Multi-Degree Reduction of Bézier Curves with Distance and Energy Optimization

In this paper, we propose a new approach to the problem of degree reduction of Bézier curves based on the given endpoint constraints. A differential term is added for the purpose of controlling the smoothness to a certain extent. Considering the adjustment of second derivative in curve design, a modified objective function including two parts is constructed here. One part is a kind of measure of the distance between original high order Bézier curve and degree-reduced curve. The other part represents the second derivative of degree-reduced curve. We tackle two kinds of conditions which are position vector constraint and tangent vector constraint respectively. The explicit representations of unknown points are presented. Some examples are illustrated to show the influence of the differential terms to approximation and smoothness effect.

A quadratic programming method for optimal degree reduction of Bézier curves with G^1-continuity

This paper presents a quadratic programming method for optimal multi-degree reduction of B6zier curves with G^1-continuity. The L2 and I2 measures of distances between the two curves are used as the objective functions. The two additional parameters, available from the coincidence of the oriented tangents, are constrained to be positive so as to satisfy the solvability condition. Finally, degree reduction is changed to solve a quadratic problem of two parameters with linear constraints. Applications of degree reduction of Bezier curves with their parameterizations close to arc-length parameterizations are also discussed.

Constrained multi-degree reduction of triangular Bézier surfaces

This paper proposes and applies a method to sort two-dimensional control points of triangular Bezier surfaces in a row vector. Using the property of bivariate Jacobi basis functions, it further presents two algorithms for multi-degree reduction of triangular Bezier surfaces with constraints, providing explicit degree-reduced surfaces. The first algorithm can obtain the explicit representation of the optimal degree-reduced surfaces and the approximating error in both boundary curve constraints and corner constraints. But it has to solve the inversion of a matrix whose degree is related with the original surface. The second algorithm entails no matrix inversion to bring about computational instability, gives stable degree-reduced surfaces quickly, and presents the error bound. In the end, the paper proves the efficiency of the two algorithms through examples and error analysis.

APPROXIMATE COMMON DIVISORS OF POLYNOMIALS AND DEGREE REDUCTION FOR RATIONAL CURVES

Abstract This paper deals with how to perturb a given set of polynomials so as to include a common linear factor. An algorithm is derived for determining such a set of perturbation polynomials which are subject to certain constrains at the endpoints of a prescribed parametric interval and minimized in a certain sense. This result can be combined with subdivision technique to obtain a continuous piecewise approximation to a rational curve.

Assessment of Nanoparticle-Enriched Solvents for Oil Recovery Enhancement

Solvents are generally used to reduce the viscosity of heavy crude oil and ultimately enhance oil recovery.Recently,a new method has been introduced where nanoparticles(NPs)are exploited to induce enhanced oil recovery owing to their ability to improve the mobility ratio,dampen the interfacial tension,and alter rock wett-ability.This study investigated the integration of nano-alumina(Al_(2)O_(3))NPs with an n-hexane solvent.In parti-cular,a Brookfield viscometer has been used to measure the crude oil viscosity and it has been found that NPs can effectively lead to a significant decrease in the overall oil viscosity(70 cp using the solvent only,45 cp when NPs are added).

Activation of small molecules over praseodymium-doped ceria

Praseodymium can modify the properties of ceria (CeO2), changing the electronic structure, reducibility and catalytic behavior. Oxygen vacancies in the ceria-based samples can activate C–O and C–H bonds of small molecules such as CO2 and propane. Partially reduced Pr/CeO2-x can selectively activate C–H of propane, giving a propylene selectivity of ca. 75% at a propane conversion of 5% to 10%. Excess reduction of Pr/CeO2-x induces coking reactions during propane dehydrogenation, resulting in fast catalyst deactivation.

Reduction mechanism of Fe2O3-Cr2O3-NiO system by carbon

Isothermal experiments on the reduction of Fe_2O_3-Cr_2O_3-NiO(molar ratio of Fe-to-Cr-to-Ni is 3:2:2)by graphite were carried out at 1350–1550°C,and effects of various factors on reduction degree were studied.The results show that the reaction rate of the Fe_2O_3-Cr_2O_3-NiO system is fast during the initial period(reduction degree,α<38%),and then the rate decreases until the end of the reduction.Factors such as temperature,carbon content,sample size have a more significant effect during the final stage(α>38%).The metallic product formed at the initial stage(a Fe-Ni alloy)greatly promotes the reduction of Cr2O3 at the final stage.Further,during the reduction of Fe_2O_3-Cr_2O_3-NiO by carbon,interfacial reaction is the rate-controlling step and g(α)=1-(1-α)0.5 is the reaction mechanism for the initial stage,whereas two-dimensional diffusion is the rate-controlling step and f(α)=α+(1-α)ln(1-α)is the reaction mechanism for the final stage.The apparent activation energies are 55.43 k J/mol and 174.54 k J/mol for the initial and the final stages,respectively.

