Tension Function

This is a hyperbolic function often used in spline interpolations that parameterises the 'tension' of a curve interpolation between points. At low tension it behaves like a cubic polynomial, while at high tension, it tends towards a piecewise smooth series of linear functions.

They are derived from the following constraints on an interval [x_i, x_i+1]:

   1   f'''' - tau^2 f'' = 0
   2   f(x_i)     = y_i
   3   f(x_i+1)   = y_i+1
   4   f''(x_i)   = z_i
   5   f''(x_i+1) = z_i+1

This yields a C2 continuous function with hyperbolic terms (not a polynomial) that ranges from looking like a cubic interpolation at low tensions (tau -> 0) and a linearly blended interpolation at high tensions.

Wiki: ecl_geometry/Tension Function (last edited 2012-01-24 11:01:02 by DanielStonier)