DarbouxVector::usage="The Darboux vector of a space curve with unit speed is the angular velocity vector of the Frenet-Serret frame. And it is calculated like this: D = τ T + κ B. Where τ is the torsion and κ the curvature. Reference: DarbouxVector, FunctionRepository; Personal. ";
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Darboux vector and Frenet-Serret frame:
Value (t1)
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ManipulateDar[t_]:=(2(512175Sin[t]-20963Sin[39t]-10557Sin[41t]+(3(25717+1040Cos[40t]+8Cos[80t])(-1339700773+1091308180Cos[40t]+6820802Cos[80t]-85280Cos[120t])(5Sin[t]+39Sin[39t]+41Sin[41t]))/(-263799072767-11488191540Cos[40t]+11733150Cos[80t]+1534880Cos[120t]+12792Cos[160t])+3120Sin[79t]+3280Sin[81t]))(25717+1040Cos[40t]+8Cos[80t]),-
2(512175Cos[t]+20963Cos[39t]-10557Cos[41t]-3120Cos[79t]+3280Cos[81t])
25717+1040Cos[40t]+8Cos[80t]
+(6(5Cos[t]-39Cos[39t]+41Cos[41t])(1339700773-1091308180Cos[40t]-6820802Cos[80t]+85280Cos[120t]))(-263799072767-11488191540Cos[40t]+11733150Cos[80t]+1534880Cos[120t]+12792Cos[160t]),
12(3209+5340Cos[40t]-1066Cos[80t])
25717+1040Cos[40t]+8Cos[80t]
+(3(3+160Cos[40t])(1339700773-1091308180Cos[40t]-6820802Cos[80t]+85280Cos[120t]))(-263799072767-11488191540Cos[40t]+11733150Cos[80t]+1534880Cos[120t]+12792Cos[160t]);​​​​p=Grid[{{DarbouxVector},{Dar[t1]}},FrameAll,ItemSize->15,ItemStyle->15];​​Show[{​​ParametricPlot3D[Dar[t],{t,0,t1},PlotStyle->{Brown,Thick},PerformanceGoal->"Quality"],Graphics3D[​​{Brown,Ball[Dar[t1],0.5]}]},​​AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange{{-45,45},{-45,45},{0,5}},ViewPoint{1.3,-2.4,2.},ImageSize700],Style["Darboux vector:",Bold,Large],{{t1,0.00002,"Value (t1)"},0.00001,5π,0.00001},Delimiter,{{p,2,"Darboux vector coordinates"}},ControlPlacementTop
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Darboux vector:
Value (t1)
Darboux vector coordinates
DarbouxVector
{21.337,34.8174,2.96011}
Darboux components graph
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​​​​Show​​Dar=(2(512175Sin[t]-20963Sin[39t]-10557Sin[41t]+(3(25717+1040Cos[40t]+8Cos[80t])(-1339700773+1091308180Cos[40t]+6820802Cos[80t]-85280Cos[120t])(5Sin[t]+39Sin[39t]+41Sin[41t]))/(-263799072767-11488191540Cos[40t]+11733150Cos[80t]+1534880Cos[120t]+12792Cos[160t])+3120Sin[79t]+3280Sin[81t]))(25717+1040Cos[40t]+8Cos[80t]),-
2(512175Cos[t]+20963Cos[39t]-10557Cos[41t]-3120Cos[79t]+3280Cos[81t])
25717+1040Cos[40t]+8Cos[80t]
+(6(5Cos[t]-39Cos[39t]+41Cos[41t])(1339700773-1091308180Cos[40t]-6820802Cos[80t]+85280Cos[120t]))(-263799072767-11488191540Cos[40t]+11733150Cos[80t]+1534880Cos[120t]+12792Cos[160t]),
12(3209+5340Cos[40t]-1066Cos[80t])
25717+1040Cos[40t]+8Cos[80t]
+(3(3+160Cos[40t])(1339700773-1091308180Cos[40t]-6820802Cos[80t]+85280Cos[120t]))(-263799072767-11488191540Cos[40t]+11733150Cos[80t]+1534880Cos[120t]+12792Cos[160t]);​​{Plot[Dar,{t,0,5π},PlotStyleThick]},​​AxesLabel->(Style[#,15,Blue]&/@{"X","Y"}),AxesOrigin{0,0},AxesTrue,AspectRatioAutomatic,PlotRangeAll,ImageSize700
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