Tangent, Normal, Binormal Directions on a Helix

Define parametric curve

x[t_]:=2*Sin[3*t]; y[t_]:=t;
z[t_]:=2*Cos[3*t];

Plot space curve

ParametricPlot3D[{x[t],y[t],z[t]},{t,0,2*Pi}, ColorFunction->Hue, PlotStyle->{Thick}, AxesLabel->{"x","y","z"}, PlotLabel->"Circular Helix With Hue Coloring"]

Define vector representation of curve and plot small section of helix near t = Pi

r[t_]:={x[t],y[t],z[t]};

HelixPlot = ParametricPlot3D[r[t],{t,Pi/2,3*Pi/2}, PlotStyle->{Thick, Black},AxesLabel->{"x","y","z"},PlotLabel->"Circular Helix Near The Point (0,Pi,-2)"]

Plot tangent line on curve near t = Pi. Tangent line is blue.

tanline[t_]:={-6*t, Pi+t,-2};


TanLinePlot = ParametricPlot3D[tanline[t],{t,-1/4,1/4},PlotStyle->{Thick,Blue}];

Show[HelixPlot,TanLinePlot, PlotRange->{-3,3},AxesLabel->{"x","y","z"},PlotLabel->"Circular Helix with Tangent Local Coordinate System"]

Plot normal line on curve near t = Pi.  Normal line is red.

normline[t_]:={0,Pi,-2+t};

NormLinePlot = ParametricPlot3D[normline[t],{t,-1,1},PlotStyle->{Thick,Red}];

Show[NormLinePlot,HelixPlot,TanLinePlot,PlotRange->{-3,3},AxesLabel->{"x","y","z"},PlotLabel->"Circular Helix with Tangent/Normal Local Coordinate System"]

Plot binormal line near t = Pi.  Binormal line is green.  Note that final graph has an orthogonal three-dimensional coordinate system “attached” to the curve at (0,Pi,-2).

binormline[t_]:={t,Pi+6*t,-2};

BiNormLinePlot = ParametricPlot3D[binormline[t],{t,-1/4,1/4},PlotStyle->{Thick,Green}];

Show[NormLinePlot,HelixPlot,TanLinePlot, BiNormLinePlot,PlotRange->{-3,3}, AxesLabel->{"x","y","z"}, PlotLabel->"Circular Helix with Tangent/Normal/BiNormal Local Coordinate System"]