Curvature and torsion of this curve

i was trying to get curvature and torsion of curve in mathematica. of this curve

r[t_]:= {t*Sqrt[2]*Sin[Pi*t]*(Sin[Pi/2*t])^2,((1-t)*Sqrt[2]*Sin[Pi*t]*(Sin[Pi/2*t])^2)+(t*Sqrt[2]*Sin[Pi*t]*(Cos[Pi/2*t])^2),(1-t)*Sqrt[2]*Sin[Pi*t]*(Cos[Pi/2*t])^2}

Simplify[FrenetSerretSystem[r[t],t]]


i wanted to get it in this way. but it is giving me strange result. can anyone help me to get it's curvature curve in any kind of way and it's plot?

• I do get a very complicated solution but it is truncated because it is too long, which error do you get? May 22, 2022 at 8:22
• Things do simplify a bit if you add assumptions to Simplify. For example, the curvatures: {c1, c2} = FrenetSerretSystem[r[t], t][[1]] And then FullSimplify[c1, Assumptions -> t > 0] gives you something that will fit on one screen. May 22, 2022 at 9:37
• craig carter , i get this error May 22, 2022 at 9:45
• craig carter , i get this error Assumptions - cannot be followed by > t>0 May 22, 2022 at 9:46
• can you help me plot the curvature and function. i was simulating a paper and my result was not following the paper. May 22, 2022 at 9:48

Since the plot of the curve is weird,

r[t_]:={t*Sqrt[2]*Sin[Pi*t]*(Sin[Pi/2*t])^2,((1-t)*Sqrt[2]*Sin[Pi*t]*(Sin[Pi/2*t])^2)+(t*Sqrt[2]*Sin[Pi*t]*(Cos[Pi/2*t])^2),(1-t)*Sqrt[2]*Sin[Pi*t]*(Cos[Pi/2*t])^2};
ParametricPlot3D[r[t], {t, 0, 2*Pi}, PlotStyle -> Thick]


, nothing strange that the result of

Simplify[FrenetSerretSystem[r[t], t]]


