Given a set of control points,, ...,, the corresponding Bézier curve (or Bernstein-Bézier curve) is given by

where is aBernstein polynomial and. Bézier splines are implemented in theWolfram Language asBezierCurve[pts].

A "rational" Bézier curve is defined by

where is the order, are theBernstein polynomials, are control points, and the weight of is the last ordinate of the homogeneous point. These curves areclosed under perspective transformations, and can representconic sections exactly.

The Bézier curve always passes through the first and last control points and lies within theconvex hull of the control points. The curve is tangent to and at the endpoints. The "variation diminishing property" of these curves is that no line can have more intersections with a Bézier curve than with the curve obtained by joining consecutive points with straight line segments. A desirable property of these curves is that the curve can betranslated and rotated by performing these operations on the control points.

Undesirable properties of Bézier curves are their numerical instability for large numbers of control points, and the fact that moving a single control point changes the global shape of the curve. The former is sometimes avoided by smoothly patching together low-order Bézier curves. A generalization of the Bézier curve is theB-spline.

See also

Explore with Wolfram|Alpha

More things to try:

• 28
• functional equation for zeta(s)
• horizontal asymptotes -2x / sqrt(1+x^2)

References

Referenced on Wolfram|Alpha

Cite this as:

Weisstein, Eric W. "Bézier Curve."FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/BezierCurve.html

Subject classifications