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]:
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.