In this paper, we show the degree reduction of Bézier curves in a matrix representation. Most degree reduction algorithms have been described as a set of recursive equations which are based on the inverse problem of degree elevation. However, degree elevation can be easily expressed in terms of matrices. Motivated by this observation, we represent most well known degree reduction algorithms in ...

As is well known, attribute reduction is one of the basic concepts in rough set theory. In this paper, a new algorithm for computing equivalence classes based on segment quick sort is provided and its time complexity is cut down to O(|C||U|) compared with traditional algorithms. Then a method for computing object number of positive region without computing positive region is given. On this basi...

(should we prove it?...) If deg(S) is significantly smaller than |S|, then it means that S has more algebraic structure than a generic set. We are going to explore the connection between combinatorial properties of a set S and its algebraic structure. We will see that interesting examples in the kind of incidence geometry questions we have been studying need to have algebraic structure. Once we...

In this paper we present a novel approach to consider the multi-degree reduction of Bézier curves with G2-continuity in L2norm. The optimal approximation is obtained by minimizing the objective function based on the L2-error between the two curves. In contrast to traditional methods, which typically consider the components of the curve separately, we use geometric information on the curve to ge...

In multiple models research, classiications or evidence is combined from several classiication models in order to form overall classiications with higher accuracy than any of the individual models. However, most previous work has concentrated on demonstrating that use of multiple models leads to statistically signiicant error reduction as compared to the error rate of the single model learned o...

A planar Bézier curve whose control points are disks is called a disk Bézier curve. It can be looked as a parametric curve with tolerance in the plane. It is an effective tool to measure or control the error. Based on minimum mean square error, this paper presents an algorithm for optimal multi-degree reduction of disk Bézier curves in L2 norm. First, applying the orthogonal property of Jacobi ...

DP curves, which possess the shape-preserving property and linear complexity evaluation algorithm, occupy a significantly important place in computer aided geometric design. Thus to do the research on the DP curves algorithms has a positive meaning. Aphirukmatakun et al. presented a degree elevation algorithm of DP curves based on the transformation between the DP basis and power basis. At the ...

Given a triangular Bézier surface of degree n, the problem of multi-degree reduction by a triangular Bézier surface of degree m with boundary constraints is investigated. This paper considers the continuity of triangular Bézier surfaces at the three corners, so that the boundary curves preserve endpoints continuity of any order α. The l2and L2-norm combined with the constrained least-squares me...

X ′ is the canonical example of a set which is definable from X but not recursive in X. The Turing degree of X ′ depends only on the Turing degree of X, so the jump induces an increasing function on the Turing degrees D. In this paper, we will discuss two aspects of the jump and its iterations. First, we will show that they are implicitly characterized by general properties of relative definabi...

x0. Introduction and Notation. Weak truth table reducibility (w-reducibility) was rst introduced by Friedberg and Rogers [FR59]. Intuitively, we say that a set A is w-reducible to a set B (written A w B) if there is a Turing reduction from A to B and a recursive function f such that, for any x, the value f(x) bounds the greatest number whose membership or nonmembership in B is used to determine...