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In the mathematical subfield of numerical analysis a Bézier curve is a parametric curve important in computer graphics. A numerically stable method to evaluate Bézier curves is de Casteljau's algorithm.

Generalizations of Bézier curves to higher dimensions are called Bézier surfaces; the Bézier triangle is a special case.

Bézier curves are also formed by many common forms of string art , where strings are looped across a frame of nails.

1 History

Bézier curves were widely publicized in 1962 by the French engineer Pierre Bézier who used them to design automobile bodies. The curves were developed in 1959 by Paul de Casteljau using de Casteljau's algorithm.

2 Definition

Given n+1 points P_i in R³ a Bézier curve of degree n is a parametric curve

composed of Bernstein basis polynomials of degree n

with the Bernstein basis polynomials defined as

P_i is called control point for the Bézier curve. A polygonA polygon (from the Greek poly for "many", and gonos for "angle") is a closed planar path composed of a finite number of sequential straight line segments. The straight line segments that make up the polygon are called its sides or edges and the points wh can be constructed by connecting the Bézier points with lineA line or straight line is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. a curve that is long and straight. Given two points, in Euclidean geometry, one can always find exactly one line that passes through thes, starting with P₀ and finishing with P_n. This polygon is called the Bézier polygon.

3 Notes

The starting point of the curve is P₀ and the ending point is P_n
The Bézier curve is completely contained in the convex hullIn mathematics, the convex hull for an object or a set of objects is the minimal convex set containing the given objects. It is the minimal convex set because the convex hull is a subset of any convex set which contains the given objects. The convex hull of the control points.
If and only if all control points lie on the curve it is a straight line.
The start (end) of the curve is tangentIn mathematics, the word tangent has two distinct, but etymologically related meanings: one in geometry, and one in trigonometry. Geometry In plane geometry, a straight line is tangent to a curve, at some point, if both line and curve pass through the poi to the first (last) section of the Bézier polygon.
A curve can be split into arbitrarily many subcurves, each of which is also a Bézier curve.
A Bézier curve cannot form a true circle.
A second Bézier curve that is truly parallel to an existing curve cannot be derived mathematically from the control points of the first (though there are heuristicFor heuristics in computer science, see heuristic (computer science Heuristic is the art and science of discovery and invention. The word comes from the same Greek root (`ευρισκω) as " eureka," meaning "to find". methods that work for practical purposes).

4 Examples

4.1 Linear Bézier curves

Given two control points P₀ and P₁ a linear Bézier curve is just a straight line between those two points. The curve is given by

4.2 Quadratic Bézier curves

A quadratic Bézier curve is the path traced by the function B(t). For points A, B, and C,

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Topics: Bezier Curve Curves Points Control Point Cubic Code Bernstein...

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