Bézier curve

1 Introduction

Bézier curves are a particular family of curves in m that parameterize a smooth path given points. They are often used to model or approximate general curved shapes because of their algebraic simplicity.

Bézier curves were originally developed by Paul de Casteljau in 1959. But it became a famous shape only when Pierre Bézier, a French engineer at Renault, used it to automobiles in the 1970’s. They are now widely used in many such as industrial and computer-aided , vector-based drawing, font design, and 3D modeling.

2 Definition

Given n+1 points 𝑷i in m, the Bézier curve for these control points is the curve 𝑩:[0,1]m defined by


The coefficients in this expression:


are the Bernstein basis polynomials.

3 Special cases

A Bézier curve between two control points is simply the line segmentMathworldPlanetmath between the two points:


For three control points, we have the quadratic Bézier curve:


For four control points, we have the cubic Bézier curve:


These are the most commonly employed versions of Bézier curves.

4 Properties

  • Except for the first and the last points, where the curve starts and ends, the Bézier curve should not pass through any of the control points. i.e. all the controls points lie on the curve if and only if the curve is a straight line.

  • The curve is tangent to the segment of line 𝑷0𝑷1¯ at the start and 𝑷n-1𝑷n¯ at the end.

  • It is easy to keep until the first order continuity when chaining curves. Moreover, every Bézier curve can be cut at any point into two new Bézier curves. In fact, the industry uses series of Bézier curves with only 4 control points (a bicubic version of the equations) to draw complex shapes.

  • The curves never do fancy stuff and always remain within the convex hull of the control points.

  • For one and two points the definition is still valid and the result became a point and a segment of line, respectively.

  • But it is impossible to draw circles or arcs of circle.

  • It is also impossible to draw two parallel curves (except in trivial cases: straight line Béziers and coincident Béziers curves)

  • If the first and the last point are the same, the curve is closed. And if 𝑷1 and 𝑷n-1 are aligned with 𝑷0 then the curve is closed with a first order continuity.

5 Curve Cut

Let 𝑩p(t) be a Bezier curve, defined by its control points 𝑷0 to 𝑷n for t[0,1].

Let a,b be such as [a,b][0,1] Let u=t-ab-a be such as u[0,1],t[a,b]

𝑩q(u) is a part of the initial bezier curve with new control points 𝑸0 to 𝑸n whose will be determined.

For n control points 𝑩(t) is a n-1-order polynom.



Let take n-2 more points, evenly spaced in [a,b] ui=k/n-1k[1..n-1]

𝑩p(u(b-a)+a)=𝑩q(ui) for each ui

For a 4 control points bezier curve If we take u in (0,13,23,1), hence t0, t1, t2, t3 respectively equal to a, b-2a3, 2b-a3, b






Title Bézier curve
Canonical name BezierCurve
Date of creation 2015-01-23 16:33:51
Last modified on 2015-01-23 16:33:51
Owner yota (10184)
Last modified by yota (10184)
Numerical id 29
Author yota (10184)
Entry type Definition
Classification msc 51N05
Synonym Bezier curve
Related topic NURBS
Related topic BSpline