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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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 |
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.
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