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A cubic spline is a Spline constructed of piecewise third-order Polynomials which pass through a set of control points. The second Derivative of each Polynomial is zero at the endpoints.
References
Burden, R. L.; Faires, J. D.; and Reynolds, A. C. Numerical Analysis, 6th ed. Boston, MA: Brooks/Cole, pp. 120-121, 1997.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Cubic Spline Interpolation.'' §3.3 in
Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England:
Cambridge University Press, pp. 107-110, 1992.