Beta-bezier curves

A new definition of Beta-Bezier curves which include classic Bezier curves as a special case is given. With the new definition, the functions of Beta-Bezier curves are easier to study. It shows that Beta-Bezier curves not only have all the basic properties of Bezier curves such as convex hull property, recursive subdivision, B-spline conversion and C² interpolation, but also the capability of modifying the shape a Bezier curve segment or a C²-continuous, composite cubic Bezier curve without changing the control points of the curve. This is because in the cubic case a Beta-Bezier curve is actually also a Bezier curve. Hence, we have a curve design technique more general than Bezier curves. Since C²-continuous, composite cubic Bezier curves are equivalent to uniform B-spline curves, this means the new curve design technique is more general than uniform B-spline curves as well.