The graduation of secondary structure elements.

A method of graduating (i.e., least-squares fitting) a smooth polynomial curve through long elements of protein secondary structure is described. It uses the Chebyshev polynomials of a discrete (integer) variable with several restraints to prevent artifactual curvatures. A new recursion formula is given which allows the evaluation of the polynomials on rational-number points as well as on the integer points. High-order splines suitable for interpolation between integer points are also discussed. The new method finds applications in graphics and in structural analysis.