Warp <war### [at] tagpovrayorg> wrote:
:   Of course the more mathematically oriented people would like an exact
: result instead of just an approximation... :)

  By the way, I think that this gives a better approximation:

  Suppose that we want to calculate the tangent of the spline at the
time value T.
  Take a small value, which we will call Epsilon (could be for example
10e-5 or something similar).
  The approximated tangent will be:

  TangentVector = Spline(T+Epsilon)-Spline(T-Epsilon)

  This given a better approximation because the resulting vector will be
closer to the true tangent, and could even be exactly the tangent.
'Spline(T+Epsilon)-Spline(T)', however, can never be exactly the tangent.

-- 
#macro M(A,N,D,L)plane{-z,-9pigment{mandel L*9translate N color_map{[0rgb x]
[1rgb 9]}scale<D,D*3D>*1e3}rotate y*A*8}#end M(-3<1.206434.28623>70,7)M(
-1<.7438.1795>1,20)M(1<.77595.13699>30,20)M(3<.75923.07145>80,99)// - Warp -

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