A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | vector X | vector Y | R | THETA | vector Fr | vector Fi | matrix Jr | matrix Ji | matrix 1/Jr | matrix 1/Ji | matrix Marie = 1/Ji*Jr + 1/Jr*Ji | matrix 1/Marie | vector Vince = 1/Ji*Fr + 1/Jr*Fi | vector Vito = 1/Ji*Fi - 1/Jr*Fr | vector DX = 1/Marie * Vince | vector DY = 1/Marie * Vito | 1 | ||||||
2 | 1 | -2 | =SQRT(A2^2 + B2^2) | =IF(B2<0, -ACOS(A2/C2), ACOS(A2/C2)) | = C2^3*C3^5*COS(3*D2 + 5*D3) - 6*C2*C3*COS(D2 + D3) - 17 | = C2^3*C3^5*SIN(3*D2 + 5*D3) - 6*C2*C3*SIN(D2 + D3) | = 3*C2^2*C3^5*COS(2*D2 + 5*D3) - 6*C3*COS(D3) | = 5*C2^3*C3^4*COS(3*D2 + 4*D3) - 6*C2*COS(D2) | = 3*C2^2*C3^5*SIN(2*D2 + 5*D3) - 6*C3*SIN(D3) | = 5*C2^3*C3^4*SIN(3*D2 + 4*D3) - 6*C2*SIN(D2) | =INDEX(MINVERSE(G2:H3), 1, 1) | =INDEX(MINVERSE(G2:H3), 1, 2) | =INDEX(MINVERSE(I2:J3), 1, 1) | =INDEX(MINVERSE(I2:J3), 1, 2) | =INDEX(MMULT(M2:N3, G2:H3), 1, 1) + INDEX(MMULT(K2:L3, I2:J3), 1, 1) | =INDEX(MMULT(M2:N3, G2:H3), 1, 2) + INDEX(MMULT(K2:L3, I2:J3), 1, 2) | =INDEX(MINVERSE(O2:P3), 1, 1) | =INDEX(MINVERSE(O2:P3), 1, 2) | =INDEX(MMULT(M2:N3, E2:E3), 1, 1) + INDEX(MMULT(K2:L3, F2:F3), 1, 1) | =INDEX(MMULT(M2:N3, F2:F3), 1, 1) - INDEX(MMULT(K2:L3, E2:E3), 1, 1) | =INDEX(MMULT(Q2:R3, S2:S3), 1, 1) | =INDEX(MMULT(Q2:R3, T2:T3), 1, 1) | 2 |
3 | 1 | -1 | =SQRT(A3^2 + B3^2) | =IF(B3<0, -ACOS(A3/C3), ACOS(A3/C3)) | = 7*C2^2*C3*COS(2*D2 + D3) - 3*C2*C3^3*COS(D2 + 3*D3) + 5 | = 7*C2^2*C3*SIN(2*D2 + D3) - 3*C2*C3^3*SIN(D2 + 3*D3) | = 14*C2*C3*COS(D2 + D3) - 3*C3^3*COS(3*D3) | = 7*C2^2*COS(2*D2) - 9*C2*C3^2*COS(D2 + 2*D3) | = 14*C2*C3*SIN(D2 + D3) - 3*C3^3*SIN(3*D3) | = 7*C2^2*SIN(2*D2) - 9*C2*C3^2*SIN(D2 + 2*D3) | =INDEX(MINVERSE(G2:H3), 2, 1) | =INDEX(MINVERSE(G2:H3), 2, 2) | =INDEX(MINVERSE(I2:J3), 2, 1) | =INDEX(MINVERSE(I2:J3), 2, 2) | =INDEX(MMULT(M2:N3, G2:H3), 2, 1) + INDEX(MMULT(K2:L3, I2:J3), 2, 1) | =INDEX(MMULT(M2:N3, G2:H3), 2, 2) + INDEX(MMULT(K2:L3, I2:J3), 2, 2) | =INDEX(MINVERSE(O2:P3), 2, 1) | =INDEX(MINVERSE(O2:P3), 2, 2) | =INDEX(MMULT(M2:N3, E2:E3), 2, 1) + INDEX(MMULT(K2:L3, F2:F3), 2, 1) | =INDEX(MMULT(M2:N3, F2:F3), 2, 1) - INDEX(MMULT(K2:L3, E2:E3), 2, 1) | =INDEX(MMULT(Q2:R3, S2:S3), 2, 1) | =INDEX(MMULT(Q2:R3, T2:T3), 2, 1) | 3 |
