Given a set of 
control points 
,
, ..., 
, the corresponding Bézier curve (or Bernstein-Bézier
curve) is given by
 |
where 
is a Bernstein polynomial and ![t in [0,1]](/images/equations/BezierCurve/Inline6.svg)
. Bézier splines are implemented in the Wolfram Language as BezierCurve[pts].
A "rational" Bézier curve is defined by
 |
where 
is the order, 
are the Bernstein polynomials, 
are control points, and the weight 
of 
is the last ordinate of the homogeneous point 
. These curves are closed under
perspective transformations, and can represent conic
sections exactly.
The Bézier curve always passes through the first and last control points and lies within the convex 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 be translated 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 the B-spline.
