' - - File: VI_Gen2.txt

' Find the Curvature, k{t],  given a parametric function by two methods.
' Then find 1 / K{t}, the radius of curvature and plot..

R<t> = <t, t^2 / 3, 0 >
T<t> = vTan[R]  :  t1 = 0
Ta{t} = sMag[vDiv[vDiff[T],sMag[vDiff[R]] ]]       ' k{t} = |dT/dt| / (ds/dt) = dT/ds
ra = 1 / Ta{t1}                                                           ' ANS: 1 / Ta{t1} = 1.5
' - - - - - - - - - - - - - - - - - - - - - - - - - - - - Method #2 - - - - - - - - - - - - - - - - - - - - - - - - - 
Ka<t> = vDiff[R]                              '  dR/dt =R'
Kb<t> = vDiff[R,2]                           '  d(dr/dt)/dt = R"

Kn{t} = sMag[vCross[Ka,Kb]]        '   | R' x R" |
Kc{t} = sMag[Ka]                            '   | R' |
Kd{t} = Kc{t}^3                                '   | R' |^3
Ke{t}= Kn{t} / Kd{t}                         '   | R' x R" |  /  | R' |^3
rb = 1/ Ke{t1}                                  '   ANS: 1/ Ke{t1} = 1.5, same answer as above.

' - - - - - - - - - - - - - - Set t1 = 0; assign r the value for the curvature - - - - - - - - - - - -
 rb = 1 / Ke{t1}                                '  Or use ra, the same value.      
' The circle with radius of curvature: :  x^2 + (y - r)^2 = (r)^2 
Cx{t} = ra* Cos[t]  :  Cy{t} = ra*Sin[t] + ra  :  Cz{t} = 0  :  C<t> = <Cx, Cy, Cz>

' Plot the parabola and circle with radius of curvature, 1/r, from the above results.
pi = 3.14159
t0 = -pi  :  tf = pi  :  n = 100
Plot[R, C, {t, t0, tf, n}]
'<t = t1>           ' Only Needed for work sheet output to verify results.

'@Radius of Curvature = @ra!