4 | vector X | vector Y | R | THETA | vector Fr | vector Fi | matrix Jr | matrix Ji | matrix 1/Jr | matrix 1/Ji | matrix Marie = 1/Ji*Jr + 1/Jr*Ji | matrix 1/Marie | vector Vince = 1/Ji*Fr + 1/Jr*Fi | vector Vito = 1/Ji*Fi - 1/Jr*Fr | vector DX = 1/Marie * Vince | vector DY = 1/Marie * Vito | 4 | ||||||
5 | = A2 - U2 | = B2 - V2 | =SQRT(A5^2 + B5^2) | =IF(B5<0, -ACOS(A5/C5), ACOS(A5/C5)) | = C5^3*C6^5*COS(3*D5 + 5*D6) - 6*C5*C6*COS(D5 + D6) - 17 | = C5^3*C6^5*SIN(3*D5 + 5*D6) - 6*C5*C6*SIN(D5 + D6) | = 3*C5^2*C6^5*COS(2*D5 + 5*D6) - 6*C6*COS(D6) | = 5*C5^3*C6^4*COS(3*D5 + 4*D6) - 6*C5*COS(D5) | = 3*C5^2*C6^5*SIN(2*D5 + 5*D6) - 6*C6*SIN(D6) | = 5*C5^3*C6^4*SIN(3*D5 + 4*D6) - 6*C5*SIN(D5) | =INDEX(MINVERSE(G5:H6), 1, 1) | =INDEX(MINVERSE(G5:H6), 1, 2) | =INDEX(MINVERSE(I5:J6), 1, 1) | =INDEX(MINVERSE(I5:J6), 1, 2) | =INDEX(MMULT(M5:N6, G5:H6), 1, 1) + INDEX(MMULT(K5:L6, I5:J6), 1, 1) | =INDEX(MMULT(M5:N6, G5:H6), 1, 2) + INDEX(MMULT(K5:L6, I5:J6), 1, 2) | =INDEX(MINVERSE(O5:P6), 1, 1) | =INDEX(MINVERSE(O5:P6), 1, 2) | =INDEX(MMULT(M5:N6, E5:E6), 1, 1) + INDEX(MMULT(K5:L6, F5:F6), 1, 1) | =INDEX(MMULT(M5:N6, F5:F6), 1, 1) - INDEX(MMULT(K5:L6, E5:E6), 1, 1) | =INDEX(MMULT(Q5:R6, S5:S6), 1, 1) | =INDEX(MMULT(Q5:R6, T5:T6), 1, 1) | 5 |
6 | = A3 - U3 | = B3 - V3 | =SQRT(A6^2 + B6^2) | =IF(B6<0, -ACOS(A6/C6), ACOS(A6/C6)) | = 7*C5^2*C6*COS(2*D5 + D6) - 3*C5*C6^3*COS(D5 + 3*D6) + 5 | = 7*C5^2*C6*SIN(2*D5 + D6) - 3*C5*C6^3*SIN(D5 + 3*D6) | = 14*C5*C6*COS(D5 + D6) - 3*C6^3*COS(3*D6) | = 7*C5^2*COS(2*D5) - 9*C5*C6^2*COS(D5 + 2*D6) | = 14*C5*C6*SIN(D5 + D6) - 3*C6^3*SIN(3*D6) | = 7*C5^2*SIN(2*D5) - 9*C5*C6^2*SIN(D5 + 2*D6) | =INDEX(MINVERSE(G5:H6), 2, 1) | =INDEX(MINVERSE(G5:H6), 2, 2) | =INDEX(MINVERSE(I5:J6), 2, 1) | =INDEX(MINVERSE(I5:J6), 2, 2) | =INDEX(MMULT(M5:N6, G5:H6), 2, 1) + INDEX(MMULT(K5:L6, I5:J6), 2, 1) | =INDEX(MMULT(M5:N6, G5:H6), 2, 2) + INDEX(MMULT(K5:L6, I5:J6), 2, 2) | =INDEX(MINVERSE(O5:P6), 2, 1) | =INDEX(MINVERSE(O5:P6), 2, 2) | =INDEX(MMULT(M5:N6, E5:E6), 2, 1) + INDEX(MMULT(K5:L6, F5:F6), 2, 1) | =INDEX(MMULT(M5:N6, F5:F6), 2, 1) - INDEX(MMULT(K5:L6, E5:E6), 2, 1) | =INDEX(MMULT(Q5:R6, S5:S6), 2, 1) | =INDEX(MMULT(Q5:R6, T5:T6), 2, 1) | 6 |
