Bézier Curve
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.gif)
. 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.
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