Sharp bounds on the approximation of a Bézier polynomial by its quasi-control polygon

Sharp, quantitative bounds on the distance between a polynomial piece and its Bézier control polygon

Sharp, Quantitative Bounds on the Distance Between a Bezier Curve and its Control Polygon

The distance between a Be-....ier segment and its control polygon is bounded in terms of the second differences of the control point sequence and a constant that depends only on the degree of the polynomial. The constant derived here is the smallest possible and is sharp for the Hausdorff distance between control polygon and curve segment. The bound provides a straightforward proof of quadratic...

Quasi-Quartic Trigonometric Bézier Curves and Surfaces with Shape Parameters

In this paper a new kind of quasi-quartic trigonometric polynomial base functions with two shape parameters λ and μ over the space Ω = span {1, sin t, cos t, sin2t, cos2t, sin3t, cos3t} is presented and the corresponding quasi-quartic trigonometric Bézier curves and surfaces are defined by the introduced base functions. Each curve segment is generated by five consecutive control points. The sha...

A Class of Quasi-Quartic Trigonometric BÉZier Curves and Surfaces

A new kind of quasi-quartic trigonometric polynomial base functions with a shape parameter λ over the space Ω=span {1, sint, cost, sint2t, cos2t} is presented, and the corresponding quasi-quartic trigonometric Bézier curves and surfaces are defined by the introduced base functions. The quasi-quartic trigonometric Bézier curves inherit most of properties similar to those of quartic Bézier curves...

Unknots with highly knotted control polygons

An example is presented of a cubic Bézier curve that is the unknot (a knot with no crossings), but whose control polygon is knotted. It is also shown that there is no upper bound on the number of crossings in the control polygon for an unknotted Bézier curve. These examples complement known upper bounds on the number of subdivisions sufficient for a control polygon to be ambient isotopic to its...