**Bezier Curve Segments and their Properties****Definition:**

A degree*n*(order*n*+1)**Bézier curve segment**iswhere the are

*k*-dimensional**control points**.**Convex Hull:**

, for

*P*(*t*) is a convex combination of the for

*P*(*t*) lies within convex hull of for .

**Affine Invariance:**

A Bézier curve is an affine combination of its control points.

Any affine transformation of a curve is the curve of the transformed control points.**This property does not hold for projective transformations!****Interpolation:**

, , , for

if , if

, .**Derivatives:**

, .

**Smoothly Joined Segments ( ):**

Let , be the last two control points of one segment.

Let , be the first two control points of the next segment.

**Smoothly Joined Segments ( ):**- :

If then segments have same derivatives at endpoints and are said to meet with continuity

**Recurrence, Subdivision:**

de Casteljau's algorithm:Use to evaluate point at

*t*, or subdivide into two new curves:- , , ... are the control points for the left half;
- , , ... are the control points for the right half

**Matrix view:**-
- Expand each Bernstein polynomial in powers of
*t* - Represent each expansion as the column of a matrix
- Quadratic example:
- In matrix format:
- is the
**monomial basis** - is a matrix containing the coefficients of the polynomials for each dimension of
- is a
**change of basis matrix**that converts a specification*P*of*P*(*t*) relative to the Bernstein basis to one relative to the monomial basis

- is the

- Expand each Bernstein polynomial in powers of

University of Waterloo

Computer Graphics Lab