In a past thesis, we proposed a homogeneous parameter, homogeneous geometric Newton-Raphson method for dealing with a rational polynomial curve and we concluded that the algorithm is robust and locally unique. After more experiments, we noticed that we have to get over not only divergence but also oscillation problems.
Oscillation happens only when the equation has complex roots. So we propose a new geometric Newton-Raphson method, which can cope with complex root solutions. The algorithm has the features that the conventional homogeneous parameter, homogeneous Newton-Raphson method has and further more can find complex roots. The results of our experiments show that the algorithm is robust with respect to both divergence and oscillation, and also has local uniqueness property.

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This paper proposes a new geometric Newton-Raphson method for dealing with a rational polynomial curve. The algorithm is robust and at the same time locally unique.
Although rational polynomial curves and surfaces have become standard forms in computer-aided design, they have many problems. For example, a Newton-Raphson algorithm for dealing with a rational polynomial curve tends to be unstable. This is a fatal problem. We propose to homogenize the coordinates of a rational curve when it is applied to the Newton-Raphson algorithm. Then it becomes very robust. Furthermore the solution point becomes locally unique with respect to an initial parameter range when the parameter is also homogenized in addition to the coordinates, because with this technique we have a freedom of controlling parameter values and we can adjust the increment of the parameter appropriately.

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Although rational polynomial curves and surfaces have become standard forms in computer-aided design, they have many problems. For example, a Newton-Raphson algorithm for dealing with rational polynomial curves tends to be non-robust. This is a fatal problem. In the past, we proposed a new Newton-Raphson method for dealing with the rational polynomial curves. In the method, we homogenize the coordinates of a rational curve when it is applied to the Newton-Raphson algorithm. Then it becomes very robust.
This paper proposes a method of adapting the algorithm to the Newton-Raphson method for dealing with the surface-surface intersection problem. Like the case of a curve, the Newton-Raphson algorithm was found to be very robust when the coordinates of a curved surface are homogenized. In this paper, we discuss the reason why the conventional method tends to be non-robust while the new method is robust. The conventional method deals with a rational function that always has a horizontal asymptote as well as vertical ones while the new method deals with a non-rational function and thus has no asymptote. The result of our analysis points out that the cause of the non-robustness lies in the horizontal asymptote of the rational function inherent in the conventional Newton-Raphson method. Our numerical experiments show that our method is very robust.

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The interference processing of curves and surfaces is an important technique in CAD/CAM applications. The lack of the stability in the interference processing is due to numerical errors caused by floating point arithmetic and the approximate computations. In the interference processing, the geometric Newton-Raphson method is generally used for approximations, because the solutions of high degree polynomial equations cannot be exactly computed. Most geometric algorithms, such as Boolean set operation algorithms, modify the topological data structure according to the existence of the solutions of polynomial equations. The number of solutions of algebraic equations can be exactly counted with Sturm's theorem and exact arithmetic. In this paper, a stable interference method for curves and surfaces using Sturm's theorem and exact arithmetic is proposed. The proposed approach can exactly evaluate the existence of intersection points in curve/surface intersection.

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The role of prostacyclin (PGI_2, 1) in the central nervous system (CNS) has still been unclear because of the lack of a specific ligand for a PGI_2 receptor in CNS. In this context, we recently elaborated 15R-TIC (2) with high binding affinity and selectivity for novel IP2 receptor which is specifically expressed in CNS neurons. The R configuration of the hydroxy-bearing C(15) in 2 is fascinating, because, in general, the configuration of biological active PGs at C(15) position is known to be S. In this symposium, we describe synthesis of the TIC derivatives for structure-binding affinity relationship in addition to biological actions. In consequence, 15-deoxy-16-m-tolyl-17,18,19,20-tetranorisocarbacyclin (5) (referred to as 15-deoxy-TIC) exhibited, among others, highest binding affinity and selectivity for a IP2 receptor. 5 has been prepared based on the combination of the Witting reaction and Pd(0)-mediated coupling of an allyl carbonate and a sulfone. Thus, Witting reaction of aldehyde 7 and a phosphorane led to 8. The reduction of 8 and subsequent methoxy carbonylation gave allyl carbonate 9 which was treated with disulfone 10 in the presence of a 1: 1 mixture of Pd(0) and diphenylphosphinoethane to give 11. Reductive removal of phenyl sulfonyl groups of 11 and subsequent deprotection of THP group gave methyl ester 12. Finally, alkaline hydrolysis of 12 led to 5. 2 and 5 prevented apoptotic cell death of hippocampal neurons induced under high oxygen (50%) atmosphere. 6, isocarabacyclin, and natural PGs except for 1 did not show such a biological effect.

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