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    Butterfly Curve

    Plot

    Plot

    Associated species

    moths and butterflies

    Equations

    x(t) = ((-13/7 sin(8/5 - 10 t) - 161/20 sin(8/5 - 5 t) - 7/3 sin(8/5 - 4 t) - 118 sin(8/5 - t) + 406/11 sin(2 t + 8/5) + 17/5 sin(3 t + 8/5) + 2/5 sin(6 t + 8/5) + 3/4 sin(7 t + 8/5) + 1/12 sin(8 t + 32/7) + 4 sin(9 t + 14/3) + 247) θ(31 π - t) θ(t - 27 π) + (-10/11 sin(3/2 - 10 t) - 11/4 sin(3/2 - 8 t) - 82/3 sin(8/5 - 2 t) + 122 sin(t + 8/5) + 48/7 sin(3 t + 14/3) + 7/8 sin(4 t + 5/3) + 31/3 sin(5 t + 8/5) + 31/8 sin(6 t + 8/5) + 6/7 sin(7 t + 8/5) + 2/3 sin(9 t + 3/2) + 3677/6) θ(27 π - t) θ(t - 23 π) + (-1/2 sin(6/7 - 20 t) - 2/3 sin(4/3 - 18 t) - 2/3 sin(4/5 - 15 t) - 2/3 sin(2/3 - 13 t) - 2/5 sin(4/5 - 12 t) - 27/4 sin(1/2 - 7 t) - 29/4 sin(6/5 - 2 t) + 335/7 sin(t + 8/3) + 36/5 sin(3 t + 5/3) + 13/2 sin(4 t + 7/4) + 47/8 sin(5 t + 1/4) + 76/11 sin(6 t + 13/4) + 4/3 sin(8 t + 5/3) + 2 sin(9 t + 18/5) + 9/5 sin(10 t + 6/7) + 3/2 sin(11 t + 4) + 1/5 sin(14 t + 18/5) + 4/5 sin(16 t + 5/4) + 2/3 sin(17 t + 21/5) + 2/3 sin(19 t + 11/6) + 1/4 sin(21 t + 11/7) + 1/3 sin(22 t + 1/2) + 861/2) θ(23 π - t) θ(t - 19 π) + (-9/5 sin(1 - 19 t) - 4/5 sin(1 - 18 t) - 2 sin(2/3 - 17 t) - 8/3 sin(4/3 - 14 t) + 5895/7 sin(t + 13/4) + 687/8 sin(2 t + 1/4) + 398/5 sin(3 t + 1/3) + 206/7 sin(4 t + 11/3) + 165/2 sin(5 t + 1/2) + 255/8 sin(6 t + 2/3) + 128/5 sin(7 t + 35/9) + 21/2 sin(8 t + 33/8) + 29/5 sin(9 t + 12/11) + 41/4 sin(10 t + 17/4) + 5 sin(11 t + 7/5) + 23/4 sin(12 t + 4/3) + 19/3 sin(13 t + 14/3) + 19/3 sin(15 t + 7/4) + 3/2 sin(16 t + 9/5) + 3/4 sin(20 t + 2/3) + 2164/5) θ(19 π - t) θ(t - 15 π) + (-43/3 sin(3/4 - 11 t) - 48/5 sin(8/7 - 10 t) + 631/3 sin(t + 14/3) + 203/4 sin(2 t + 8/3) + 473/4 sin(3 t + 3/2) + 139/4 sin(4 t + 1) + 237/5 sin(5 t + 32/7) + 535/6 sin(6 t + 11/6) + 1863/19 sin(7 t + 14/3) + 229/5 sin(8 t + 2/3) + 93/4 sin(9 t + 5/4) + 163/9 sin(12 t + 2/5) + 133/5 sin(13 t + 1) + 18 sin(14 t + 5/3) + 41/8 sin(15 t + 17/9) + sin(16 t + 8/5) - 205/3) θ(15 π - t) θ(t - 11 π) + (-46/3 sin(4/3 - 12 t) - 113/4 sin(2/3 - 2 t) - 297/2 sin(3/5 - t) + 223/3 sin(3 t + 1/4) + 