{{(2 Sqrt[ 2] \[Pi] \[Sqrt](240 + 591 \[Pi]^2 + 80 \[Pi]^4 - 492 \[Pi]^2 t - 320 \[Pi]^4 t + 492 \[Pi]^2 t^2 + 560 \[Pi]^4 t^2 - 480 \[Pi]^4 t^3 + 240 \[Pi]^4 t^4 + 126 \[Pi]^2 (-1 + 2 t) Cos[\[Pi] t] - 1/2 (720 + \[Pi]^2 (-325 - 972 t + 972 t^2) + 48 \[Pi]^4 (1 - 4 t + 7 t^2 - 6 t^3 + 3 t^4)) Cos[ 2 \[Pi] t] + 120 \[Pi]^2 Cos[3 \[Pi] t] - 240 \[Pi]^2 t Cos[3 \[Pi] t] + 144 Cos[4 \[Pi] t] + 195 \[Pi]^2 Cos[4 \[Pi] t] - 48 \[Pi]^4 Cos[4 \[Pi] t] + 108 \[Pi]^2 t Cos[4 \[Pi] t] + 192 \[Pi]^4 t Cos[4 \[Pi] t] - 108 \[Pi]^2 t^2 Cos[4 \[Pi] t] - 336 \[Pi]^4 t^2 Cos[4 \[Pi] t] + 288 \[Pi]^4 t^3 Cos[4 \[Pi] t] - 144 \[Pi]^4 t^4 Cos[4 \[Pi] t] - 12 \[Pi]^2 Cos[5 \[Pi] t] + 24 \[Pi]^2 t Cos[5 \[Pi] t] - 24 Cos[6 \[Pi] t] + 139/2 \[Pi]^2 Cos[6 \[Pi] t] - 8 \[Pi]^4 Cos[6 \[Pi] t] - 102 \[Pi]^2 t Cos[6 \[Pi] t] + 32 \[Pi]^4 t Cos[6 \[Pi] t] + 102 \[Pi]^2 t^2 Cos[6 \[Pi] t] - 56 \[Pi]^4 t^2 Cos[6 \[Pi] t] + 48 \[Pi]^4 t^3 Cos[6 \[Pi] t] - 24 \[Pi]^4 t^4 Cos[6 \[Pi] t] + 18 \[Pi]^2 Cos[7 \[Pi] t] - 36 \[Pi]^2 t Cos[7 \[Pi] t] + 6 \[Pi]^2 Cos[8 \[Pi] t] + 568 \[Pi] Sin[\[Pi] t] + 264 \[Pi]^3 Sin[\[Pi] t] - 536 \[Pi]^3 t Sin[\[Pi] t] + 536 \[Pi]^3 t^2 Sin[\[Pi] t] - 180 \[Pi] Sin[2 \[Pi] t] + 4 \[Pi]^3 Sin[2 \[Pi] t] + 360 \[Pi] t Sin[2 \[Pi] t] + 244 \[Pi]^3 t Sin[2 \[Pi] t] - 756 \[Pi]^3 t^2 Sin[2 \[Pi] t] + 504 \[Pi]^3 t^3 Sin[2 \[Pi] t] - 240 \[Pi] Sin[3 \[Pi] t] + 96 \[Pi]^3 Sin[3 \[Pi] t] - 144 \[Pi]^3 t Sin[3 \[Pi] t] + 144 \[Pi]^3 t^2 Sin[3 \[Pi] t] + 144 \[Pi] Sin[4 \[Pi] t] + 136 \[Pi]^3 Sin[4 \[Pi] t] - 288 \[Pi] t Sin[4 \[Pi] t] - 344 \[Pi]^3 t Sin[4 \[Pi] t] + 216 \[Pi]^3 t^2 Sin[4 \[Pi] t] - 144 \[Pi]^3 t^3 Sin[4 \[Pi] t] + 64 \[Pi] Sin[5 \[Pi] t] + 32 \[Pi]^3 Sin[5 \[Pi] t] - 80 \[Pi]^3 t Sin[5 \[Pi] t] + 80 \[Pi]^3 t^2 Sin[5 \[Pi] t] - 36 \[Pi] Sin[6 \[Pi] t] + 36 \[Pi]^3 Sin[6 \[Pi] t] + 72 \[Pi] t Sin[6 \[Pi] t] - 108 \[Pi]^3 t Sin[6 \[Pi] t] + 108 \[Pi]^3 t^2 Sin[6 \[Pi] t] - 72 \[Pi]^3 t^3 Sin[6 \[Pi] t] - 24 \[Pi] Sin[7 \[Pi] t] + 8 \[Pi]^3 Sin[7 \[Pi] t] - 24 \[Pi]^3 t Sin[7 \[Pi] t] + 24 \[Pi]^3 t^2 Sin[7 \[Pi] t]))/(7 + 8 \[Pi]^2 - 16 \[Pi]^2 t + 16 \[Pi]^2 t^2 + 4 (-1 + \[Pi]^2 (1 - t + t^2)) Cos[ 2 \[Pi] t] + (-3 + 4 \[Pi]^2 (1 - 3 t + 3 t^2)) Cos[ 4 \[Pi] t] + 6 \[Pi] Sin[\[Pi] t] - 4 \[Pi] Sin[2 \[Pi] t] + 8 \[Pi] t Sin[2 \[Pi] t] - 2 \[Pi] Sin[3 \[Pi] t] - 6 \[Pi] Sin[4 \[Pi] t] + 12 \[Pi] t Sin[4 \[Pi] t])^( 3/2), (16 Sqrt[ 2] \[Pi]^2 (9 \[Pi] - 18 \[Pi] t - 34 \[Pi] Cos[\[Pi] t] - 14 \[Pi] (-1 + 2 t) Cos[2 \[Pi] t] + 9 \[Pi] Cos[3 \[Pi] t] + \[Pi] Cos[4 \[Pi] t] - 2 \[Pi] t Cos[4 \[Pi] t] + \[Pi] Cos[5 \[Pi] t] - 62 Sin[2 \[Pi] t] - 5 Sin[4 \[Pi] t]))/(480 + 1182 \[Pi]^2 + 160 \[Pi]^4 - 984 \[Pi]^2 t - 640 \[Pi]^4 t + 984 \[Pi]^2 t^2 + 1120 \[Pi]^4 t^2 - 960 \[Pi]^4 t^3 + 480 \[Pi]^4 t^4 + 252 \[Pi]^2 (-1 + 2 t) Cos[\[Pi] t] + (-720 + \[Pi]^2 (325 + 972 t - 972 t^2) - 48 \[Pi]^4 (1 - 4 t + 7 t^2 - 6 t^3 + 3 t^4)) Cos[ 2 \[Pi] t] + 240 \[Pi]^2 Cos[3 \[Pi] t] - 480 \[Pi]^2 t Cos[3 \[Pi] t] + 288 Cos[4 \[Pi] t] + 390 \[Pi]^2 Cos[4 \[Pi] t] - 96 \[Pi]^4 Cos[4 \[Pi] t] + 216 \[Pi]^2 t Cos[4 \[Pi] t] + 384 \[Pi]^4 t Cos[4 \[Pi] t] - 216 \[Pi]^2 t^2 Cos[4 \[Pi] t] - 672 \[Pi]^4 t^2 Cos[4 \[Pi] t] + 576 \[Pi]^4 t^3 Cos[4 \[Pi] t] - 288 \[Pi]^4 t^4 Cos[4 \[Pi] t] - 24 \[Pi]^2 Cos[5 \[Pi] t] + 48 \[Pi]^2 t Cos[5 \[Pi] t] - 48 Cos[6 \[Pi] t] + 139 \[Pi]^2 Cos[6 \[Pi] t] - 16 \[Pi]^4 Cos[6 \[Pi] t] - 204 \[Pi]^2 t Cos[6 \[Pi] t] + 64 \[Pi]^4 t