' - - File: IX_Extra2.txt
'  Radius of curvature = | R'{t} |^3  /  |R'{t} x R''{t}  |

Fa{t} = Cosh[t] - t^2  :  F<t> = <t, Fa, 0>
Df{t} = Diff[Fa,t]        :  Dp<t> = <t, Df, 0>

R{t} = sDiv[sPwr[sMag[vDiff[F]],3], sMag[vCross[vDiff[F],vDiff[F,2]]]]
' - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

'  - - - - - - - - - - - - - - - - - - Step 1:  Plot to estimate the zeros.- - - - - - - - - - - - - - - - - - - - -
pi = 3.14159: t0 = -pi :  tf = pi : n = 200
'Plot[F, Dp,{t, t0,tf,n}]              ' Write down brackets for zeros of interest. Comment out.

' - - - - - - - - - - - - - - - - - - - Step 2: ID the zeroes of interest. - - - - - - - - - - - - - - - - - - - - - 
to = 0.001  ' Tolerance for the zero accuracy.
Zero[Df,{-3,-2,z0}to]               ' Assign z0, zerocrossing; leave undefined.
Za{t} = z0
Zero[Df,{2,3,z1}to]                  ' Assign z1, zero crossing; leave undefined.
Zb{t} = z1

' - - - - - - - - - - - - - - - - - - - Step3: 2nd Derivative for min/max. -  - - - - - - - - - - - - - - - - - -
c1 = Fa{z0}  :  c2 = Fa{z1}     '  Check the graph if values are reasonable.

' - -  - - - - - - - - - - - - - - - - - Step 4: Draw the circles with the proper offsets. - - - - - - - - -
'c1 (& c2) as centers, compute both R{c1} and R{c2} for both circles..

ra = R{z0}  :  rb = R{z1}  :   rc = R{0}  :  c3 = Fa{0}
Ra<t> = <ra*Cos[t] +z0, ra*Sin[t] + (c1 + ra), 0 >      ' First parametric equation for circle.
Rb<t> = <rb*Cos[t] +z1, rb*Sin[t] + (c2 + rb), 0>       ' Second equation.
Rc<t> = <rc*Cos[t]  + 0 , rc*Sin[t] + (c3  - rc), 0>       ' Center equation.

Plot[F, Ra, Rb, Rc, {t, t0, tf, n}]                                         ' Comment out above Plot command.
<t=1>   ' Any value to enable worksheet display for checking.