167/2 sin(4 t + 34/11) + 851/10 sin(5 t + 22/5) + 557/6 sin(6 t + 11/6) + 44/3 sin(7 t + 19/10) + 114/5 sin(8 t + 9/4) + 133/5 sin(9 t + 7/3) + 80/3 sin(10 t + 19/6) + 16 sin(11 t + 17/4) + 37/3 sin(13 t + 7/8) + 16/3 sin(14 t + 2) + 33/8 sin(15 t + 5/6) + 13/2 sin(16 t + 7/4) + 373/3) θ(11 π - t) θ(t - 7 π) + (-35/3 sin(3/5 - 14 t) - 414/5 sin(1/13 - 6 t) - 129/4 sin(1/4 - 5 t) - 26 sin(7/5 - 4 t) + 647/5 sin(t + 14/5) + 279/5 sin(2 t + 1/7) + 1191/10 sin(3 t + 17/7) + 77/2 sin(7 t + 21/11) + 152/3 sin(8 t + 29/10) + 39 sin(9 t + 4/3) + 53/2 sin(10 t + 3) + 25/4 sin(11 t + 2) + 40/3 sin(12 t + 5/4) + 71/4 sin(13 t + 40/13) + 261/20 sin(15 t + 3/2) + 16/3 sin(16 t + 4/5) + 2951/4) θ(7 π - t) θ(t - 3 π) + (-52/5 sin(5/6 - 9 t) - 578/5 sin(6/7 - 6 t) - 721/18 sin(4/5 - 3 t) + 29/4 sin(10 t) + 1690/7 sin(t + 5/3) + 95 sin(2 t + 9/5) + 38 sin(4 t + 15/4) + 92 sin(5 t + 7/4) + 283/4 sin(7 t + 2) + 161/3 sin(8 t + 10/3) + 47/4 sin(11 t + 8/3) + 114/5 sin(12 t + 23/8) + 41/3 sin(13 t + 7/2) + 14/3 sin(14 t + 5/4) + 58/3 sin(15 t + 1/2) + 41/4 sin(16 t + 17/4) + 2908/3) θ(3 π - t) θ(t + π)) θ(sqrt(sgn(sin(t/2)))) y(t) = ((-5/4 sin(11/7 - 10 t) - 19/3 sin(8/5 - 6 t) - 1321/20 sin(8/5 - 2 t) + 599/4 sin(t + 8/5) + 58/5 sin(3 t + 8/5) + 11/3 sin(4 t + 14/3) + 29/3 sin(5 t + 8/5) + 9/5 sin(7 t + 8/5) + 8/5 sin(8 t + 14/3) + 11/3 sin(9 t + 8/5) + 1559/2) θ(31 π - t) θ(t - 27 π) + (-1/3 sin(3/2 - 10 t) - 8/3 sin(8/5 - 8 t) - 29/14 sin(8/5 - 6 t) - 22/3 sin(8/5 - 4 t) - 589/10 sin(8/5 - 2 t) + 481/3 sin(t + 8/5) + 37/4 sin(3 t + 8/5) + 26/3 sin(5 t + 8/5) + 22/7 sin(7 t + 8/5) + 17/16 sin(9 t + 8/5) + 3874/5) θ(27 π - t) θ(t - 23 π) + (-1/20 sin(7/6 - 22 t) - 1/4 sin(4/3 - 17 t) - 1/3 sin(4/3 - 15 t) - 3/5 sin(3/4 - 13 t) - 15/7 sin(1/2 - 11 t) - 12/5 sin(1/12 - 9 t) + 1001/4 sin(t + 17/4) + 32/3 sin(2 t + 57/14) + 23 sin(3 t + 28/9) + 16/3 sin(4 t + 17/5) + 33/5 sin(5 t + 7/4) + 4 sin(6 t + 8/3) + 23/5 sin(7 t + 3/4) + 2 sin(8 t + 27/13) + 2/5 sin(10 t + 8/5) + 1/2 sin(12 t + 15/4) + 10/9 sin(14 t + 39/10) + 7/8 sin(16 t + 32/11) + 3/4 sin(18 t + 7/3) + 1/3 sin(19 t + 3/2) + 1/3 sin(20 t + 5/3) + 2/5 sin(21 t + 4/5) + 1415/6) θ(23 π - t) θ(t - 19 