2D curve representations usually take algebraic forms in ways not related to visual perception. This poses great difficulties in connecting curve representation with object recognition where information computed from raw images must be manipulated in a perceptually meaningful way and compared to the representation. In this paper we show that 2D curves can be represented compactly by imposing sh...

A method is proposed for constructing a spline curve of the Bezier type, which is continuous along with its first derivative by a piecewise polynomial function. Conditions for its existence and uniqueness are given. The constructed curve lies inside the convex hull of the control points, and the segments of the broken line connecting the control points are tangent to the curve. To construct the...

Shape modeling using planar cubic algebraic curves calls for computing the real inflection points of these curves since inflection points represents important shape feature. A real inflection point is also required for transforming projectively a planar cubic algebraic curve to the normal form, in order to facilitate further analysis of the curve. However, the naive method for computing the inf...

Derivatives and normals of rational Bézier curves and surface patches are discussed. A nonuniformly scaled hodograph of a degree m × n tensor-product rational surface, which provides correct derivative direction but not magnitude, can be written as a degree (2m−2) × 2n or 2m× (2n−2) vector function in polynomial Bézier form. Likewise, the scaled normal direction is degree (3m−2)× (3n−2). Effici...

A novel extension of the cubic Bézier curve with two shape parameters is presented in this work. The proposed curve is still a cubic polynomial model, which has simpler structure than other similar models. The proposed curve has the same properties with the usual cubic Bézier curve and its shape can be adjusted by altering values of the two shape parameters while the control points are fixed. W...

Vercauteren [27] introduced a notion of optimal pairings. Up to know the only known optimal pairing is the optimal Ate pairing. In this paper, we give some properties of optimal pairing and provide an algorithm for finding an optimal pairing if there exists one which is defined on the given elliptic curve. Applying the cyclotomic polynomial, we construct some new optimal pairings and provide a ...

We present a new approach toward the rational parametrization of canal surfaces. According to our previous work, every canal surface with rational (respectively polynomial) spine curve and rational (respectively polynomial) radius function is a rational (respectively polynomial) Pythagorean hodograph curve in R3,1. Drawing upon this formalism and utilizing the underlying Lorentzian geometry, th...

This paper is devoted to the comparison of various shape properties of triangular Bézier surfaces and of their Bézier nets, such as polyhedral convexity, axial convexity and subharmonicity. In order to better compare these properties, the different notations used by different authors are unified. It also includes counterexamples for the results that are not true.  1999 Elsevier Science B.V. Al...

Observational data hint at a finite universe, with spherical manifolds such as the Poincaré dodecahedral space tentatively providing the best fit. Simulating the physics of a model universe requires knowing the eigenmodes of the Laplace operator on the space. The present article provides explicit polynomial eigenmodes for all globally homogeneous 3-manifolds: the Poincaré dodecahedral space S3/...

In this paper, a class of rational spatial curves that have a rational binormal is introduced. Such curves (called PB curves) play an important role in the derivation of rational rotation-minimizing conformal frames. The PB curve construction proposed is based upon the dual curve representation and the Euler-Rodrigues frame obtained from quaternion polynomials. The construction significantly si...