7 | vector X | vector Y | R | THETA | vector Fr | vector Fi | matrix Jr | matrix Ji | matrix 1/Jr | matrix 1/Ji | matrix Marie = 1/Ji*Jr + 1/Jr*Ji | matrix 1/Marie | vector Vince = 1/Ji*Fr + 1/Jr*Fi | vector Vito = 1/Ji*Fi - 1/Jr*Fr | vector DX = 1/Marie * Vince | vector DY = 1/Marie * Vito | 7 | ||||||
8 | = A5 - U5 | = B5 - V5 | =SQRT(A8^2 + B8^2) | =IF(B8<0, -ACOS(A8/C8), ACOS(A8/C8)) | = C8^3*C9^5*COS(3*D8 + 5*D9) - 6*C8*C9*COS(D8 + D9) - 17 | = C8^3*C9^5*SIN(3*D8 + 5*D9) - 6*C8*C9*SIN(D8 + D9) | = 3*C8^2*C9^5*COS(2*D8 + 5*D9) - 6*C9*COS(D9) | = 5*C8^3*C9^4*COS(3*D8 + 4*D9) - 6*C8*COS(D8) | = 3*C8^2*C9^5*SIN(2*D8 + 5*D9) - 6*C9*SIN(D9) | = 5*C8^3*C9^4*SIN(3*D8 + 4*D9) - 6*C8*SIN(D8) | =INDEX(MINVERSE(G8:H9), 1, 1) | =INDEX(MINVERSE(G8:H9), 1, 2) | =INDEX(MINVERSE(I8:J9), 1, 1) | =INDEX(MINVERSE(I8:J9), 1, 2) | =INDEX(MMULT(M8:N9, G8:H9), 1, 1) + INDEX(MMULT(K8:L9, I8:J9), 1, 1) | =INDEX(MMULT(M8:N9, G8:H9), 1, 2) + INDEX(MMULT(K8:L9, I8:J9), 1, 2) | =INDEX(MINVERSE(O8:P9), 1, 1) | =INDEX(MINVERSE(O8:P9), 1, 2) | =INDEX(MMULT(M8:N9, E8:E9), 1, 1) + INDEX(MMULT(K8:L9, F8:F9), 1, 1) | =INDEX(MMULT(M8:N9, F8:F9), 1, 1) - INDEX(MMULT(K8:L9, E8:E9), 1, 1) | =INDEX(MMULT(Q8:R9, S8:S9), 1, 1) | =INDEX(MMULT(Q8:R9, T8:T9), 1, 1) | 8 |
9 | = A6 - U6 | = B6 - V6 | =SQRT(A9^2 + B9^2) | =IF(B9<0, -ACOS(A9/C9), ACOS(A9/C9)) | = 7*C8^2*C9*COS(2*D8 + D9) - 3*C8*C9^3*COS(D8 + 3*D9) + 5 | = 7*C8^2*C9*SIN(2*D8 + D9) - 3*C8*C9^3*SIN(D8 + 3*D9) | = 14*C8*C9*COS(D8 + D9) - 3*C9^3*COS(3*D9) | = 7*C8^2*COS(2*D8) - 9*C8*C9^2*COS(D8 + 2*D9) | = 14*C8*C9*SIN(D8 + D9) - 3*C9^3*SIN(3*D9) | = 7*C8^2*SIN(2*D8) - 9*C8*C9^2*SIN(D8 + 2*D9) | =INDEX(MINVERSE(G8:H9), 2, 1) | =INDEX(MINVERSE(G8:H9), 2, 2) | =INDEX(MINVERSE(I8:J9), 2, 1) | =INDEX(MINVERSE(I8:J9), 