Cos[6 \[Pi] t] + 204 \[Pi]^2 t^2 Cos[6 \[Pi] t] - 112 \[Pi]^4 t^2 Cos[6 \[Pi] t] + 96 \[Pi]^4 t^3 Cos[6 \[Pi] t] - 48 \[Pi]^4 t^4 Cos[6 \[Pi] t] + 36 \[Pi]^2 Cos[7 \[Pi] t] - 72 \[Pi]^2 t Cos[7 \[Pi] t] + 12 \[Pi]^2 Cos[8 \[Pi] t] + 1136 \[Pi] Sin[\[Pi] t] + 528 \[Pi]^3 Sin[\[Pi] t] - 1072 \[Pi]^3 t Sin[\[Pi] t] + 1072 \[Pi]^3 t^2 Sin[\[Pi] t] - 360 \[Pi] Sin[2 \[Pi] t] + 8 \[Pi]^3 Sin[2 \[Pi] t] + 720 \[Pi] t Sin[2 \[Pi] t] + 488 \[Pi]^3 t Sin[2 \[Pi] t] - 1512 \[Pi]^3 t^2 Sin[2 \[Pi] t] + 1008 \[Pi]^3 t^3 Sin[2 \[Pi] t] - 480 \[Pi] Sin[3 \[Pi] t] + 192 \[Pi]^3 Sin[3 \[Pi] t] - 288 \[Pi]^3 t Sin[3 \[Pi] t] + 288 \[Pi]^3 t^2 Sin[3 \[Pi] t] + 288 \[Pi] Sin[4 \[Pi] t] + 272 \[Pi]^3 Sin[4 \[Pi] t] - 576 \[Pi] t Sin[4 \[Pi] t] - 688 \[Pi]^3 t Sin[4 \[Pi] t] + 432 \[Pi]^3 t^2 Sin[4 \[Pi] t] - 288 \[Pi]^3 t^3 Sin[4 \[Pi] t] + 128 \[Pi] Sin[5 \[Pi] t] + 64 \[Pi]^3 Sin[5 \[Pi] t] - 160 \[Pi]^3 t Sin[5 \[Pi] t] + 160 \[Pi]^3 t^2 Sin[5 \[Pi] t] - 72 \[Pi] Sin[6 \[Pi] t] + 72 \[Pi]^3 Sin[6 \[Pi] t] + 144 \[Pi] t Sin[6 \[Pi] t] - 216 \[Pi]^3 t Sin[6 \[Pi] t] + 216 \[Pi]^3 t^2 Sin[6 \[Pi] t] - 144 \[Pi]^3 t^3 Sin[6 \[Pi] t] - 48 \[Pi] Sin[7 \[Pi] t] + 16 \[Pi]^3 Sin[7 \[Pi] t] - 48 \[Pi]^3 t Sin[7 \[Pi] t] + 48 \[Pi]^3 t^2 Sin[7 \[Pi] t])}, {{(4 Sin[(\[Pi] t)/ 2]^2 (\[Pi] t + 2 \[Pi] t Cos[\[Pi] t] + Sin[\[Pi] t]))/(\[Sqrt](7 + 8 \[Pi]^2 - 16 \[Pi]^2 t + 16 \[Pi]^2 t^2 + 4 (-1 + \[Pi]^2 (1 - t + t^2)) Cos[ 2 \[Pi] t] + (-3 + 4 \[Pi]^2 (1 - 3 t + 3 t^2)) Cos[ 4 \[Pi] t] + 6 \[Pi] Sin[\[Pi] t] - 4 \[Pi] Sin[2 \[Pi] t] + 8 \[Pi] t Sin[2 \[Pi] t] - 2 \[Pi] Sin[3 \[Pi] t] - 6 \[Pi] Sin[4 \[Pi] t] + 12 \[Pi] t Sin[ 4 \[Pi] t])), (2 (\[Pi] Cos[\[Pi] t] + \[Pi] (-1 + 2 t) Cos[ 2 \[Pi] t] + Sin[2 \[Pi] t]))/(\[Sqrt](7 + 8 \[Pi]^2 - 16 \[Pi]^2 t + 16 \[Pi]^2 t^2 + 4 (-1 + \[Pi]^2 (1 - t + t^2)) Cos[ 2 \[Pi] t] + (-3 + 4 \[Pi]^2 (1 - 3 t + 3 t^2)) Cos[ 4 \[Pi] t] + 6 \[Pi] Sin[\[Pi] t] - 4 \[Pi] Sin[2 \[Pi] t] + 8 \[Pi] t Sin[2 \[Pi] t] - 2 \[Pi] Sin[3 \[Pi] t] - 6 \[Pi] Sin[4 \[Pi] t] + 12 \[Pi] t Sin[4 \[Pi] t])), -((4 Cos[(\[Pi] t)/ 2]^2 (\[Pi] - \[Pi] t + 2 \[Pi] (-1 + t) Cos[\[Pi] t] + Sin[\[Pi] t]))/(\[Sqrt](7 + 8 \[Pi]^2 - 16 \[Pi]^2 t + 16 \[Pi]^2 t^2 + 4 (-1 + \[Pi]^2 (1 - t + t^2)) Cos[ 2 \[Pi] t] + (-3 + 4 \[Pi]^2 (1 - 3 t + 3 t^2)) Cos[ 4 \[Pi] t] + 6 \[Pi] Sin[\[Pi] t] - 4 \[Pi] Sin[2 \[Pi] t] + 8 \[Pi] t Sin[2 \[Pi] t] - 2 \[Pi] Sin[3 \[Pi] t] - 6 \[Pi] Sin[4 \[Pi] t] + 12 \[Pi] t Sin[4 \[Pi] t])))}, {(Sin[(\[Pi] t)/ 2] (2 \[Pi] (9 + 4 (1 + 7 \[Pi]^2) t - 44 \[Pi]^2 t^2 + 44 \[Pi]^2 t^3) Cos[(\[Pi] t)/2] + 4 \[Pi] (-6 + 17 t + 18 \[Pi]^2 t - 34 \[Pi]^2 t^2 + 34 \[Pi]^2 t^3) Cos[(3 \[Pi] t)/2] - 54 \[Pi] Cos[(5 \[Pi] t)/2] + 28 \[Pi] t Cos[(5 \[Pi] t)/2] + 40 \[Pi]^3 t Cos[(5 \[Pi] t)/2] - 104 \[Pi]^3 t^2 Cos[(5 \[Pi] t)/2] + 104 \[Pi]^3 t^3 Cos[(5 \[Pi] t)/2] + 31 \[Pi] Cos[(7 \[Pi] t)/2] - 62 \[Pi] t Cos[(7 \[Pi] t)/2] + 16 \[Pi]^3 t Cos[(7 \[Pi] t)/2] - 32 \[Pi]^3 t^2 Cos[(7 \[Pi] t)/2] + 32 \[Pi]^3 t^3 Cos[(7 \[Pi] t)/2] + 23 \[Pi] Cos[(9 \[Pi] t)/2] - 42 \[Pi] t Cos[(9 \[Pi] t)/2] + 8 \[Pi]^3 t Cos[(9 \[Pi] t)/2] - 24 \[Pi]^3 t^2 Cos[(9 \[Pi] t)/2] + 24 \[Pi]^3 t^3 Cos[(9 \[Pi] t)/2] + 6 \[Pi] Cos[(11 \[Pi] t)/2] + 24 Sin[(\[Pi] t)/2] - 20 \[Pi]^2 Sin[(\[Pi] t)/2] - 26 \[Pi]^2 t Sin[(\[Pi] t)/2] + 36 \[Pi]^2 t^2 Sin[(\[Pi] t)/2] + 48 Sin[(3 \[Pi] t)/2] + 68 \[Pi]^2 Sin[(3 \[Pi] t)/2] - 110 \[Pi]^2 t Sin[(3 \[Pi] t)/2] + 84 \[Pi]^2 t^2 Sin[(3 \[Pi] t)/2] + 