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Maximum coefficients for Extended Polynomial

  • December 3, 2020
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Katsumoto Ikeda
Ultraviolet
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In the Manual and Help File, the Extended Polynomial function is given, but it would be helpful to have the table of which order corresponds to the value needed to fill all terms of a given radial order (similar to the table for the Zernikes). This would provide a quick reference for the Maximum Term Number to enter.

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Katsumoto Ikeda
Ultraviolet
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Below is a table for the Extended Polynomial orders, Parameter values are there for reference.

 

Maximum Term # Name X Y Par
1 X1Y0 1 0 15
2 X0Y1 0 1 16
3 X2Y0 2 0 17
4 X1Y1 1 1 18
5 X0Y2 0 2 19
6 X3Y0 3 0 20
7 X2Y1 2 1 21
8 X1Y2 1 2 22
9 X0Y3 0 3 23
10 X4Y0 4 0 24
11 X3Y1 3 1 25
12 X2Y2 2 2 26
13 X1Y3 1 3 27
14 X0Y4 0 4 28
15 X5Y0 5 0 29
16 X4Y1 4 1 30
17 X3Y2 3 2 31
18 X2Y3 2 3 32
19 X1Y4 1 4 33
20 X0Y5 0 5 34
21 X6Y0 6 0 35
22 X5Y1 5 1 36
23 X4Y2 4 2 37
24 X3Y3 3 3 38
25 X2Y4 2 4 39
26 X1Y5 1 5 40
27 X0Y6 0 6 41
28 X7Y0 7 0 42
29 X6Y1 6 1 43
30 X5Y2 5 2 44
31 X4Y3 4 3 45
32 X3Y4 3 4 46
33 X2Y5 2 5 47
34 X1Y6 1 6 48
35 X0Y7 0 7 49
36 X8Y0 8 0 50
37 X7Y1 7 1 51
38 X6Y2 6 2 52
39 X5Y3 5 3 53
40 X4Y4 4 4 54
41 X3Y5 3 5 55
42 X2Y6 2 6 56
43 X1Y7 1 7 57
44 X0Y8 0 8 58
45 X9Y0 9 0 59
46 X8Y1 8 1 60
47 X7Y2 7 2 61
48 X6Y3 6 3 62
49 X5Y4 5 4 63
50 X4Y5 4 5 64
51 X3Y6 3 6 65
52 X2Y7 2 7 66
53 X1Y8 1 8 67
54 X0Y9 0 9 68
55 X10Y0 10 0 69
56 X9Y1 9 1 70
57 X8Y2 8 2 71
58 X7Y3 7 3 72
59 X6Y4 6 4 73
60 X5Y5 5 5 74
61 X4Y6 4 6 75
62 X3Y7 3 7 76
63 X2Y8 2 8 77
64 X1Y9 1 9 78
65 X0Y10 0 10 79
66 X11Y0 11 0 80
67 X10Y1 10 1 81
68 X9Y2 9 2 82
69 X8Y3 8 3 83
70 X7Y4 7 4 84
71 X6Y5 6 5 85
72 X5Y6 5 6 86
73 X4Y7 4 7 87
74 X3Y8 3 8 88
75 X2Y9 2 9 89
76 X1Y10 1 10 90
77 X0Y11 0 11 91
78 X12Y0 12 0 92
79 X11Y1 11 1 93
80 X10Y2 10 2 94
81 X9Y3 9 3 95
82 X8Y4 8 4 96
83 X7Y5 7 5 97
84 X6Y6 6 6 98
85 X5Y7 5 7 99
86 X4Y8 4 8 100
87 X3Y9 3 9 101
88 X2Y10 2 10 102
89 X1Y11 1 11 103
90 X0Y12 0 12 104
91 X13Y0 13 0 105
92 X12Y1 12 1 106
93 X11Y2 11 2 107
94 X10Y3 10 3 108
95 X9Y4 9 4 109
96 X8Y5 8 5 110
97 X7Y6 7 6 111
98 X6Y7 6 7 112
99 X5Y8 5 8 113
100 X4Y9 4 9 114
101 X3Y10 3 10 115
102 X2Y11 2 11 116
103 X1Y12 1 12 117
104 X0Y13 0 13 118
105 X14Y0 14 0 119
106 X13Y1 13 1 120
107 X12Y2 12 2 121
108 X11Y3 11 3 122
109 X10Y4 10 4 123
110 X9Y5 9 5 124
111 X8Y6 8 6 125
112 X7Y7 7 7 126
113 X6Y8 6 8 127
114 X5Y9 5 9 128
115 X4Y10 4 10 129
116 X3Y11 3 11 130
