' - - File XII_FamilyTNB.txt
pi = 3.14159

'  Find the tangent lines at pi/6 position

R{t} = 3 + 8 * Sin[2*pi*t]
a0=1.6                                                   ' Original constant. Set pi/6 to, a NUMERICAL value.
 
X{t} = R{t} * Cos[2*pi*t]/2.                     ' Divide by 2 to scale.
Y{t} = R{t} * Sin[2*pi*t]/2
F<t> = <X,Y,0>
' - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
D<t> = vDiff[F]

T<t> = vDiv[D,sMag[D] ]                        ' dR/dt / |dR/dt |  =  dR/ds  = T (tangent vector)
Rn<t> =vEval[F,a0]
Tn<t> = vEval[T,a0]
L<t> = vAdd[vMpy[Tn,#],Rn]                 ' Tangent line; # is parameter position.
' - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Td<t> = vDiff[T]                                      ' dT / dt
A<t> = vDiv[Td,sMag[D]]                      ' (dT/dt) / |dR/dt|  =  (dT/dt) / (ds/dt) = dT/ds
N<t> = vDiv[A,sMag[A]]                        ' N = (dT/ds) / |dT/ds|
Nn<t> = vEval[N,a0]
P<t> = vAdd[vMpy[Nn,#],Rn]
' - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
B<t> = vCross[T,N]                               ' biNormal is vector cross product, T x N.
Bn<t> = vEval[B,a0]
H<t> = vAdd[vMpy[Bn,#],Rn]
' - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

t0 = 0  :  tf = 2  :  n = 100
Plot[F, L, P, H, {t, t0, tf, n}]







