A polynomial basis with a shape parameter for curve and surface modeling

Abstract

Based on Bernstein polynomials, a system of functions with a free parameter is proposed in the space of polynomials of degree at most $n$. The system inherits several properties of Bernstein polynomials, such as linear independence, non-negativity, partition of unity and symmetry. This new family of functions are employed to construct control point based parametric curves. The free parameter serves as a shape adjustment parameter, by means of which a one-parameter family of polynomial curves is obtained. The new family of curves is in common with Bézier curves in most of the geometric properties, providing a smooth transition between the Bézier curve and the straight line segment joining the first and last control points. Shape preserving properties, such as monotonicity preservation, as well as length, hodograph and variation diminishing are studied. The proposed basis can also be used to create tensor product surfaces. The extent to which the suggested basis generation method can be applied to other (non-polynomial) function spaces is also being investigated.