2.4211 0.3718 0.0023 −0.0082 0.0061 −0.0096 −0.3719 2.4211
6. These points are different from the original star points of the spherical CCD clearly.
• n0 = 4
When n0 = 4, the twelve locally D-optimal star points are
±
−0.1639 0.2468 −1.5123 −1.9040 0.0051 0.0006 2.2762 −0.8702 −0.1072 −0.2236 −0.0016 −0.0090
−0.8419 −2.1003 0.6268 −0.6976 0.0066 0.0086
−0.2880 −0.8777 −1.8189 1.3557 0.0079 −0.0035 0.0066 0.0077 0.0065 0.0018 2.4247 0.3475 0.0101 0.0018 −0.0058 0.0038 −0.3475 2.4247
6. These points are different from the original star points of the spherical CCD clearly.
4.1.6 27 Full Factorial Design
Since a locally D-optimal design depends on b(0), we choose b(0) by fixing the values of R and φj, j = 1, 2, · · · , 6, as R = 0.25√ and one radius r in our objective function, and then we employ the BAR sampler with Nt = 10, T (t) ∝ t−2, and 400 times iterations to find the fourteen locally D-optimal star points numerically.
• n0 = 1
When n0 = 1, the fourteen locally D-optimal star points are
±
−0.0031 −0.0035 0.0193 0.0407 −2.6453 0.0086 0.0103 0.5452 −0.4518 −0.1086 −2.5466 −0.0400 0.0048 0.0119 2.2701 1.1807 0.6246 0.2498 0.0041 −0.0056 0.0002 1.0505 −0.9765 −2.1684 0.4908 −0.0082 −0.0001 0.0120 0.6676 −2.1091 1.3769 0.4581 0.0190 −0.0108 0.0028 0.0052 −0.0040 0.0083 0.0082 0.0103 2.6146 0.4045
−0.0091 0.0094 0.0076 0.0074 0.0091 −0.4046 2.6146
7. These points are different from the original star points of the spherical CCD clearly.
• n0 = 2
When n0 = 2, the fourteen locally D-optimal star points are
±
−0.2880 −0.5690 0.3767 −0.3176 −2.5200 −0.0038 0.0028 0.0669 1.3541 −2.0876 −0.7197 −0.5347 −0.0075 0.0089
−0.4026 0.2904 −0.5781 2.4984 −0.4209 0.0033 0.0059 0.9311 2.0169 1.3760 0.1703 −0.3776 0.0088 0.0047
−2.4256 0.8309 0.5219 −0.3314 0.2094 −0.0004 −0.0044
−0.0038 −0.0049 −0.0082 −0.0067 −0.0024 2.6187 0.3775
−0.0042 −0.0061 0.0076 −0.0027 0.0069 −0.3775 2.6187
7. These points are different from the original star points of the spherical CCD clearly.
• n0 = 3
When n0 = 3, the fourteen locally D-optimal star points are
±
0.0004 −0.0015 −0.0013 0.1463 2.6417 −0.0071 0.0098
−0.4337 0.0447 −0.5725 2.5421 −0.1410 −0.0074 −0.0016
−1.2164 2.1792 0.8768 −0.0481 0.0045 0.0095 0.0032
−0.6485 −1.2411 2.2074 0.4071 −0.0221 0.0073 0.0090
−2.2162 −0.8417 −1.0152 −0.5901 0.0320 0.0037 0.0054 0.0090 −0.0027 −0.0102 0.0075 0.0054 2.6248 0.3323 0.0068 0.0037 −0.0057 −0.0001 −0.0106 −0.3323 2.6248
and the radius, r, is √
7. These points are different from the original star points of the spherical CCD clearly.
• n0 = 4
When n0 = 4, the fourteen locally D-optimal star points are
±
−0.3179 −0.1713 0.3766 0.1114 2.5914 −0.0071 0.0098 1.3429 0.6922 −1.9489 −0.8002 0.5281 −0.0074 −0.0016 1.1791 −1.2834 −0.4307 1.9431 0.0389 0.0095 0.0032 1.8795 0.7400 1.6852 −0.2792 0.0465 0.0073 0.0090 0.4157 −2.0729 0.1863 −1.5791 −0.0453 0.0037 0.0054
−0.0079 0.0075 −0.0088 −0.0058 0.0070 2.6248 0.3323
−0.0057 0.0034 −0.0071 0.0016 −0.0103 −0.3323 2.6248
,
and the radius, r, is √
7. These points are different from the original star points of the spherical CCD clearly.
4.1.7 28 Full Factorial Design
Since a locally D-optimal design depends on b(0), we choose b(0) by fixing the values of R and φj, j = 1, 2, · · · , 7, as R = 0.25√
8, 0.5√
8, 0.75√
8, and √
8, and φj = 0, 10π,
2π
10, 3π10, and 4π10. Thus totally there are 4 × 57 = 312500 cases here.