2, 2) | =INDEX(MMULT(M8:N9, G8:H9), 2, 1) + INDEX(MMULT(K8:L9, I8:J9), 2, 1) | =INDEX(MMULT(M8:N9, G8:H9), 2, 2) + INDEX(MMULT(K8:L9, I8:J9), 2, 2) | =INDEX(MINVERSE(O8:P9), 2, 1) | =INDEX(MINVERSE(O8:P9), 2, 2) | =INDEX(MMULT(M8:N9, E8:E9), 2, 1) + INDEX(MMULT(K8:L9, F8:F9), 2, 1) | =INDEX(MMULT(M8:N9, F8:F9), 2, 1) - INDEX(MMULT(K8:L9, E8:E9), 2, 1) | =INDEX(MMULT(Q8:R9, S8:S9), 2, 1) | =INDEX(MMULT(Q8:R9, T8:T9), 2, 1) | 9 |
SHEET 4B. Values returned by the above functions.
A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | vector X | vector Y | R | THETA | vector Fr | vector Fi | matrix Jr | matrix Ji | matrix 1/Jr | matrix 1/Ji | matrix Marie = 1/Ji*Jr + 1/Jr*Ji | matrix 1/Marie | vector Vince = 1/Ji*Fr + 1/Jr*Fi | vector Vito = 1/Ji*Fi - 1/Jr*Fr | vector DX = 1/Marie * Vince | vector DY = 1/Marie * Vito | 1 | ||||||
2 | 1 | -2 | 2.2360680 | -1.1071487 | 25 | -34 | 78 | 214 | 18 | -28 | 0.0052 | -0.0743 | 0.0084 | -0.0236 | 3.6119 | 2.0446 | 0.3711 | 0.1075 | 1.6116 | -2.0405 | 0.5110 | -0.5571 | 2 |
3 | 1 | -1 | 1.4142136 | -0.7853982 | -26 | -13 | -8 | 15 | -36 | -10 | 0.0028 | 0.0271 | -0.0303 | -0.0152 | -3.1668 | -7.0605 | -0.1664 | -0.1898 | -0.8099 | 1.8616 | -0.1145 | -0.0138 | 3 |
4 | vector X | vector Y | R | THETA | vector Fr | vector Fi | matrix Jr | matrix Ji | matrix 1/Jr | matrix 1/Ji | matrix Marie = 1/Ji*Jr + 1/Jr*Ji | matrix 1/Marie | vector Vince = 1/Ji*Fr + 1/Jr*Fi | vector Vito = 1/Ji*Fi - 1/Jr*Fr | vector DX = 1/Marie * Vince | vector DY = 1/Marie * Vito | 4 | ||||||
5 | 0.489014027 | -1.442856945 | 1.5234733 | -1.2440256 | 0.6379 | -10.1120 | 43.3812 | 78.6293 | 14.6679 | -20.8000 | 0.0125 | -0.0682 | -0.0084 | -0.0532 | 1.6139 | -1.9179 | 0.3654 | -0.1464 | 0.2766 | -0.1108 | 0.1025 | -0.1585 | 5 |
6 | 1.114485968 | -0.986232727 | 1.4881982 | -0.7244218 | -4.6227 | -2.3883 | -6.6887 | 14.4616 | -21.1175 | 3.2955 | 0.0058 | 0.0376 | -0.0540 | -0.0375 | -2.8018 | -4.7870 | -0.2139 | -0.1232 | -0.0095 | 0.8061 | -0.0580 | -0.0756 | 6 |
