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Interpolation with cubic splines between eight points. Hand-drawn technical drawings for shipbuilding are a historical example of spline interpolation; drawings were constructed using flexible rulers that were bent to follow pre-defined points.
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]
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.
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.
Spline interpolation uses low-degree polynomials in each of the intervals, and chooses the polynomial pieces such that they fit smoothly together. The resulting function is called a spline. For instance, the natural cubic spline is piecewise cubic and twice continuously differentiable. Furthermore, its second derivative is zero at the end points.
The cubic B-spline is the cardinal B-spline of order 4, ... is known as the cardinal spline interpolation problem. ... As a concrete example, ...