We propose an algorithm based on Newton's method and subdivision for finding all zeros of a polynomial system in a bounded region of the plane. This algorithm can be used to find the intersections between a line and a surface, which has applications in graphics and computer-aided geometric design. The algorithm can operate on polynomials represented in any basis that satisfies a few conditions. The power basis, the Bernstein basis, and the first-kind Chebyshev basis are among those compatible with the algorithm. The main novelty of our algorithm is an analysis showing that its running is bounded only in terms of the condition number of the polynomial's zeros and a constant depending on the polynomial basis. Introduction.The problem of line-surface intersection has many applications in areas such as geometric modeling, robotics, collision avoidance, manufacturing simulation, scientific visualization, and computer graphics. It is also a basis for considering intersections between more complicated objects. This article deals with intersections of a line and a parametric surface. The parametric method of surface representation is a very convenient way of approximating and designing curved surfaces, and computation using parametric representation is often much more efficient than other types of surface representations.Typically, intersection problems reduce to solving systems of nonlinear equations. Subdivision methods introduced by Whitted [21, 14] were the first to be used for this problem. In these methods, a simple shape such as rectangular box or sphere is used to bound the surface and is tested whether the line intersects the bounding volume. If it does, the surface patch is subdivided, and the bounding volumes are found for each subpatch. The process repeats until no bounding volumes intersect the line or the volumes are smaller than a specified size where the intersection between such volumes and the line are taken as the intersections between the surface and the line. Subdivision methods are robust and simple, but normally not efficient when high accuracy of the solutions are required. They also cannot indicate if there are more than one zero inside the remaining subdomains.Regardless, a variation of subdivision methods known as Bézier clipping by Nishita, Sederberg, and Kakimoto should be noted for its efficient subdivision [12]. For a nonrational Bézier surface, Bézier clipping uses the intersection between the convex hull of the orthographic projection of the surface along the line and a parameter axis
In [5], Srijuntongsiri and Vavasis propose the Kantorovich-Test Subdivision algorithm, or KTS, which is an algorithm for finding all zeros of a polynomial system in a bounded region of the plane. This algorithm can be used to find the intersections between a line and a surface. The main features of KTS are that it can operate on polynomials represented in any basis that satisfies certain conditions and that its efficiency has an upper bound that depends only on the conditioning of the problem and the choice of the basis representing the polynomial system. This article explores in detail the dependence of the efficiency of the KTS algorithm on the choice of basis. Three bases are considered: the power, the Bernstein, and the Chebyshev bases. These three bases satisfy the basis properties required by KTS. Theoretically, Chebyshev case has the smallest upper bound on its running time. The computational results, however, do not show that Chebyshev case performs better than the other two. * Supported in part by NSF DMS 0434338 and NSF CCF 0085969.
Species-specific markers are crucial for infectious disease diagnostics. Mutations within a marker sequence can lead to false-negative results, inappropriate treatment, and economic loss.
Reducing the kinematic errors is an important problem in five-axis machining. Errors of this type substantially affect the quality of the five-axis manufacturing. In this paper, we propose and analyse a new numerical algorithm to reduce the kinematic errors of a five-axis tool path using minimisation of the variation of the rotation angles. Our algorithm finds the locations of the cutter contact points (CC points) in the angular space and the orientation/location of the target surface relative to the mounting table that minimise the angle variation. We show through the numerical experiments and cutting simulations that the proposed method is efficient and accurate.
To become a self-regulated learner, one needs to have a skill required to induce himself to comprehend their own cognition. In this paper, we provided a definition of Seed skill to become a self-regulated learner (S2SRL) as a basis terminology for developing our proposed framework, CREMA—Computer-Supported Meta-Reflective Learning Model via MWP in order to design an environment to encourage learners to use intrinsic comprehension of metacognitive questioning to acquire S2SRL in mathematical word problem (MWP) learning. To assess our proposed framework, we addressed these questions: (i) Can CREMA really support learner to gain S2SRL and (ii) How does it work in a practical environment? To answer these two questions, three classes of low performance students of grade 9 (total 101 students) were assigned into three different learning groups: (i) a group of students who learnt MWP with our proposed method by implementing CREMA, (ii) a group of students who learnt MWP in traditional method combining MetaQ—metacognitive questions and motivational statements, and (iii) a class of students who learnt MWP in traditional method. The result from our investigation showed that MetaQ played an important role in CREMA, while integrating computer and technology enhanced students’ learning sense and empowered methodology to facilitate learning objects in the implementation of CREMA to effectively support students to gain S2SRL in MWP learning.
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