7 | vector X | vector Y | R | THETA | vector Fr | vector Fi | matrix Jr | matrix Ji | matrix 1/Jr | matrix 1/Ji | matrix Marie = 1/Ji*Jr + 1/Jr*Ji | matrix 1/Marie | vector Vince = 1/Ji*Fr + 1/Jr*Fi | vector Vito = 1/Ji*Fi - 1/Jr*Fr | vector DX = 1/Marie * Vince | vector DY = 1/Marie * Vito | 7 | ||||||
8 | 0.386532948 | -1.284334111 | 1.3412389 | -1.2784592 | -1.2703 | -1.9698 | 28.4282 | 55.1863 | 21.4933 | -3.5676 | 0.0179 | -0.0804 | 0.0797 | 0.0419 | 3.7608 | 4.3037 | 0.3737 | -0.1119 | -0.1493 | -0.1855 | -8.82E-03 | -2.54E-02 | 8 |
9 | 1.172485927 | -0.910628350 | 1.4845765 | -0.6603492 | -0.5669 | -0.1354 | -6.1139 | 12.2847 | -17.0086 | 6.7837 | 0.0089 | 0.0414 | 0.1998 | 0.2525 | 3.6237 | 14.3777 | -0.0942 | 0.0977 | -0.4201 | -0.3930 | -2.70E-02 | -2.09E-02 | 9 |
10 | vector X | vector Y | R | THETA | vector Fr | vector Fi | matrix Jr | matrix Ji | matrix 1/Jr | matrix 1/Ji | matrix Marie = 1/Ji*Jr + 1/Jr*Ji | matrix 1/Marie | vector Vince = 1/Ji*Fr + 1/Jr*Fi | vector Vito = 1/Ji*Fi - 1/Jr*Fr | vector DX = 1/Marie * Vince | vector DY = 1/Marie * Vito | 10 | ||||||
11 | 0.395349375 | -1.258978595 | 1.3195940 | -1.2665237 | -0.1496 | 0.0794 | 25.6986 | 54.6416 | 25.9262 | 3.6068 | 0.0193 | -0.0888 | 0.0299 | -0.0135 | 2.8242 | 0.8344 | 0.3672 | -0.0620 | -4.35E-03 | 3.21E-03 | -9.89E-04 | 6.18E-04 | 11 |
12 | 1.199483241 | -0.889685430 | 1.4934190 | -0.6381834 | -2.02E-02 | 1.90E-02 | -5.6747 | 11.8796 | -16.6583 | 7.9595 | 0.0092 | 0.0418 | 0.0625 | 0.0973 | 0.5979 | 4.9379 | -0.0445 | 0.2100 | -9.80E-03 | 9.04E-03 | -7.86E-03 | 1.76E-03 | 12 |
13 | vector X | vector Y | R | THETA | vector Fr | vector Fi | matrix Jr | matrix Ji | matrix 1/Jr | matrix 1/Ji | matrix Marie = 1/Ji*Jr + 1/Jr*Ji | matrix 1/Marie | vector Vince = 1/Ji*Fr + 1/Jr*Fi | vector Vito = 1/Ji*Fi - 1/Jr*Fr | vector DX = 1/Marie * Vince | vector DY = 1/Marie * Vito | 13 | ||||||
14 | 0.396338818 | -1.259596682 | 1.3204803 | -1.2659491 | 7.09E-04 | -4.08E-04 | 26.0099 | 55.1444 | 26.1455 | 3.6287 | 0.0192 | -0.0885 | 0.0297 | -0.0135 | 2.8247 | 0.8368 | 0.3673 | -0.0623 | 1.55E-05 | -1.51E-05 | 2.62E-06 | -3.00E-06 | 14 |
