The uniqueness of the rational B?zier polygon is unique

Given a properly parameterized rational Bezier curve of irreducible degree, it is well-known that its control polygon is uniquely defined. We prove that this property is peculiar to the Bezier model. In the rational form associated with any other normalized polynomial basis, Moebius reparameterizations change the control polygon, so the same segment admits different polygons. This feature stems from identifying a remarkable algebraic property of Bernstein polynomials, namely that they provide the only normalized eigenbasis of the linear map, transforming homogeneous control points, that a Moebius reparameterization with fixed endpoints defines. Indeed, preserving the affine control polygon boils down to stretching the homogeneous points, i.e., a map with a diagonal matrix. Our result carries over to the alternative representation of rational curves using trigonometric polynomials so that, once again, the control polygon associated with the corresponding normalized B-basis is the only one enjoying uniqueness. (c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Automated summary

Research shows that in the Bezier model, the control polygon of a properly parameterized rational Bezier curve of irreducible degree is unique. However, in the rational form associated with other normalized polynomial bases, Moebius reparameterizations can change the control polygon, resulting in different polygons for the same curve segment.