Bezier Curves

Bezier curves is another method used for the construction of curves.  Properties:

1) Bezier curves always passes through first and last control points.
2) Basic functions are real.
3) Curve follows the shape of defining polygon.
4) Degree of polynomial defining the curve is one less than number of control points.
5) Direction of tangent vector at end points is same as that of the vector determined by first and last segments.
6) Curve lies entirely within the convex hull formed by 4 control points.
7) Curve exhibits variation diminishing property.
8) Due to the convex hull property polynomial follows the control points.
9) Curve is invariant under an affine transformation.

A cubic Bezier curve has 4 control points:

A cubic Bezier curve has 4 control points

Bezier Matrix:

Bezier Matrix

P(u) = (I – u)³ P1 +3u (1 – u)² P2 + 3u² (1-u) P3 + u³ P4

Leave a Reply