16 Sin[(5 \[Pi] t)/2] - 20 \[Pi]^2 Sin[(5 \[Pi] t)/2] + 14 \[Pi]^2 t Sin[(5 \[Pi] t)/2] + 68 \[Pi]^2 t^2 Sin[(5 \[Pi] t)/2] - 20 Sin[(7 \[Pi] t)/2] + 20 \[Pi]^2 Sin[(7 \[Pi] t)/2] - 56 \[Pi]^2 t Sin[(7 \[Pi] t)/2] + 68 \[Pi]^2 t^2 Sin[(7 \[Pi] t)/2] - 12 Sin[(9 \[Pi] t)/2] + 8 \[Pi]^2 Sin[(9 \[Pi] t)/2] - 38 \[Pi]^2 t Sin[(9 \[Pi] t)/2] + 48 \[Pi]^2 t^2 Sin[(9 \[Pi] t)/2] + 8 \[Pi]^2 Sin[(11 \[Pi] t)/2] - 12 \[Pi]^2 t Sin[(11 \[Pi] t)/2]))/(\[Sqrt](7/2 + 4 \[Pi]^2 - 8 \[Pi]^2 t + 8 \[Pi]^2 t^2 + 2 (-1 + \[Pi]^2 (1 - t + t^2)) Cos[2 \[Pi] t] + 1/2 (-3 + 4 \[Pi]^2 (1 - 3 t + 3 t^2)) Cos[4 \[Pi] t] + 3 \[Pi] Sin[\[Pi] t] - 2 \[Pi] Sin[2 \[Pi] t] + 4 \[Pi] t Sin[2 \[Pi] t] - \[Pi] Sin[3 \[Pi] t] - 3 \[Pi] Sin[4 \[Pi] t] + 6 \[Pi] t Sin[4 \[Pi] t]) \[Sqrt](480 + 1182 \[Pi]^2 + 160 \[Pi]^4 - 984 \[Pi]^2 t - 640 \[Pi]^4 t + 984 \[Pi]^2 t^2 + 1120 \[Pi]^4 t^2 - 960 \[Pi]^4 t^3 + 480 \[Pi]^4 t^4 + 252 \[Pi]^2 (-1 + 2 t) Cos[\[Pi] t] + (-720 + \[Pi]^2 (325 + 972 t - 972 t^2) - 48 \[Pi]^4 (1 - 4 t + 7 t^2 - 6 t^3 + 3 t^4)) Cos[ 2 \[Pi] t] + 240 \[Pi]^2 Cos[3 \[Pi] t] - 480 \[Pi]^2 t Cos[3 \[Pi] t] + 288 Cos[4 \[Pi] t] + 390 \[Pi]^2 Cos[4 \[Pi] t] - 96 \[Pi]^4 Cos[4 \[Pi] t] + 216 \[Pi]^2 t Cos[4 \[Pi] t] + 384 \[Pi]^4 t Cos[4 \[Pi] t] - 216 \[Pi]^2 t^2 Cos[4 \[Pi] t] - 672 \[Pi]^4 t^2 Cos[4 \[Pi] t] + 576 \[Pi]^4 t^3 Cos[4 \[Pi] t] - 288 \[Pi]^4 t^4 Cos[4 \[Pi] t] - 24 \[Pi]^2 Cos[5 \[Pi] t] + 48 \[Pi]^2 t Cos[5 \[Pi] t] - 48 Cos[6 \[Pi] t] + 139 \[Pi]^2 Cos[6 \[Pi] t] - 16 \[Pi]^4 Cos[6 \[Pi] t] - 204 \[Pi]^2 t Cos[6 \[Pi] t] + 64 \[Pi]^4 t Cos[6 \[Pi] t] + 204 \[Pi]^2 t^2 Cos[6 \[Pi] t] - 112 \[Pi]^4 t^2 Cos[6 \[Pi] t] + 96 \[Pi]^4 t^3 Cos[6 \[Pi] t] - 48 \[Pi]^4 t^4 Cos[6 \[Pi] t] + 36 \[Pi]^2 Cos[7 \[Pi] t] - 72 \[Pi]^2 t Cos[7 \[Pi] t] + 12 \[Pi]^2 Cos[8 \[Pi] t] + 1136 \[Pi] Sin[\[Pi] t] + 528 \[Pi]^3 Sin[\[Pi] t] - 1072 \[Pi]^3 t Sin[\[Pi] t] + 1072 \[Pi]^3 t^2 Sin[\[Pi] t] - 360 \[Pi] Sin[2 \[Pi] t] + 8 \[Pi]^3 Sin[2 \[Pi] t] + 720 \[Pi] t Sin[2 \[Pi] t] + 488 \[Pi]^3 t Sin[2 \[Pi] t] - 1512 \[Pi]^3 t^2 Sin[2 \[Pi] t] + 1008 \[Pi]^3 t^3 Sin[2 \[Pi] t] - 480 \[Pi] Sin[3 \[Pi] t] + 192 \[Pi]^3 Sin[3 \[Pi] t] - 288 \[Pi]^3 t Sin[3 \[Pi] t] + 288 \[Pi]^3 t^2 Sin[3 \[Pi] t] + 288 \[Pi] Sin[4 \[Pi] t] + 272 \[Pi]^3 Sin[4 \[Pi] t] - 576 \[Pi] t Sin[4 \[Pi] t] - 688 \[Pi]^3 t Sin[4 \[Pi] t] + 432 \[Pi]^3 t^2 Sin[4 \[Pi] t] - 288 \[Pi]^3 t^3 Sin[4 \[Pi] t] + 128 \[Pi] Sin[5 \[Pi] t] + 64 \[Pi]^3 Sin[5 \[Pi] t] - 160 \[Pi]^3 t Sin[5 \[Pi] t] + 160 \[Pi]^3 t^2 Sin[5 \[Pi] t] - 72 \[Pi] Sin[6 \[Pi] t] + 72 \[Pi]^3 Sin[6 \[Pi] t] + 144 \[Pi] t Sin[6 \[Pi] t] - 216 \[Pi]^3 t Sin[6 \[Pi] t] + 216 \[Pi]^3 t^2 Sin[6 \[Pi] t] - 144 \[Pi]^3 t^3 Sin[6 \[Pi] t] - 48 \[Pi] Sin[7 \[Pi] t] + 16 \[Pi]^3 Sin[7 \[Pi] t] - 48 \[Pi]^3 t Sin[7 \[Pi] t] + 48 \[Pi]^3 t^2 Sin[7 \[Pi] t])), (-24 - 3 \[Pi]^2 + 36 \[Pi]^2 t - 36 \[Pi]^2 t^2 - 34 \[Pi]^2 (-1 + 2 t) Cos[\[Pi] t] + 2 (16 + \[Pi]^2 (25 - 8 t + 8 t^2)) Cos[2 \[Pi] t] + 17 \[Pi]^2 Cos[3 \[Pi] t] - 34 \[Pi]^2 t Cos[3 \[Pi] t] - 8 Cos[4 \[Pi] t] + 15 \[Pi]^2 Cos[4 \[Pi] t] - 20 \[Pi]^2 t Cos[4 \[Pi] t] + 20 \[Pi]^2 t^2 Cos[4 \[Pi] t] + 13 \[Pi]^2 Cos[5 \[Pi] t] - 26 \[Pi]^2 t Cos[5 \[Pi] t] + 2 \[Pi]^2 Cos[6 \[Pi] t] - 64 \[Pi] Sin[\[Pi] t] - 8 \[Pi]^3 Sin[\[Pi] t] + 24 \[Pi]^3 t Sin[\[Pi] t] - 24 \[Pi]^3 t^2 Sin[\[Pi] t] + 20 \[Pi] Sin[2 \[Pi] t] + 16 \[Pi]^3 Sin[2 \[Pi] t] - 40 \[Pi] t Sin[2 \[Pi] t] - 48 \[Pi]^3 t Sin[2 \[Pi] t] + 48 \[Pi]^3 t^2 Sin[2 \[Pi] t] - 32 \[Pi]^3 t^3 Sin[2 \[Pi] t] + 2 \[Pi] Sin[3 \[Pi] t] + 12 \[Pi]^3 Sin[3 \[Pi] t] - 36 \[Pi]^3 t Sin[3 \[Pi] t] + 36 \[Pi]^3 t^2 Sin[3 \[Pi] t] - 10 \[Pi] Sin[4 \[Pi] t] + 4 \[Pi]^3 Sin[4 \[Pi] t] + 20 \[Pi] t Sin[4 \[Pi] t] - 12 \[Pi]^3 t Sin[4 \[Pi] t] + 12 \[Pi]^3 t^2 Sin[4 \[Pi] t] - 8 \[Pi]^3 t^3 Sin[4 \[Pi] t] - 14 \[Pi] Sin[5 \[Pi] t] + 4 \[Pi]^3 Sin[5 \[Pi] t] - 12 \[Pi]^3 t Sin[5 \[Pi] t] + 12 \[Pi]^3 t^2 Sin[ 5 \[Pi] t])/(\[Sqrt](7/2 + 4 \[Pi]^2 - 8 \[Pi]^2 t + 8 \[Pi]^2 t^2 + 2 (-1 + \[Pi]^2 (1 - t + t^2)) Cos[2 \[Pi] t] + 1/2 (-3 + 4 \[Pi]^2 (1 - 3 t + 3 t^2)) Cos[4 \[Pi] t] + 3 \[Pi] Sin[\[Pi] t] - 2 \[Pi] Sin[2 \[Pi] t] + 4 \[Pi] t Sin[2 \[Pi] t] - \[Pi] Sin[3 \[Pi] t] - 3 \[Pi] Sin[4 \[Pi] t] + 6 \[Pi] t Sin[4 \[Pi] t]) \[Sqrt](480 + 1182 \[Pi]^2 + 160 \[Pi]^4 - 984 \[Pi]^2 t - 640 \[Pi]^4 t + 984 \[Pi]^2 t^2 + 1120 \[Pi]^4 t^2 - 960 \[Pi]^4 t^3 + 480 \[Pi]^4 t^4 + 252 \[Pi]^2 (-1 + 2 t) Cos[\[Pi] t] + (-720 + \[Pi]^2 (325 + 972 t - 972 t^2) - 48 \[Pi]^4 (1 - 4 t + 7 t^2 - 6 t^3 + 3 t^4)) Cos[ 2 \[Pi] t] + 240 \[Pi]^2 Cos[3 \[Pi] t] - 480 \[Pi]^2 t Cos[3 \[Pi] t] + 288 Cos[4 \[Pi] t] + 390 \[Pi]^2 Cos[4 \[Pi] t] - 96 \[Pi]^4 Cos[4 \[Pi] t] + 216 \[Pi]^2 t Cos[4 \[Pi] t] + 384 \[Pi]^4 t Cos[4 \[Pi] t] - 216 \[Pi]^2 t^2 Cos[4 \[Pi] t] - 672 \[Pi]^4 t^2 Cos[4 \[Pi] t] + 576 \[Pi]^4 t^3 Cos[4 \[Pi] t] - 288 \[Pi]^4 t^4 Cos[4 \[Pi] t] - 24 \[Pi]^2 Cos[5 \[Pi] t] + 48 \[Pi]^2 t Cos[5 \[Pi] t] - 48 Cos[6 \[Pi] t] + 139 \[Pi]^2 Cos[6 \[Pi] t] - 16 \[Pi]^4 Cos[6 \[Pi] t] - 204 \[Pi]^2 t Cos[6 \[Pi] t] + 64 \[Pi]^4 t Cos[6 \[Pi] t] + 204 \[Pi]^2 t^2 Cos[6 \[Pi] t] - 112 \[Pi]^4 t^2 Cos[6 \[Pi] t] + 96 \[Pi]^4 t^3 Cos[6 \[Pi] t] - 48 \[Pi]^4 t^4 Cos[6 \[Pi] t] + 36 \[Pi]^2 Cos[7 \[Pi] t] - 72 \[Pi]^2 t Cos[7 \[Pi] t] + 12 \[Pi]^2 Cos[8 \[Pi] t] + 1136 \[Pi] Sin[\[Pi] t] + 528 \[Pi]^3 Sin[\[Pi] t] - 1072 \[Pi]^3 t Sin[\[Pi] t] + 1072 \[Pi]^3 t^2 Sin[\[Pi] t] - 360 \[Pi] Sin[2 \[Pi] t] + 8 \[Pi]^3 Sin[2 \[Pi] t] + 720 \[Pi] t Sin[2 \[Pi] t] + 488 \[Pi]^3 t Sin[2 \[Pi] t] - 1512 \[Pi]^3 t^2 Sin[2 \[Pi] t] + 1008 \[Pi]^3 t^3 Sin[2 \[Pi] t] - 480 \[Pi] Sin[3 \[Pi] t] + 192 \[Pi]^3 Sin[3 \[Pi] t] - 288 \[Pi]^3 t Sin[3 \[Pi] t] + 288 \[Pi]^3 t^2 Sin[3 \[Pi] t] + 288 \[Pi] Sin[4 \[Pi] t] + 272 \[Pi]^3 Sin[4 \[Pi] t] - 576 \[Pi] t Sin[4 \[Pi] t] - 688 \[Pi]^3 t Sin[4 \[Pi] t] + 432 \[Pi]^3 t^2 Sin[4 \[Pi] t] - 288 \[Pi]^3 t^3 Sin[4 \[Pi] t] + 128 \[Pi] Sin[5 \[Pi] t] + 64 \[Pi]^3 Sin[5 \[Pi] t] - 160 \[Pi]^3 t Sin[5 \[Pi] t] + 160 \[Pi]^3 t^2 Sin[5 \[Pi] t] - 72 \[Pi] Sin[6 \[Pi] t] + 72 \[Pi]^3 Sin[6 \[Pi] t] + 144 \[Pi] t Sin[6 \[Pi] t] - 216 \[Pi]^3 t Sin[6 \[Pi] t] + 216 \[Pi]^3 t^2 Sin[6 \[Pi] t] - 144 \[Pi]^3 t^3 Sin[6 \[Pi] t] - 48 \[Pi] Sin[7 \[Pi] t] + 16 \[Pi]^3 Sin[7 \[Pi] t] - 48 \[Pi]^3 t Sin[7 \[Pi] t] + 48 \[Pi]^3 t^2 Sin[7 \[Pi] t])), -((Cos[(\[Pi] t)/ 2] ((-24 + \[Pi]^2 (10 + 46 t - 36 t^2)) Cos[(\[Pi] t)/ 2] + (48 + \[Pi]^2 (42 - 58 t + 84 t^2)) Cos[(3 \[Pi] t)/ 2] - 16 Cos[(5 \[Pi] t)/2] - 62 \[Pi]^2 Cos[(5 \[Pi] t)/2] + 150 \[Pi]^2 t Cos[(5 \[Pi] t)/2] - 68 \[Pi]^2 t^2 Cos[(5 \[Pi] t)/2] - 20 Cos[(7 \[Pi] t)/2] + 32 \[Pi]^2 Cos[(7 \[Pi] t)/2] - 80 \[Pi]^2 t Cos[(7 \[Pi] t)/2] + 68 \[Pi]^2 t^2 Cos[(7 \[Pi] t)/2] + 12 Cos[(9 \[Pi] t)/2] - 18 \[Pi]^2 Cos[(9 \[Pi] t)/2] + 58 \[Pi]^2 t Cos[(9 \[Pi] t)/2] - 48 \[Pi]^2 t^2 Cos[(9 \[Pi] t)/2] - 4 \[Pi]^2 Cos[(11 \[Pi] t)/2] + 12 \[Pi]^2 t Cos[(11 \[Pi] t)/2] - 26 \[Pi] Sin[(\[Pi] t)/2] - 56 \[Pi]^3 Sin[(\[Pi] t)/2] + 8 \[Pi] t Sin[(\[Pi] t)/2] + 144 \[Pi]^3 t Sin[(\[Pi] t)/2] - 176 \[Pi]^3 t^2 Sin[(\[Pi] t)/2] + 88 \[Pi]^3 t^3 Sin[(\[Pi] t)/2] + 44 \[Pi] Sin[(3 \[Pi] t)/2] + 72 \[Pi]^3 Sin[(3 \[Pi] t)/2] - 68 \[Pi] t Sin[(3 \[Pi] t)/2] - 208 \[Pi]^3 t Sin[(3 \[Pi] t)/2] + 272 \[Pi]^3 t^2 Sin[(3 \[Pi] t)/2] - 136 \[Pi]^3 t^3 Sin[(3 \[Pi] t)/2] + 26 \[Pi] Sin[(5 \[Pi] t)/2] - 40 \[Pi]^3 Sin[(5 \[Pi] t)/2] + 28 \[Pi] t Sin[(5 \[Pi] t)/2] + 144 \[Pi]^3 t Sin[(5 \[Pi] t)/2] - 208 \[Pi]^3 t^2 Sin[(5 \[Pi] t)/2] + 104 \[Pi]^3 t^3 Sin[(5 \[Pi] t)/2] - 31 \[Pi] Sin[(7 \[Pi] t)/2] + 16 \[Pi]^3 Sin[(7 \[Pi] t)/2] + 62 \[Pi] t Sin[(7 \[Pi] t)/2] - 48 \[Pi]^3 t Sin[(7 \[Pi] t)/2] + 64 \[Pi]^3 t^2 Sin[(7 \[Pi] t)/2] - 32 \[Pi]^3 t^3 Sin[(7 \[Pi] t)/2] + 19 \[Pi] Sin[(9 \[Pi] t)/2] - 8 \[Pi]^3 Sin[(9 \[Pi] t)/2] - 42 \[Pi] t Sin[(9 \[Pi] t)/2] + 32 \[Pi]^3 t Sin[(9 \[Pi] t)/2] - 48 \[Pi]^3 t^2 Sin[(9 \[Pi] t)/2] + 24 \[Pi]^3 t^3 Sin[(9 \[Pi] t)/2] + 6 \[Pi] Sin[(11 \[Pi] t)/2]))/(\[Sqrt](7/2 + 4 \[Pi]^2 - 8 \[Pi]^2 t + 8 \[Pi]^2 t^2 + 2 (-1 + \[Pi]^2 (1 - t + t^2)) Cos[2 \[Pi] t] + 1/2 (-3 + 4 \[Pi]^2 (1 - 3 t + 3 t^2)) Cos[4 \[Pi] t] + 3 \[Pi] Sin[\[Pi] t] - 2 \[Pi] Sin[2 \[Pi] t] + 4 \[Pi] t Sin[2 \[Pi] t] - \[Pi] Sin[3 \[Pi] t] - 3 \[Pi] Sin[4 \[Pi] t] + 6 \[Pi] t Sin[4 \[Pi] t]) \[Sqrt](480 + 1182 \[Pi]^2 + 160 \[Pi]^4 - 984 \[Pi]^2 t - 640 \[Pi]^4 t + 984 \[Pi]^2 t^2 + 1120 \[Pi]^4 t^2 - 960 \[Pi]^4 t^3 + 480 \[Pi]^4 t^4 + 252 \[Pi]^2 (-1 + 2 t) Cos[\[Pi] t] + (-720 + \[Pi]^2 (325 + 972 t - 972 t^2) - 48 \[Pi]^4 (1 - 4 t + 7 t^2 - 6 t^3 + 3 t^4)) Cos[ 2 \[Pi] t] + 240 \[Pi]^2 Cos[3 \[Pi] t] - 480 \[Pi]^2 t Cos[3 \[Pi] t] + 288 Cos[4 \[Pi] t] + 390 \[Pi]^2 Cos[4 \[Pi] t] - 96 \[Pi]^4 Cos[4 \[Pi] t] + 216 \[Pi]^2 t Cos[4 \[Pi] t] + 384 \[Pi]^4 t Cos[4 \[Pi] t] - 216 \[Pi]^2 t^2 Cos[4 \[Pi] t] - 672 \[Pi]^4 t^2 Cos[4 \[Pi] t] + 576 \[Pi]^4 t^3 Cos[4 \[Pi] t] - 288 \[Pi]^4 t^4 Cos[4 \[Pi] t] - 24 \[Pi]^2 Cos[5 \[Pi] t] + 48 \[Pi]^2 t Cos[5 \[Pi] t] - 48 Cos[6 \[Pi] t] + 139 \[Pi]^2 Cos[6 \[Pi] t] - 16 \[Pi]^4 Cos[6 \[Pi] t] - 204 \[Pi]^2 t Cos[6 \[Pi] t] + 64 \[Pi]^4 t Cos[6 \[Pi] t] + 204 \[Pi]^2 t^2 Cos[6 \[Pi] t] - 112 \[Pi]^4 t^2 Cos[6 \[Pi] t] + 96 \[Pi]^4 t^3 Cos[6 \[Pi] t] - 48 \[Pi]^4 t^4 Cos[6 \[Pi] t] + 36 \[Pi]^2 Cos[7 \[Pi] t] - 72 \[Pi]^2 t Cos[7 \[Pi] t] + 12 \[Pi]^2 Cos[8 \[Pi] t] + 1136 \[Pi] Sin[\[Pi] t] + 528 \[Pi]^3 Sin[\[Pi] t] - 1072 \[Pi]^3 t Sin[\[Pi] t] + 1072 \[Pi]^3 t^2 Sin[\[Pi] t] - 360 \[Pi] Sin[2 \[Pi] t] + 8 \[Pi]^3 Sin[2 \[Pi] t] + 720 \[Pi] t Sin[2 \[Pi] t] + 488 \[Pi]^3 t Sin[2 \[Pi] t] - 1512 \[Pi]^3 t^2 Sin[2 \[Pi] t] + 1008 \[Pi]^3 t^3 Sin[2 \[Pi] t] - 480 \[Pi] Sin[3 \[Pi] t] + 192 \[Pi]^3 Sin[3 \[Pi] t] - 288 \[Pi]^3 t Sin[3 \[Pi] t] + 288 \[Pi]^3 t^2 Sin[3 \[Pi] t] + 288 \[Pi] Sin[4 \[Pi] t] + 272 \[Pi]^3 Sin[4 \[Pi] t] - 576 \[Pi] t Sin[4 \[Pi] t] - 688 \[Pi]^3 t Sin[4 \[Pi] t] + 432 \[Pi]^3 t^2 Sin[4 \[Pi] t] - 288 \[Pi]^3 t^3 Sin[4 \[Pi] t] + 128 \[Pi] Sin[5 \[Pi] t] + 64 \[Pi]^3 Sin[5 \[Pi] t] - 160 \[Pi]^3 t Sin[5 \[Pi] t] + 160 \[Pi]^3 t^2 Sin[5 \[Pi] t] - 72 \[Pi] Sin[6 \[Pi] t] + 72 \[Pi]^3 Sin[6 \[Pi] t] + 144 \[Pi] t Sin[6 \[Pi] t] - 216 \[Pi]^3 t Sin[6 \[Pi] t] + 216 \[Pi]^3 t^2 Sin[6 \[Pi] t] - 144 \[Pi]^3 t^3 Sin[6 \[Pi] t] - 48 \[Pi] Sin[7 \[Pi] t] + 16 \[Pi]^3 Sin[7 \[Pi] t] - 48 \[Pi]^3 t Sin[7 \[Pi] t] + 48 \[Pi]^3 t^2 Sin[7 \[Pi] t])))}, {-((2 