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Paper which explains step by step how cubic spline interpolation is done, but only for equidistant knots. Numerical Recipes in C, Go to Chapter 3 Section 3-3; A note on cubic splines; Information about spline interpolation (including code in Fortran 77) TinySpline:Open source C-library for splines which implements cubic spline interpolation
The space of all natural cubic splines, for instance, is a subspace of the space of all cubic C 2 splines. The literature of splines is replete with names for special types of splines. These names have been associated with: The choices made for representing the spline, for example:
Discrete cubic splines were originally introduced as solutions of certain minimization problems. [1] [2] They have applications in computing nonlinear splines. [1] [3] They are used to obtain approximate solution of a second order boundary value problem. [4] Discrete interpolatory splines have been used to construct biorthogonal wavelets. [5]
Cubic polynomial splines are extensively used in computer graphics and geometric modeling to obtain curves or motion trajectories that pass through specified points of the plane or three-dimensional space. In these applications, each coordinate of the plane or space is separately interpolated by a cubic spline function of a separate parameter t.
Bicubic interpolation can be accomplished using either Lagrange polynomials, cubic splines, or cubic convolution algorithm. In image processing , bicubic interpolation is often chosen over bilinear or nearest-neighbor interpolation in image resampling , when speed is not an issue.
Example showing non-monotone cubic interpolation (in red) and monotone cubic interpolation (in blue) of a monotone data set. Monotone interpolation can be accomplished using cubic Hermite spline with the tangents m i {\displaystyle m_{i}} modified to ensure the monotonicity of the resulting Hermite spline.
Lagrange and other interpolation at equally spaced points, as in the example above, yield a polynomial oscillating above and below the true function. This behaviour tends to grow with the number of points, leading to a divergence known as Runge's phenomenon; the problem may be eliminated by choosing interpolation points at Chebyshev nodes. [5]
The most familiar example is the cubic smoothing spline, but there are many other possibilities, including for the case where is a vector quantity. Cubic spline definition [ edit ]