Splines

Introduction

A spline is a mathematical function that is piecewise-defined by polynomial functions, and which can approximate complex shapes in computer graphics, computer-aided design (CAD), and in the mathematical field of numerical analysis.

History

The term 'spline' originates from the flexible spline devices used by shipbuilders and draftsmen to draw smooth shapes. These mechanical devices were made of flexible metal or wood strips, or 'splines'. The modern mathematical definition of splines was developed in the 1960s and 1970s, primarily as a tool for computer graphics and CAD.

Mathematical Definition

In the mathematical field of numerical analysis, a spline is a special function defined piecewise by polynomials. The function's domain is divided into sub-intervals, and on each sub-interval, the function's values are defined by a separate polynomial function.

Types of Splines

There are several types of splines, each with its own characteristics and uses. These include linear splines, quadratic splines, cubic splines, B-splines, NURBS, and others.

Linear Splines

Linear splines are the simplest type of spline. They are piecewise linear functions that connect a set of points with straight lines.

Quadratic Splines

Quadratic splines are piecewise quadratic functions. They provide a smoother approximation to the data than linear splines.

Cubic Splines

Cubic splines are piecewise cubic polynomials. They are the most commonly used type of spline in applications due to their balance between computational efficiency and approximation accuracy.

B-Splines

B-splines are a generalization of the basic spline concept, allowing for greater flexibility and control over the shape of the curve.

NURBS

NURBS are a type of B-spline that includes a weighting factor for each control point, allowing for the representation of both standard B-splines and conic sections like circles and ellipses.

Applications

Splines have a wide range of applications in various fields. They are used in computer graphics to generate smooth curves and surfaces, in CAD to design complex shapes, in image and signal processing for interpolation and smoothing, and in numerical analysis for function approximation.