UNIFYING REPRESENTATION OF BZIER CURVE AND GENERALIZED BALL CURVES

This paper presents two new families of the generalized Ball curves which include the Be ′zier curve, the generalized Ball curves defined by Wang and Said independently and some intermediate curves. The relative degree elevation and reduction schemes, recursive algorithms and the Bernstein\|Be ′zier representation are also given.

Approximate Degree Reduction of Bezier Curves

Degree reduction of parametric curves and surfaces is an important process in data communication between CAD systems. The degenerate condition of Bezier curves and the constrained optimization method are used to develop a new degree reduction method for Bezier curves. An error analysis of the degree reduction is also given. The degree reduction scheme is combined with a subdivision algorithm to generate lower degree approximations which are within some preset error tolerance of the prescribed Bezier curve. Geometric continuity between adjacent curve segments is also considered in the subdivision/degree reduction process.

Approximate Degree Reduction of Triangular Bezier Surfaces

Degree reduction of parametric curves and surfaces is an important process in the exchange of product model data between various CAD systems. In this paper the degenerate conditions of triangular Bezier surface patches are derived. The degenerate conditions and constrained optimization methods are used to develop a degree reduction method for triangular Bezier surface patches. The error in the degree reduction of a triangular Bezier surface is also shown to depend on some geometric invariants which decrease exponentially in the subdivision process. Therefore, the degree reduction method can be combined with a subdivision algorithm to generate lower degree approximations which are within some preset error tolerance.

Optimal multi-degree reduction of C-Bezier surfaces with constraints

We propose an optimal approach to solve the problem of multi-degree reduction of C-Brzier surfaces in the norm L2 with prescribed constraints. The control points of the degree-reduced C-Brzier surfaces can be explicitly obtained by using a matrix operation that is based on the transfer matrix of the C-Brzier basis. With prescribed boundary constraints, this method can be applied to piecewise continuous patches or to a single patch with the combination of surface subdivision. The resulting piecewise approximating patches are globally G1 continuous. Finally, numerical examples are presented to show the effectiveness of the method.

Mineralization characteristics of pellets prepared by ultrafine magnetite concentrate:influence on metallurgical property and improving method

The reducing property of pellets prepared by ultrafine magnetite concentrate(UMC)and improving method were revealed.The results show that the reduction degree of UMC pellets is only about 56%compared with that of pellets prepared from ordinary iron ore concentrate with relatively coarse particle size,which is significantly lower than the general reduction degree of about 70%.When the composite binder composed of bentonite and organic binder was added,the reduction degree was significantly increased to 69.66%.The revealed mechanism shows that the reduced pellets with common bentonite have a concentric structure,the oxidation gap between the inner and outer layers is obvious,and the outer dense oxide layer hinders the oxidation and reduction of the inner layer.After adding the composite binder,the organic components significantly improved the internal porosity of the pellets and the aggregation degree of ultrafine iron ore concentrate particles in the granulation process,forming a porous structure.The non-uniform double-layer structure is eliminated,and the increased pores are conducive to the internal diffusion of CO,and finally the reduction degree of pellets is increased to the level equivalent to that of ordinary magnetite pellets.

Softening and Melting Behavior of Mixed Burden for Oxygen Blast Furnace

The behaviors of mixed burden in the cohesive zone of oxygen blast furnace were studied by softening and melting tests, and the influence of reducing gas and burden basicity on the softening and melting behaviors of mixed burden was also investigated. The results indicated that the softening range became wide, however, the melting range narrowed sharply in the atmosphere of oxygen blast furnace. The permeability of burden in the oxygen blast furnace was obviously improved comparing with the conventional blast furnace. In addition, the content of sulphur in the dripping iron of oxygen blast furnace was much lower than that of conventional blast furnace, however, the content of carbon increased. An optimum basicity of burden, which could lead to the appearance of the narrower melting range and better permeability of burden, was obtained in the atmosphere of oxygen blast furnace.

An orthogonal basis for the hyperbolic hybrid polynomial space

Motivated by the wlde usage of the Tchebyshev basis and Legendre basis in the algebra polynomial space, we construct an orthogonal basis with the properties of the H-Bézier basis In the hyperbolic hybrid polynomial space, which is similar to the Legendre basis and holds remarkable properties. Moreover, we derive the transformation matrices that map the H-Bézler basis and the orthogonal basis forms into each other. An example for approximating the degree reduction of the H- Bézier curves Is sketched to Illustrate the utility of the orthogonal basis.