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Spline interpolation is a mathematical technique used to create a continuous curve or surface from a set of discrete data points. This technique is commonly employed in various fields such as computer-aided design (CAD), engineering, and computer graphics to create smooth and accurate representations of complex shapes.

The process of spline interpolation involves determining the coefficients of a set of polynomials that pass through each data point. These polynomials, known as basis splines or B-splines, form the basis of the spline curve or surface. The B-splines are constructed such that they are piecewise continuous, meaning that they are defined in segments, or pieces, and are connected to form a single, smooth curve.

Spline interpolation can be used for both curve and surface interpolation. In curve interpolation, the algorithm determines the coefficients of a single B-spline curve that passes through the data points. In surface interpolation, multiple B-spline curves are created to form a surface that passes through the data points.

Spline interpolation has several advantages over other interpolation techniques, including its ability to produce smooth curves and surfaces. It is also relatively simple to implement and is well-suited for large datasets. However, it can be computationally intensive for complex shapes and large numbers of data points.

In addition to its practical applications, spline interpolation has also been widely studied in the field of mathematics. It is used as a tool to study the properties of polynomial curves and surfaces, and has led to various results in topology and geometry.