π) + (-59/4 sin(1/18 - 13 t) - 39/5 sin(1/6 - 12 t) - 55/3 sin(1/4 - 11 t) - 15 sin(1/3 - 10 t) - 13/5 sin(1/9 - 9 t) - 85/4 sin(3/5 - 8 t) - 22/5 sin(2/3 - 7 t) - 591/4 sin(4/3 - 2 t) - 536 sin(3/2 - t) + 295 sin(3 t + 15/8) + 27/2 sin(4 t + 7/4) + 209/4 sin(5 t + 2) + 201/8 sin(6 t + 11/5) + 21/2 sin(14 t + 1/10) + 35/6 sin(15 t + 1/3) + 14/3 sin(16 t + 3/4) + 13 sin(17 t + 1/3) + 2 sin(18 t + 6/7) + 36/5 sin(19 t + 2/3) + 14/3 sin(20 t + 1) + 775/2) θ(19 π - t) θ(t - 15 π) + (-76/5 sin(1/3 - 15 t) - 4 sin(5/4 - 13 t) - 31/3 sin(7/5 - 12 t) - 49/3 sin(12/11 - 11 t) - 122/5 sin(1/2 - 8 t) - 526/7 sin(5/6 - 6 t) - 561/8 sin(1 - 3 t) - 55/2 sin(2/5 - 2 t) + 1027/5 sin(t + 6/7) + 323/5 sin(4 t + 31/10) + 215/6 sin(5 t + 4/3) + 71/3 sin(7 t + 7/3) + 47/4 sin(9 t + 17/5) + 85/4 sin(10 t + 13/3) + 72/5 sin(14 t + 14/3) + 87/8 sin(16 t + 4/3) + 3168/5) θ(15 π - t) θ(t - 11 π) + (-35/6 sin(3/4 - 13 t) - 7 sin(7/5 - 12 t) - 46/3 sin(8/7 - 8 t) - 181/4 sin(1/8 - 4 t) - 327/4 sin(1/7 - 2 t) + 320/3 sin(t + 23/6) + 169/5 sin(3 t + 15/4) + 347/4 sin(5 t + 29/7) + 239/3 sin(6 t + 12/5) + 47/4 sin(7 t + 8/3) + 35/4 sin(9 t + 5/2) + 35/2 sin(10 t + 18/5) + 118/7 sin(11 t + 23/6) + 10 sin(14 t + 7/4) + 25/4 sin(15 t + 32/11) + 39/5 sin(16 t + 7/4) + 442/5) θ(11 π - t) θ(t - 7 π) + (-12 sin(1/5 - 12 t) - 61/2 sin(1/2 - 11 t) - 804/11 sin(3/2 - 3 t) + 167/3 sin(2 t) + 275/4 sin(8 t) + 29/4 sin(15 t) + 373/3 sin(t + 19/5) + 66/5 sin(4 t + 5/2) + 161/3 sin(5 t + 9/4) + 102/5 sin(6 t + 8/3) + 143/4 sin(7 t + 1/12) + 222/5 sin(9 t + 22/5) + 47/2 sin(10 t + 2/3) + 7/5 sin(13 t + 3/2) + 7 sin(14 t + 7/6) + 181/10 sin(16 t + 10/3) + 223/3) θ(7 π - t) θ(t - 3 π) + (-31/6 sin(1/6 - 15 t) - 12 sin(8/5 - 14 t) - 541/10 sin(11/10 - 6 t) + 640/3 sin(t + 15/14) + 306/5 sin(2 t + 3/2) + 221/4 sin(3 t + 1/4) + 131/3 sin(4 t + 8/3) + 47/3 sin(5 t + 13/7) + 38 sin(7 t + 18/7) + 181/5 sin(8 t + 2/3) + 49/4 sin(9 t + 3/4) + 103/6 sin(10 t + 22/5) + 43/5 sin(11 t + 3/4) + 14/3 sin(12 t + 1/5) + 55/3 sin(13 t + 4/3) + 34/7 sin(16 t + 8/3) + 3182/5) θ(3 π - t) θ(t + π)) θ(sqrt(sgn(sin(t/2))))

    Property

    animal curves

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