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 ...

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 ...

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 ...

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 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 study bivariate dual 2-subdivision schemes using one-dimensional (1D) stencils and reproducing bivariate polynomials of high orders. We show that such dual 2-subdivision schemes always possess the quasi-interpolating ...

• Quasi-interpolating bivariate dual 2-subdivision schemes using 1D stencils are studied.

We extend the notion of intrinsic supersmoothness previously defined for splines on triangulations to that for semialgebraic splines: piecewise polynomials on partitions defined by algebraic curves.

Cell complex with one interior vertex defined by irreducible algebraic arcs.

• intrinsic supersmoothness is not prescribed in the definition of spline spaces.

The research subject of this paper is the spatial Pythagorean hodograph (PH) curves whose projections to the horizontal plane are planar PH curves. Because of this geometric configuration, we name them PH curves over PH curves, or ...

• Spatial Pythagorean hodograph curves over planer Pythagorean hodograph (PHoPH) are introduced.

• Construction of G1 continuous PH cubic biarcs is done.
• Thorough analysis of ...

The interpolation of two points and two tangent directions by planar parametric cubic curves with prescribed arc lengths is considered. It is well known that this problem is highly nonlinear if standard cubic curves are used. However, ...

We present an adaptive refinement algorithm for T-splines on unstructured 2D meshes. While for structured 2D meshes, one can refine elements alternatingly in horizontal and vertical direction, such an approach cannot be generalized ...

• Refinement algorithm for T-splines on unstructured meshes.
• Procedure is based ...

• Use polynomials of Wachspress GBCs to construct smooth vertex splines over convex quadrilaterals.

In this paper we construct smooth bivariate spline functions over a polygonal partition, e.g. a convex quadrilateral partition by using vertex spline techniques. Vertex splines, introduced in Chui and Lai (1985) are smooth piecewise ...

• We construct a new representation of CSRBF, which unites interpolation and approximation.

Interpolation and approximation radial basis function (RBF) has been widely used in the implicit surface reconstruction. However, a simple interpolation or approximation method may either have unwanted oscillation around the scattered ...

Spaces of polynomial splines defined on curved triangulations of curved domains are introduced and studied. In addition, some applications to interpolation and data fitting problems are discussed.

Curved triangulations of several curved domains.

• Defines splines on curved triangulations.
• Shows how to store, evaluate, and ...

• C2-continuous cubic spline based streaming interpolation of Hermite data not only is possible but also can be effective.

C 2-continuous Hermite interpolation of streaming data with the use of cubic splinelets – building blocks for C 2-continuous cubic splines – is presented. Two specific types of splinelets, denoted as A and B, are ...

• The connection between geometric continuity of PH curves and their associated ER frames is examined.

The problem of constructing a curve that interpolates given initial/final positions along with orientational frames is addressed. In more detail, the resulting interpolating curve is a PH curve of degree 7 and among the adaptive frames ...

Exponential polynomials are essential in subdivision for the reconstruction of specific families of curves and surfaces, such as conic sections and quadric surfaces. It is well known that if a linear subdivision scheme is able to ...

• Closed spatial loops are constructed using quintic Pythagorean hodograph curves.

The construction of continuous adapted orthonormal frames along C 1 closed–loop spatial curves is addressed. Such frames are important in the design of periodic spatial rigid–body motions along smooth closed paths. The construction is ...