CUBIC SPLINE INTERPOLATION METHODS

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This class of spline is most often used to set up paths for object motions or to provide a representation for an existing object or drawing, but interpolation splines are also used sometimes to design object shapes. Cubic polynomials offer a reasonable compromise between flexibility and speed of computation. Compared to higher – order polynomials, cubic splines require less calculations and memory and they are more stable. Compared to lower-order polynomials, cubic splines are more flexible for modeling arbitrary curve shapes.
Given a set of control points, cubic interpolation splines are obtained by fitting the input points with a piecewise cubic polynomial curve that passes through every control point. Suppose we have n+1 control points specified with coordinates


A cubic interpolation fit of these points is illustrated in figure below. We can describe the parametric cubic polynomial that is to be fitted between each pair of control points with the following set of equations:


For each of these three equations, we need to determine the values of the four coefficients a, b, c, and d in the polynomial representation for each of the n curve sections between the n+1 control points. We do this by setting enough boundary conditions at the “joints” between curves sections so that we can obtain numerical values for all the coefficients. In the following sections, we discuss common methods for setting the boundary conditions for cubic interpolation splines.