The uniqueness of the rational Bézier polygon is unique

Given a properly parameterized rational Bézier curve of irreducible degree, it is well-known that its control polygon is uniquely defined. We prove that this property is peculiar to the Bézier 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.