We revisit the rational cubic Bézier representation of conics, simplifying and expanding previous works, elucidating their connection, and making them more accessible. The key ingredient is the concept of conic associated with a given (planar) cubic Bézier polygon, resulting from an intuitive geometric construction: Take a cubic semicircle, whose control polygon forms a square, and apply the perspective that maps this square to the given polygon. Since cubic conics come from a quadratic version by inserting a base point, this conic admitting the polygon turns out to be unique. Therefore, detecting whether a cubic is a conic boils down to checking out whether it coincides with the conic associated with its control polygon. These two curves coincide if they have the same shape factors (aka, shape invariants) or, equivalently, the same oriented curvatures at the endpoints. Our results hold for any cubic polygon (with no three points collinear), irrespective of its convexity. However, only polygons forming a strictly convex quadrilateral define conics whose cubic form admits positive weights. Also, we provide a geometric interpretation for the added expressive power (over quadratics) that such cubics with positive weights offer. In addition to semiellipses, they encompass elliptical segments with rho-values over the negative unit interval.

The weights of rational Bézier curves cannot be regarded as true independent shape factors since they do not enjoy invariance with respect to Moebius (i.e., rational linear) reparametrizations, which do not change the curve shape. However, the existence of such shape factors, also called shape invariants, is well-known. They are associated with each inner control point and are computed as the ratio of weight ratios for three consecutive control points. We show that these shape factors, in addition to their invariance to Moebius reparameterization, provide a more convenient shape control than the customary weights since they exert a more localized push/pull. Each shape factor amounts to that of the conic defined by a triplet of consecutive control points and weights. Thus, shape factors can be controlled in a geometric way using existing techniques for conics by setting the conic rho-factor via moving the associated shoulder point. Each shoulder point moves along a radial direction through its corresponding control point, furnishing a more practical shape handle than sliding the traditional weight points (aka Farin points) on the polygon legs.

Recently, Bizzarri et al. (2021) discussed the so-called Pythagorean-hodograph curves of Tschirnhaus type, a generalization to higher degrees of Tschirnhausen cubic. We recall that these curves in Bézier form coincide with the typical curves introduced by Mineur et al. (1998), as well as with a classical family of sinusoidal spirals. Therefore, they all enjoy the same properties, such as the rational character of their offsets or the existence of only one curve (up to similarities) for each degree. By elucidating this connection among curves of Tschirnhaus type, typical curves, and sinusoidal spirals, we rederive several relevant results found by Bizzarri et al. (2021).

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.

Most Escher-like impossible figures admit a realization as 3D objects, when seen from a specific viewpoint. To avoid the restrictions of polyhedral models, we explore how to build these 3D objects with curved surfaces, through the prism of CAGD. Such impossible objects can be regarded as optical anamorphisms, a visual illusion based on depth misperception, which can be achieved by using standard CAGD techniques, in particular homogeneous coordinates and NURBS. The main tool is an anamorphic deformation, i.e., along radial lines through the viewpoint. Deforming NURBS boils down to the straightforward exercise of displacing the control points along such radial directions and simultaneously changing their weights. Thus, we change the distance to the observer of different parts of an object, which leads to depth misperception and apparent contradiction. The deformation alters shading, but the introduction of textures in parameter space, unaffected by the deformation, deceives the eye and helps conceal this artifact, as 3D printouts corroborate. Examples confirm the superiority of our textured NURBS over simpler polyhedral models.

In an optical anamorphosis, an object is seen distorted unless the viewer is positioned at a specific point, where the object appears normal. We describe how to endow a rational Free-Form Deformation with an anamorphic character in a simple manner, obtaining an AFFD (Anamorphic Free-Form Deformation). Given a (planar or 3D) initial object, which will appear normal from the desired viewpoint, we deform the object with a rational Bézier surface or volume as deformation tool. To achieve the desired deformation, the user input amounts to displacing the control points of the deformation tool along radial directions through the viewpoint, whereas the weights come as a byproduct. Mathematically, the deformation means changing the last homogeneous coordinate of the control points. An AFFD defined by a linear Bézier tetrahedron can be regarded as a user-friendly way to construct a perspective collineation. In this case, or when the deformation tool is a Bézier triangle of degree one, the AFFD transforms NURBS to NURBS keeping the original degree. With a deformation tool of higher degree, the rational composition required to obtain the exact result yields NURBS also of higher degree. For a quadratic Bézier triangle as deformation tool, our AFFD coincides with the inverse of a stereographic projection.

We propose a unified methodology to tackle geometry processing operations admitting explicit algebraic expressions. This new approach is based on representing and manipulating polynomials algebraically in a recently basis, the symmetric analogue of the power form (s-power basis for brevity), so called because it is associated with a “Hermite two-point expansion” instead of a Taylor expansion. Given the expression of a polynomial in this basis over the unit interval u ε[0, 1], degree reduction is trivally obtained by truncation, which yields the He many terms as desired of the corresponding Hermite interpolant and build “s-power series,” akin to Taylor series. Applications include computing integral approximations of rational polynomials, or approximations of offset curves.

A new polynomial basis over the unit interval t∈[0,1] is proposed. The work is motivation by the fact that the monomial (power) form is not suitable in CAGD, as it suffers from serious numerical problems, and the monomial coefficients have no geometric meaning. The new form is the symmetric analogue of the power form, because it can be regarded as an “Hermite two-point expansion” instead of a Taylor expansion. This form enjoys good numerical properties and admits a Horner-like evaluation algorithm that is almost as fast as that of the power form. In a ddition, the symmetric power coeddicients convey a geometric meaning, and therefore they can be used as shape handles. A polynomial expressed in the symmetric power basis is decomposed into linear, cubic, quintic, and successive components. In consequence, this basis is better suited to handle polynomials of different degrees than the Bernstein basis, and those algorithms involving degree operations have extremely simple formulations. The minimum degree of a polynomial is immediately obtained by inspecting its coefficients. Degree reduction of a curve or surface reduces to drooping the desired high degree terms.