We describe a new de Casteljau–type algorithm for complex rational Bézier curves. After proving that these curves exhibit the maximal possible circularity, we construct their points via a de Casteljau–type algorithm over complex numbers. ...

• Discussion of the circularity of complex rational Bézier curves.
• New de Casteljau-type algorithm based on linear factors of the denominator.
• Geometric construction of Farin points for Sanchez-Reyes' de Casteljau algorithm.

Moving planes have been widely recognized as a potent algebraic tool in various fundamental problems of geometric modeling, including implicitization, intersection computation, singularity calculation, and point inversion of parametric surfaces. ...

• An efficient approach to computing a set of moving planes that follow the given rational parametric surface is proposed.
• The moving planes are significant to surface implicitization, surface intersection, inversion formula, etc.
• ...

The multirational blossom of order k and degree −n of a k-times differentiable function f ( t ) is defined as a multivariate function f ( u 1 , … , u k / v 1 , … , v k + n ) characterized by four axioms: bisymmetry in the u and v parameters, ...

• A constructive proof that the multirational blossom of a differentiable function is uniquely characterized by four axioms.
• This uniqueness clarifies an explicit formula for the multirational blossom in terms of the divided differences.

We study the smoothness of envelopes generated by motions of rotational rigid bodies in the context of 5-axis Computer Numerically Controlled (CNC) machining. A moving cutting tool, conceptualized as a rotational solid, forms a surface, called ...

• C 0 tools still have plenty of room for sweeping continuous envelopes.
• Tools and motions designed with quadratic B-splines are C 1.
• Singularities in envelopes could be hard to avoid and stable.
• An Inverse Function Theorem for ...

This paper summarizes interesting results on systematic backward and forward error analyses performed for corner cutting algorithms providing evaluation of univariate and multivariate functions defined in terms of Bernstein and Bernstein related ...

• De Casteljau-type algorithms are numerically stable under imprecise computer arithmetic or perturbed input data.
• Bernstein-type bases are optimally stable for evaluation among all bases of the generated spaces formed by nonnegative ...

The quintics are the lowest–order planar Pythagorean–hodograph (PH) curves suitable for free–form design, since they can exhibit inflections. A quintic PH curve r ( t ) may be constructed from a complex quadratic pre–image polynomial w ( t ) by ...

Left: The four canonical–form quintic PH curves r ( t ) determined by the two end derivatives d i = 2.0 + 1.5 i, d f = 1.5 – 4.5 i. Right: the corresponding pre–image polynomials w ( t ), showing the footpoints of the origin on them.

• The presence or absence of inflections on planar quintic PH curves is characterized.
• The space of planar PH quintics is partitioned by curves that possess points of infinite curvature.
• Proximity of the pre-image parabola to the ...

5-axis CNC (Computer Numerical Control) milling is widely used to create complex part geometries in the industrial area. The cutting tool creates the swept volume as it moves along the defined path. The swept volume is subtracted from the initial ...

• A swept volume model is suggested for 5-axis CNC milling with a flat-end mill.
• The swept volume is generated in four steps.
• Swept boundaries are derived from a parameterized circle along the tool axis.

The Riemannian center of mass (a.k.a. Karcher mean or Fréchet mean) provides the equivalent to the Euclidean affine average on manifolds. In spite of its many potential applications in computer graphics and geometric modeling, there exist ...

• The definition of a curve through weighted averages can be extended to the manifold setting using the Riemannian center of mass (RCM).
• The RCM can be estimated on a triangle mesh using a discretization of differential operators.
• An ...

In the first part of the paper a planar generalization of offset curves is introduced and some properties are derived. In particular, it is seen that these curves exhibit good regularity properties and a study on self-intersection avoidance is ...

• Generalized plane offsets have good regularity properties.
• Generalized plane offsets to rational Pythagorean-hodograph curves are rational.
• A bound on the offset distance prevents self-intersections on generalized offsets.
• Any ...

The space of C 1 cubic Clough–Tocher splines is a classical finite element approximation space over triangulations for solving partial differential equations. However, for such a space there is no B-spline basis available, which is a preferred ...

Bézier curves are an essential tool for curve design. Due to their properties, common operations such as translation, rotation, or scaling can be applied to the curve by simply modifying the control polygon of the curve. More flexibility, and ...

In the construction and analysis of a planar Pythagorean–hodograph (PH) quintic curve r ( t ), t ∈ [ 0 , 1 ] using the complex representation, it is convenient to invoke a translation/rotation/scaling transformation so r ( t ) is in canonical ...

• The construction of quintic PH curves with two prescribed control-polygon legs is formulated.
• For given control-polygon legs, only the solution of a quadratic or quartic equation is required.
• Certain families of quintic PH curves ...

We generate hierarchical T-meshes by repeatedly inserting new line segments, in order to adapt both the size and the shape of the cells to the specific requirements of the underlying application. The associated spaces of bilinear spline functions ...

In this paper, the well-known Savitzky–Golay filter is generalized for three-dimensional triangular meshes. Savitzky–Golay filter is designed for smoothing noisy measurement signals, initially, a locally polynomial model is fitted to ...

• Savitzky-Golay filter can be generalized for 3D triangle meshes.
• Mesh smoothing ...

We propose a new method for constructing rational spatial Pythagorean Hodograph (PH) curves based on determining a suitable rational framing motion. While the spherical component of the framing motion is arbitrary, the translation part ...

• Compute rational PH curves from framing motions plus denominator polynomial.
• ...

Despite the fact that the orthogonal projection of a spatial Pythagorean hodograph (PH) curve into the plane is not a planar PH curve in general, we can find special cases such that the PH property is preserved when the curve is ...

• PH curves are characterized via intersection multiplicity of a suitable associated curve with the absolute conic.

In this paper a construction of a globally G 1 family of Bézier surfaces, defined by smoothing masks approximating the well-known Catmull-Clark (CC) subdivision surface is presented. The resulting surface is a collection of Bézier ...

• We propose geometrically smooth approximation schemes for Catmull-Clark limit surfaces by means of biquintic Bézier patches

Unrefinement is a tool that allows to perform faster numerical simulations by controlling the level of precision in the specified area. We introduce an algorithm that creates a coarser geometry from an initial regular geometry, which ...

• We generate a coarser THB-spline geometry representation from a given finer one.

We propose a novel local approximation method for analysis-suitable T-spline (AS T-spline) spaces via quasi-interpolation. The quasi-interpolants are defined as linear combination of the approximated function's values at appropriately ...

• A quasi-interpolation for analysis-suitable T-spline spaces is proposed.
• The ...

In this paper we deal with the following problem: given an algebraic plane curve C, implicitly defined, we determine its “asymptotic family”, that is, the set of algebraic curves that have the same asymptotic behavior as C.

• Given a plane curve C, we determine the set of curves having the same asymptotic behavior as C.