2-2

Illustrative application of the Frenet-Serret formulae

Needs["VectorAnalysis`"]

Clear[x,v,a]

The equations

x=r Cos[t]
y=r Sin[t]
z=h t

provide parametric description (relative to a right-handed Cartesian frame) of a right handed helix, that spirals about the z-axis.

x={r Cos[t],r Sin[t],h t};
v=(∂_t) x;
a=(∂_(t,t))x;

x
v
a

Manipulate[ParametricPlot3D[{ Cos[t], Sin[t],h t}, {t,0,10π}, PlotRange->{{-1,1},{-1,1},{0,10}}, Ticks->{{-1,1},{-1,1},{5,10}}],{h,0,0.3}]

T=(v/(Sqrt[DotProduct[v,v]]))//Simplify

DotProduct[T,T]//Simplify

((CrossProduct[v,CrossProduct[a,v]])/(Sqrt[DotProduct[v,v]DotProduct[CrossProduct[a,v],CrossProduct[a,v]]]))//Simplify

U={- Cos[t],- Sin[t],0};

DotProduct[T,U]//Simplify
DotProduct[U,U]//Simplify

CrossProduct[T, U]//Simplify

V={((h  Sin[t])/(Sqrt[ ((h^2)+(r^2))])),-((h  Cos[t])/(Sqrt[ ((h^2)+(r^2))])),(r/(Sqrt[ ((h^2)+(r^2))]))};

DotProduct[T,V]//Simplify
DotProduct[U,V]//Simplify
DotProduct[V,V]//Simplify

Working from the Frenet-Serret formula

(∂_t) U=(Sqrt[DotProduct[(∂_t) x,(∂_t) x]])(-κ T+τ V)

—we have (by the orthonormality of T and V)

κ =- DotProduct[(∂_t) U,T]//Simplify
τ =+ DotProduct[(∂_t) U,V]//Simplify

—both fo which turn out, in this instance, to be constant.

Look now to the case of a left-handed helix:

Clear[x,v,a,T,U,V, κ, τ]

x={r Cos[t],-r Sin[t],h t};
v=(∂_t) x;
a=(∂_(t,t))x;

x
v
a

Manipulate[ParametricPlot3D[{ Cos[t], -Sin[t],h t}, {t,0,10π}, PlotRange->{{-1,1},{-1,1},{0,10}}, Ticks->{{-1,1},{-1,1},{5,10}}],{h,0,0.3}]

T=(v/(Sqrt[DotProduct[v,v]]))//Simplify

DotProduct[T,T]//Simplify

((CrossProduct[v,CrossProduct[a,v]])/(Sqrt[DotProduct[v,v]DotProduct[CrossProduct[a,v],CrossProduct[a,v]]]))//Simplify

U={- Cos[t],Sin[t],0};

DotProduct[T,U]//Simplify
DotProduct[U,U]//Simplify

CrossProduct[T, U]//Simplify

V={-((h  Sin[t])/(Sqrt[ ((h^2)+(r^2))])),-((h  Cos[t])/(Sqrt[ ((h^2)+(r^2))])),-(r/(Sqrt[ ((h^2)+(r^2))]))};

DotProduct[T,V]//Simplify
DotProduct[U,V]//Simplify
DotProduct[V,V]//Simplify

κ =- DotProduct[(∂_t) U,T]//Simplify
τ =+ DotProduct[(∂_t) U,V]//Simplify

Reversing the helicity has reversed the sign of the torsion.

Manipulate[Graphics3D[
{Thick,{Red,Line[{{0,0,0},{-(( Sin[t])/(Sqrt[(h^2)+1])),(( Cos[t])/(Sqrt[(h^2)+1])),(h/(Sqrt[(h^2)+1]))}}]},{RGBColor[0,.5,.5],Line[{{0,0,0},{- Cos[t],- Sin[t],0}}]},{Blue,Line[{{0,0,0},{((h  Sin[t])/(Sqrt[ ((h^2)+1)])),-((h  Cos[t])/(Sqrt[ ((h^2)+1)])),(1/(Sqrt[ ((h^2)+1)]))}}]}}, PlotRange->{{-1.5,1.5},{-1.5,1.5},{-1.5,1.5}}, PlotLabel->"T, U, V",LabelStyle->Directive[Bold]],{h,0,0.3}, {t,0,2π}]

Hyperlink["Curvature & Torsion","http://demonstrations.wolfram.com/CurvatureAndTorsion/"]