We show one case with R =√
8, φ1 = 0, φ2 = 0, φ3 = 0, φ4 = 0, φ5 = 0, φ6 = 0, and φ7 = 10π as an example here. The numbers of the center points, n0, are 1, 2, 3, and 4. For k = 8, based on (5), we need to have twenty-eight rotation angles, θ1, θ2, · · · , θ28, and one radius r in our objective function, and then we employ the BAR sampler with Nt = 10, T (t) ∝ t−2, and 400 times iterations to find the sixteen locally D-optimal star points numerically.
• n0 = 1
When n0 = 1, the sixteen locally D-optimal star points are
±
1.1995 0.8808 0.0373 −1.3777 −0.5529 −1.8921 −0.0118 0.0001 1.1563 0.8180 0.1100 −1.3618 0.3876 1.9943 −0.0012 −0.0025 1.3185 −0.7656 −0.2194 0.7979 −2.1706 0.5284 −0.0002 −0.0085 1.3574 1.0112 0.9888 1.7806 0.9668 −0.2284 0.0041 0.0073 0.1129 1.0743 −2.5338 0.6171 0.1776 0.0205 0.0056 −0.0073
−1.2770 1.9473 0.7351 0.2437 −1.3661 0.3330 0.0107 0.0070 0.0087 −0.0077 −0.0004 −0.0112 0.0005 −0.0075 2.8078 0.3406 0.0038 −0.0054 −0.0115 −0.0010 −0.0050 0.0042 −0.3406 2.8078
8. These points are different from the original star points of the spherical CCD clearly.
• n0 = 2
When n0 = 2, the sixteen locally D-optimal star points are
±
−0.0360 0.0220 −0.1349 0.0816 2.4172 1.4595 0.0116 −0.0124
−0.0215 −0.6300 −0.3645 0.3305 1.3831 −2.3340 0.0118 0.0047 1.1974 0.3041 −2.3252 0.9634 −0.2893 0.2351 0.0138 −0.0101 1.4587 1.4277 0.0731 −1.8673 0.3558 −0.4638 −0.0012 −0.0044
−0.6285 −1.5862 −1.2502 −1.8510 −0.1650 0.2693 0.0053 −0.0146
−2.0104 1.7191 −0.9348 −0.2075 0.0795 −0.2818 −0.0127 0.0129
−0.0114 0.0111 0.0104 −0.0055 −0.0127 0.0020 2.8123 0.3006 0.0137 −0.0128 −0.0116 −0.0077 0.0080 0.0129 −0.3005 2.8123
8. These points are different from the original star points of the spherical CCD clearly.
• n0 = 3
When n0 = 3, the sixteen locally D-optimal star points are
±
−0.0007 0.0009 −0.0002 0.0018 −0.1415 −2.8248 0.0124 −0.0157
−0.4159 0.2216 −0.1157 −0.0479 2.7825 −0.1393 0.0049 0.0138
−2.4337 1.3484 −0.1752 0.0168 −0.4769 0.0249 0.0002 −0.0009 0.7495 1.2588 −0.5757 2.3497 0.0282 0.0003 0.0001 0.0111
−1.0179 −2.0270 −1.2860 1.0958 −0.0252 0.0016 −0.0055 −0.0015
−0.5533 −0.6630 2.4434 1.1293 0.0910 −0.0039 0.0134 −0.0137 0.0009 −0.0024 −0.0123 −0.0036 −0.0062 0.0110 2.8119 0.3046
−0.0050 −0.0097 0.0154 −0.0025 −0.0136 −0.0163 −0.3045 2.8118
8. These points are different from the original star points of the spherical CCD clearly.
• n0 = 4
When n0 = 4, the sixteen locally D-optimal star points are
±
−0.0313 0.0386 0.1082 −0.7810 −2.7070 0.2180 0.0116 −0.0124
−0.1654 −0.2605 −0.3118 2.0839 −0.7526 −1.7025 0.0118 0.0047 2.3783 0.3584 −1.4309 −0.2535 −0.0321 −0.3199 0.0138 −0.0101
−0.0052 0.2733 −0.7050 1.6007 −0.3101 2.1840 −0.0012 −0.0044 1.5215 −0.5745 2.2057 0.6131 −0.0882 0.3256 0.0053 −0.0146 0.0071 −2.7198 −0.6938 −0.2127 0.0168 0.2747 −0.0127 0.0129
−0.0119 −0.0106 0.0012 −0.0058 0.0134 0.0099 2.8153 0.2715 0.0177 0.0129 0.0092 −0.0006 −0.0131 0.0055 −0.2714 2.8152
8. These points are different from the original star points of the spherical CCD clearly.