A Bezier curve is a weighted sum of *n*+1 control points,
,
where the weights
are the Bernstein polynomials:

(74) |

The Bezier curve of order

**Linear interpolation.**- Equation 75 is obviously
a linear interpolation between two points. Equation 76 can
be rewritten as a linear interpolation between linear interpolations
between points:

(78)

Equation 77 can be rewritten as a linear interpolation between linear interpolations between linear interpolations between points. This is left as an exercise for the reader. **Weighted average.**- A Bezier curve can be seen as a weighted average of all of its control points. Because all of the weights are positive, and because the weights sum to one, the Bezier curve is guaranteed to lie within the convex hull of its control points.
**Refinement of the control polygon.**- A Bezier curve can be seen
as some sort of refinement of the polygon made by connecting its
control points in order. The Bezier curve starts and ends at the two
end points and its shape is determined by the relative positions of
the
*n*-1 other control points, although it will generally not pass through these other control points. The tangent vectors at the start and end of the curve pass through the end point and the immediately adjacent point.

Any Bezier curve is infinitely differentiable within itself, and is
therefore continuous to any degree (*C*^{n}-continuous, ). We
therefore only need concern ourselves with continuity across the joins
between curves. Assume that we have two Bezier curves of the same
order:
,
defined by
,
and
,
defined by
.
*C*^{0}-continuity (continuity of position) can be achieved by setting
.
This gives a formula for
in
terms of the s:

Similarly for

Combining Equations 80 and 79 gives a formula for in terms of the s:

= | (81) | ||

= | (82) |

Continuing in this vein, we find that the requirements for

Combining Equations 83, 80, and 79 gives a formula for in terms of the s:

= | (84) | ||

= | (85) |

(86) |

You can think about this as moving the control points of one Bezier curve along a set of Bezier curves to sweep out a surface. Continuity across a boundary between two Bezier patches is only guaranteed if each of the Bezier curves across the join obey the curve continuity conditions. Again, this was covered in the IB course.