15 | 1.201347402 | -0.891440836 | 1.4959620 | -0.6383835 | 1.14E-04 | -4.34E-05 | -5.6635 | 11.9609 | -16.6775 | 8.0034 | 0.0091 | 0.0417 | 0.0618 | 0.0969 | 0.6008 | 4.9352 | -0.0447 | 0.2102 | 4.93E-05 | -4.06E-05 | 9.68E-06 | -7.87E-06 | 15 |
16 | vector X | vector Y | R | THETA | vector Fr | vector Fi | matrix Jr | matrix Ji | matrix 1/Jr | matrix 1/Ji | matrix Marie = 1/Ji*Jr + 1/Jr*Ji | matrix 1/Marie | vector Vince = 1/Ji*Fr + 1/Jr*Fi | vector Vito = 1/Ji*Fi - 1/Jr*Fr | vector DX = 1/Marie * Vince | vector DY = 1/Marie * Vito | 16 | ||||||
17 | 0.396336198 | -1.259593684 | 1.3204767 | -1.2659503 | 1.50E-08 | -1.04E-08 | 26.0084 | 55.1420 | 26.1445 | 3.6287 | 0.0192 | -0.0885 | 0.0297 | -0.0135 | 2.8247 | 0.8368 | 0.3673 | -0.0623 | 2.50E-10 | -3.49E-10 | 2.47E-11 | -7.01E-11 | 17 |
18 | 1.201337727 | -0.891432968 | 1.4959496 | -0.6383831 | 2.71E-09 | -4.55E-10 | -5.6635 | 11.9606 | -16.6775 | 8.0031 | 0.0091 | 0.0417 | 0.0618 | 0.0969 | 0.6008 | 4.9352 | -0.0447 | 0.2102 | 1.08E-09 | -9.34E-10 | 2.15E-10 | -1.81E-10 | 18 |
19 | vector X | vector Y | R | THETA | vector Fr | vector Fi | matrix Jr | matrix Ji | matrix 1/Jr | matrix 1/Ji | matrix Marie = 1/Ji*Jr + 1/Jr*Ji | matrix 1/Marie | vector Vince = 1/Ji*Fr + 1/Jr*Fi | vector Vito = 1/Ji*Fi - 1/Jr*Fr | vector DX = 1/Marie * Vince | vector DY = 1/Marie * Vito | 19 | ||||||
20 | 0.396336198 | -1.259593684 | 1.3204767 | -1.2659503 | 0 | 2.13E-14 | 26.0084 | 55.1420 | 26.1445 | 3.6287 | 0.0192 | -0.0885 | 0.0297 | -0.0135 | 2.8247 | 0.8368 | 0.3673 | -0.0623 | 7.04E-16 | 6.77E-16 | 2.55E-16 | 1.87E-16 | 20 |
21 | 1.201337727 | -0.891432968 | 1.4959496 | -0.6383831 | 0 | -3.33E-15 | -5.6635 | 11.9606 | -16.6775 | 8.0031 | 0.0091 | 0.0417 | 0.0618 | 0.0969 | 0.6008 | 4.9352 | -0.0447 | 0.2102 | 5.47E-17 | 9.95E-16 | -2.00E-17 | 1.79E-16 | 21 |
A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V |
URL http://members.aceweb.com/patrussell/approximations/Sheet_4.htm
Unpublished Work. © Copyright 2001 Pat Russell. Updated April 17, 2009.