splines

In numerical analysis, a B-spline (short for basis spline) is a type of spline function designed to have minimal support (overlap) for a given degree, smoothness, and set of breakpoints (knots that partition its domain), making it a fundamental building block for all spline functions of that degree. A B-spline is defined as a piecewise polynomial of order



n


{\displaystyle n}

, meaning a degree of



n
−
1


{\displaystyle n-1}

. It is built from sections that meet at these knots, where the continuity of the function and its derivatives depends on how often each knot repeats (its multiplicity). Any spline function of a specific degree can be uniquely expressed as a linear combination of B-splines of that degree over the same knots, a property that makes them versatile in mathematical modeling. A special subtype, cardinal B-splines, uses equidistant knots.
The concept of B-splines traces back to the 19th century, when Nikolai Lobachevsky explored similar ideas at Kazan University in Russia, though the term "B-spline" was coined by Isaac Jacob Schoenberg in 1967, reflecting their role as basis functions.

B-splines are widely used in fields like computer-aided design (CAD) and computer graphics, where they shape curves and surfaces through a set of control points, as well as in data analysis for tasks like curve fitting and numerical differentiation of experimental data. From designing car bodies to smoothing noisy measurements, B-splines offer a flexible way to represent complex shapes and functions with precision.

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