Cos[(\[Pi] t)/ 2] (12 \[Pi] Cos[(\[Pi] t)/2] + \[Pi] (-5 + 6 t) Cos[( 3 \[Pi] t)/2] + 7 \[Pi] Cos[(5 \[Pi] t)/2] - 6 \[Pi] t Cos[(5 \[Pi] t)/2] + 2 \[Pi] Cos[(7 \[Pi] t)/2] + 8 Sin[(\[Pi] t)/2] + 16 \[Pi]^2 Sin[(\[Pi] t)/2] - 32 \[Pi]^2 t Sin[(\[Pi] t)/2] + 16 \[Pi]^2 t^2 Sin[(\[Pi] t)/2] + 4 Sin[(3 \[Pi] t)/2] - 4 \[Pi]^2 Sin[(3 \[Pi] t)/2] + 8 \[Pi]^2 t Sin[(3 \[Pi] t)/2] - 4 \[Pi]^2 t^2 Sin[(3 \[Pi] t)/2] - 4 Sin[(5 \[Pi] t)/2] + 4 \[Pi]^2 Sin[(5 \[Pi] t)/2] - 8 \[Pi]^2 t Sin[(5 \[Pi] t)/2] + 4 \[Pi]^2 t^2 Sin[(5 \[Pi] t)/2]))/(\[Sqrt](240 + 591 \[Pi]^2 + 80 \[Pi]^4 - 492 \[Pi]^2 t - 320 \[Pi]^4 t + 492 \[Pi]^2 t^2 + 560 \[Pi]^4 t^2 - 480 \[Pi]^4 t^3 + 240 \[Pi]^4 t^4 + 126 \[Pi]^2 (-1 + 2 t) Cos[\[Pi] t] - 1/2 (720 + \[Pi]^2 (-325 - 972 t + 972 t^2) + 48 \[Pi]^4 (1 - 4 t + 7 t^2 - 6 t^3 + 3 t^4)) Cos[ 2 \[Pi] t] + 120 \[Pi]^2 Cos[3 \[Pi] t] - 240 \[Pi]^2 t Cos[3 \[Pi] t] + 144 Cos[4 \[Pi] t] + 195 \[Pi]^2 Cos[4 \[Pi] t] - 48 \[Pi]^4 Cos[4 \[Pi] t] + 108 \[Pi]^2 t Cos[4 \[Pi] t] + 192 \[Pi]^4 t Cos[4 \[Pi] t] - 108 \[Pi]^2 t^2 Cos[4 \[Pi] t] - 336 \[Pi]^4 t^2 Cos[4 \[Pi] t] + 288 \[Pi]^4 t^3 Cos[4 \[Pi] t] - 144 \[Pi]^4 t^4 Cos[4 \[Pi] t] - 12 \[Pi]^2 Cos[5 \[Pi] t] + 24 \[Pi]^2 t Cos[5 \[Pi] t] - 24 Cos[6 \[Pi] t] + 139/2 \[Pi]^2 Cos[6 \[Pi] t] - 8 \[Pi]^4 Cos[6 \[Pi] t] - 102 \[Pi]^2 t Cos[6 \[Pi] t] + 32 \[Pi]^4 t Cos[6 \[Pi] t] + 102 \[Pi]^2 t^2 Cos[6 \[Pi] t] - 56 \[Pi]^4 t^2 Cos[6 \[Pi] t] + 48 \[Pi]^4 t^3 Cos[6 \[Pi] t] - 24 \[Pi]^4 t^4 Cos[6 \[Pi] t] + 18 \[Pi]^2 Cos[7 \[Pi] t] - 36 \[Pi]^2 t Cos[7 \[Pi] t] + 6 \[Pi]^2 Cos[8 \[Pi] t] + 568 \[Pi] Sin[\[Pi] t] + 264 \[Pi]^3 Sin[\[Pi] t] - 536 \[Pi]^3 t Sin[\[Pi] t] + 536 \[Pi]^3 t^2 Sin[\[Pi] t] - 180 \[Pi] Sin[2 \[Pi] t] + 4 \[Pi]^3 Sin[2 \[Pi] t] + 360 \[Pi] t Sin[2 \[Pi] t] + 244 \[Pi]^3 t Sin[2 \[Pi] t] - 756 \[Pi]^3 t^2 Sin[2 \[Pi] t] + 504 \[Pi]^3 t^3 Sin[2 \[Pi] t] - 240 \[Pi] Sin[3 \[Pi] t] + 96 \[Pi]^3 Sin[3 \[Pi] t] - 144 \[Pi]^3 t Sin[3 \[Pi] t] + 144 \[Pi]^3 t^2 Sin[3 \[Pi] t] + 144 \[Pi] Sin[4 \[Pi] t] + 136 \[Pi]^3 Sin[4 \[Pi] t] - 288 \[Pi] t Sin[4 \[Pi] t] - 344 \[Pi]^3 t Sin[4 \[Pi] t] + 216 \[Pi]^3 t^2 Sin[4 \[Pi] t] - 144 \[Pi]^3 t^3 Sin[4 \[Pi] t] + 64 \[Pi] Sin[5 \[Pi] t] + 32 \[Pi]^3 Sin[5 \[Pi] t] - 80 \[Pi]^3 t Sin[5 \[Pi] t] + 80 \[Pi]^3 t^2 Sin[5 \[Pi] t] - 36 \[Pi] Sin[6 \[Pi] t] + 36 \[Pi]^3 Sin[6 \[Pi] t] + 72 \[Pi] t Sin[6 \[Pi] t] - 108 \[Pi]^3 t Sin[6 \[Pi] t] + 108 \[Pi]^3 t^2 Sin[6 \[Pi] t] - 72 \[Pi]^3 t^3 Sin[6 \[Pi] t] - 24 \[Pi] Sin[7 \[Pi] t] + 8 \[Pi]^3 Sin[7 \[Pi] t] - 24 \[Pi]^3 t Sin[7 \[Pi] t] + 24 \[Pi]^3 t^2 Sin[7 \[Pi] t]))), (2 Sin[\[Pi] t] (-4 + 8 \[Pi]^2 t - 8 \[Pi]^2 t^2 + (4 - 4 \[Pi]^2 (-1 + t) t) Cos[2 \[Pi] t] - 3 \[Pi] (-1 + 2 t) Sin[2 \[Pi] t] + 2 \[Pi] Sin[3 \[Pi] t]))/(\[Sqrt](240 + 591 \[Pi]^2 + 80 \[Pi]^4 - 492 \[Pi]^2 t - 320 \[Pi]^4 t + 492 \[Pi]^2 t^2 + 560 \[Pi]^4 t^2 - 480 \[Pi]^4 t^3 + 240 \[Pi]^4 t^4 + 126 \[Pi]^2 (-1 + 2 t) Cos[\[Pi] t] - 1/2 (720 + \[Pi]^2 (-325 - 972 t + 972 t^2) + 48 \[Pi]^4 (1 - 4 t + 7 t^2 - 6 t^3 + 3 t^4)) Cos[ 2 \[Pi] t] + 120 \[Pi]^2 Cos[3 \[Pi] t] - 240 \[Pi]^2 t Cos[3 \[Pi] t] + 144 Cos[4 \[Pi] t] + 195 \[Pi]^2 Cos[4 \[Pi] t] - 48 \[Pi]^4 Cos[4 \[Pi] t] + 108 \[Pi]^2 t Cos[4 \[Pi] t] + 192 \[Pi]^4 t Cos[4 \[Pi] t] - 108 \[Pi]^2 t^2 Cos[4 \[Pi] t] - 336 \[Pi]^4 t^2 Cos[4 \[Pi] t] + 288 \[Pi]^4 t^3 Cos[4 \[Pi] t] - 144 \[Pi]^4 t^4 Cos[4 \[Pi] t] - 12 \[Pi]^2 Cos[5 \[Pi] t] + 24 \[Pi]^2 t Cos[5 \[Pi] t] - 