A Bezier Curve is a parametric curve used in computer graphics and related fields to model smooth curves that can be scaled indefinitely. Named after the French engineer Pierre Bézier, who used these curves to design automobile bodies at Renault in the 1960s, Bezier Curves are fundamental in the field of vector graphics, CAD (Computer-Aided Design), and font design.
The concept of Bezier Curves was independently developed by Paul de Casteljau at Citroën in 1959, though Pierre Bézier's name became synonymous with these curves due to his extensive work and publications on the subject. Bézier published his findings in the early 1960s, and his approach to curve design was soon adopted in various applications, particularly in the design of automobile bodies and later in the development of computer graphics.
A Bezier Curve of degree n is defined by:
where \( t \) is the parameter ranging from 0 to 1, and \( \binom{n}{i} \) is the binomial coefficient.
The curve itself can be expressed as:
\[ B(t) = \sum_{i=0}^{n} B_{i,n}(t) \cdot P_i \]While Bezier Curves are versatile, they have limitations:
Extensions like B-Splines and NURBS (Non-Uniform Rational B-Splines) were developed to overcome these issues by providing local control and the ability to represent conic sections and surfaces more accurately.