We have learned that projecting a 4-dimensional B-spline curve to
hyperplane *w*=1 yields a 3-dimensional NURBS curve. What if this
B-spline curve is a Bézier curve? The result is a
**Rational Bézier** curve!
The left image below shows a rational Bézier curve of degree 4,
and the right one shows the relationship between a 3-dimensional
Bézier curve of degree 4 (in red) and its projection rational
Bézier curve (in blue) in hyperplane *w*=1.

What is the curve defined by
a set of *n*+1 control points **P**_{0},
**P**_{1}, ..., **P**_{n}, each of which is
associated with a non-negative weight *w*_{i}
(*i.e.*, **P**_{i} has weight
*w*_{i} >= 0), and knots 0 (multiplicity *n*+1)
and 1 (multiplicity *n*+1)?
Since the 4-dimensional B-spline curve defined by the lifted control points
**P**^{w}_{i} (0 <= *i* <= *n*)
reduces to a Bézier curve of degree *n*, its basis functions are
*B*_{n,0}(*u*), *B*_{n,1}(*u*),
..., *B*_{n,n}(*u*). Projecting this
Bézier curve to hyperplane *w* = 1, we have the following:

where *R*_{i,n}(*u*) is defined as follows:

This is a special case of NURBS curves and is referred to as a
*rational Bézier* curve.
Since a rational Bézier curve is a special case of NURBS curves,
rational Bézier curves satisfy all important properties that NURBS curves
have. This is similar to the fact that Bézier curves have all the
important properties
of B-spline curves. However, since there is no internal knots, rational
Bézier curves do not have the local modification property, which means
modifying a control point or its weight will cause a global change. Moreover,
the curve is now contained in the convex hull defined by the whole set of
control points, and modifying the weight of a control point will push or pull
the curve away from or toward the control point. Of course, rational
Bézier curves are projective invariant rather than affine invariant!
See **NURBS: Important Properties**
for more details.