24 Cos[6 \[Pi] t] + 139/2 \[Pi]^2 Cos[6 \[Pi] t] - 8 \[Pi]^4 Cos[6 \[Pi] t] - 102 \[Pi]^2 t Cos[6 \[Pi] t] + 32 \[Pi]^4 t Cos[6 \[Pi] t] + 102 \[Pi]^2 t^2 Cos[6 \[Pi] t] - 56 \[Pi]^4 t^2 Cos[6 \[Pi] t] + 48 \[Pi]^4 t^3 Cos[6 \[Pi] t] - 24 \[Pi]^4 t^4 Cos[6 \[Pi] t] + 18 \[Pi]^2 Cos[7 \[Pi] t] - 36 \[Pi]^2 t Cos[7 \[Pi] t] + 6 \[Pi]^2 Cos[8 \[Pi] t] + 568 \[Pi] Sin[\[Pi] t] + 264 \[Pi]^3 Sin[\[Pi] t] - 536 \[Pi]^3 t Sin[\[Pi] t] + 536 \[Pi]^3 t^2 Sin[\[Pi] t] - 180 \[Pi] Sin[2 \[Pi] t] + 4 \[Pi]^3 Sin[2 \[Pi] t] + 360 \[Pi] t Sin[2 \[Pi] t] + 244 \[Pi]^3 t Sin[2 \[Pi] t] - 756 \[Pi]^3 t^2 Sin[2 \[Pi] t] + 504 \[Pi]^3 t^3 Sin[2 \[Pi] t] - ...+ 6 \[Pi] t Sin[(5 \[Pi] t)/2] - 2 \[Pi] Sin[(7 \[Pi] t)/2]))/(\[Sqrt](240 + 591 \[Pi]^2 + 80 \[Pi]^4 - 492 \[Pi]^2 t - 320 \[Pi]^4 t + 492 \[Pi]^2 t^2 + 560 \[Pi]^4 t^2 - 480 \[Pi]^4 t^3 + 240 \[Pi]^4 t^4 + 126 \[Pi]^2 (-1 + 2 t) Cos[\[Pi] t] - 1/2 (720 + \[Pi]^2 (-325 - 972 t + 972 t^2) + 48 \[Pi]^4 (1 - 4 t + 7 t^2 - 6 t^3 + 3 t^4)) Cos[ 2 \[Pi] t] + 120 \[Pi]^2 Cos[3 \[Pi] t] - 240 \[Pi]^2 t Cos[3 \[Pi] t] + 144 Cos[4 \[Pi] t] + 195 \[Pi]^2 Cos[4 \[Pi] t] - 48 \[Pi]^4 Cos[4 \[Pi] t] + 108 \[Pi]^2 t Cos[4 \[Pi] t] + 192 \[Pi]^4 t Cos[4 \[Pi] t] - 108 \[Pi]^2 t^2 Cos[4 \[Pi] t] - 336 \[Pi]^4 t^2 Cos[4 \[Pi] t] + 288 \[Pi]^4 t^3 Cos[4 \[Pi] t] - 144 \[Pi]^4 t^4 Cos[4 \[Pi] t] - 12 \[Pi]^2 Cos[5 \[Pi] t] + 24 \[Pi]^2 t Cos[5 \[Pi] t] - 24 Cos[6 \[Pi] t] + 139/2 \[Pi]^2 Cos[6 \[Pi] t] - 8 \[Pi]^4 Cos[6 \[Pi] t] - 102 \[Pi]^2 t Cos[6 \[Pi] t] + 32 \[Pi]^4 t Cos[6 \[Pi] t] + 102 \[Pi]^2 t^2 Cos[6 \[Pi] t] - 56 \[Pi]^4 t^2 Cos[6 \[Pi] t] + 48 \[Pi]^4 t^3 Cos[6 \[Pi] t] - 24 \[Pi]^4 t^4 Cos[6 \[Pi] t] + 18 \[Pi]^2 Cos[7 \[Pi] t] - 36 \[Pi]^2 t Cos[7 \[Pi] t] + 6 \[Pi]^2 Cos[8 \[Pi] t] + 568 \[Pi] Sin[\[Pi] t] + 264 \[Pi]^3 Sin[\[Pi] t] - 536 \[Pi]^3 t Sin[\[Pi] t] + 536 \[Pi]^3 t^2 Sin[\[Pi] t] - 180 \[Pi] Sin[2 \[Pi] t] + 4 \[Pi]^3 Sin[2 \[Pi] t] + 360 \[Pi] t Sin[2 \[Pi] t] + 244 \[Pi]^3 t Sin[2 \[Pi] t] - 756 \[Pi]^3 t^2 Sin[2 \[Pi] t] + 504 \[Pi]^3 t^3 Sin[2 \[Pi] t] - 240 \[Pi] Sin[3 \[Pi] t] + 96 \[Pi]^3 Sin[3 \[Pi] t] - 144 \[Pi]^3 t Sin[3 \[Pi] t] + 144 \[Pi]^3 t^2 Sin[3 \[Pi] t] + 144 \[Pi] Sin[4 \[Pi] t] + 136 \[Pi]^3 Sin[4 \[Pi] t] - 288 \[Pi] t Sin[4 \[Pi] t] - 344 \[Pi]^3 t Sin[4 \[Pi] t] + 216 \[Pi]^3 t^2 Sin[4 \[Pi] t] - 144 \[Pi]^3 t^3 Sin[4 \[Pi] t] + 64 \[Pi] Sin[5 \[Pi] t] + 32 \[Pi]^3 Sin[5 \[Pi] t] - 80 \[Pi]^3 t Sin[5 \[Pi] t] + 80 \[Pi]^3 t^2 Sin[5 \[Pi] t] - 36 \[Pi] Sin[6 \[Pi] t] + 36 \[Pi]^3 Sin[6 \[Pi] t] + 72 \[Pi] t Sin[6 \[Pi] t] - 108 \[Pi]^3 t Sin[6 \[Pi] t] + 108 \[Pi]^3 t^2 Sin[6 \[Pi] t] - 72 \[Pi]^3 t^3 Sin[6 \[Pi] t] - 24 \[Pi] Sin[7 \[Pi] t] + 8 \[Pi]^3 Sin[7 \[Pi] t] - 24 \[Pi]^3 t Sin[7 \[Pi] t] + 24 \[Pi]^3 t^2 Sin[7 \[Pi] t])))}}}

is such huge (The one is shortened (see ...) to be presented here.) .

Addition. You ask "help me to get it's curvature curve in any kind of way and it's plot?" Here it is.

fs = FrenetSerretSystem[r[t], t];
fs[[1]][[1]] (*Curvature*)


A very long expression

Plot[fs[[1]][[1]], {t, 0, 2*Pi}, PlotRange -> All]


• What is wrong in my answer? It is not a good practice to down vote without any explanation. May 22, 2022 at 19:23
• I am not the down vote. But I think it is recommend to only post Simplify[FrenetSerretSystem[r[t], t]] // LeafCount instead of the long result. May 23, 2022 at 5:44
• thanks. i will try to use it. May 23, 2022 at 19:22