117 X2Y12 2 12 131
118 X1Y13 1 13 132
119 X0Y14 0 14 133
120 X15Y0 15 0 134
121 X14Y1 14 1 135
122 X13Y2 13 2 136
123 X12Y3 12 3 137
124 X11Y4 11 4 138
125 X10Y5 10 5 139
126 X9Y6 9 6 140
127 X8Y7 8 7 141
128 X7Y8 7 8 142
129 X6Y9 6 9 143
130 X5Y10 5 10 144
131 X4Y11 4 11 145
132 X3Y12 3 12 146
133 X2Y13 2 13 147
134 X1Y14 1 14 148
135 X0Y15 0 15 149
136 X16Y0 16 0 150
137 X15Y1 15 1 151
138 X14Y2 14 2 152
139 X13Y3 13 3 153
140 X12Y4 12 4 154
141 X11Y5 11 5 155
142 X10Y6 10 6 156
143 X9Y7 9 7 157
144 X8Y8 8 8 158
145 X7Y9 7 9 159
146 X6Y10 6 10 160
147 X5Y11 5 11 161
148 X4Y12 4 12 162
149 X3Y13 3 13 163
150 X2Y14 2 14 164
151 X1Y15 1 15 165
152 X0Y16 0 16 166
153 X17Y0 17 0 167
154 X16Y1 16 1 168
155 X15Y2 15 2 169
156 X14Y3 14 3 170
157 X13Y4 13 4 171
158 X12Y5 12 5 172
159 X11Y6 11 6 173
160 X10Y7 10 7 174
161 X9Y8 9 8 175
162 X8Y9 8 9 176
163 X7Y10 7 10 177
164 X6Y11 6 11 178
165 X5Y12 5 12 179
166 X4Y13 4 13 180
167 X3Y14 3 14 181
168 X2Y15 2 15 182
169 X1Y16 1 16 183
170 X0Y17 0 17 184
171 X18Y0 18 0 185
172 X17Y1 17 1 186
173 X16Y2 16 2 187
174 X15Y3 15 3 188
175 X14Y4 14 4 189
176 X13Y5 13 5 190
177 X12Y6 12 6 191
178 X11Y7 11 7 192
179 X10Y8 10 8 193
180 X9Y9 9 9 194
181 X8Y10 8 10 195
182 X7Y11 7 11 196
183 X6Y12 6 12 197
184 X5Y13 5 13 198
185 X4Y14 4 14 199
186 X3Y15 3 15 200
187 X2Y16 2 16 201
188 X1Y17 1 17 202
189 X0Y18 0 18 203
190 X19Y0 19 0 204
191 X18Y1 18 1 205
192 X17Y2 17 2 206
193 X16Y3 16 3 207
194 X15Y4 15 4 208
195 X14Y5 14 5 209
196 X13Y6 13 6 210
197 X12Y7 12 7 211
198 X11Y8 11 8 212
199 X10Y9 10 9 213
200 X9Y10 9 10 214
201 X8Y11 8 11 215
202 X7Y12 7 12 216
203 X6Y13 6 13 217
204 X5Y14 5 14 218
205 X4Y15 4 15 219
206 X3Y16 3 16 220
207 X2Y17 2 17 221
208 X1Y18 1 18 222
209 X0Y19 0 19 223
210 X20Y0 20 0 224
211 X19Y1 19 1 225
212 X18Y2 18 2 226
213 X17Y3 17 3 227
214 X16Y4 16 4 228
215 X15Y5 15 5 229
216 X14Y6 14 6 230
217 X13Y7 13 7 231
218 X12Y8 12 8 232
219 X11Y9 11 9 233
220 X10Y10 10 10 234
221 X9Y11 9 11 235
222 X8Y12 8 12 236
223 X7Y13 7 13 237
224 X6Y14 6 14 238
225 X5Y15 5 15 239
226 X4Y16 4 16 240
227 X3Y17 3 17 241
228 X2Y18 2 18 242
229 X1Y19 1 19 243
230 X0Y20 0 20 244
Katsumoto Ikeda
Ultraviolet
Forum|alt.badge.img

Alternately, if you want a formula to fill all the terms for a given order, the value in the Maximum Term Number should be 

N = 1/2 y^2 + 3/2 y

For example, if you wanted all the orders for 7 up to X0Y7, 

N = 1/2*(7^2) + 3/2*(7) = 49/2 + 21/2 = 35 

So you would type 35 for the Maximum Term number. 

Michael.Young
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  • Michael.Young
    Michael.Young

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