從圓錐曲線作圖出發,放寬條件產生新圖形並研究其性質。設圓O的半徑為a ,點F 坐標為(k , 0),Q在圓O上,R在 上滿足=t(t?R)〈其中FR 及FQ 表有向線段〉,若直線L過點R與 垂直並與交於點P ,當Q 在圓O上移動時,用極座標推導動點P 方程式, F 點在圓內,t=1/2 時為橢圓; F 點在圓外, t=1/2 時為雙曲線。稱F 點在圓內為E 型曲線為封閉曲線;稱F 點在圓外為H 型曲線為開放曲線。仿圓錐曲線令擬離心率e 為 k / a 。兩曲線e 值相同時有相似性, 以討論圖形。
旋轉直線L , 使過點R 與交於P , 用極座標推導動點P 方程式,當csco ≧時可為開放圖形,稱F 點在圓內為E' 型曲線,在圓外為H' 型曲線。
拋物線仿橢圓作法,將中垂線作圖改為t 比值,改變後圖形仍為拋物線,方程式為 (1-t)y2 - kx + (1-t)k2 = 0 。旋轉後, 拋物線變為雙曲線, 方程式為 (1-t)y2 + cotoxy - kx + (1-t)k2 = 0 。
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Reactions of Bis(oxy)enamines with O-Nucleophiles in the Presence of Metal Salts
NO donors are an emerging class of pharmaceutical compounds, with many important functions in the cardiovascular, nervous and immune systems. With great therapeutic potential, the development of new NO donor compounds would be of great medicinal value, potentially opening a whole class of drugs to be used to treat various ailments. This project studies a specific class of compounds, substituted cyclic oxime ethers, which have proven to be useful intermediates in fields such as medicine and biochemistry. The cyclic structure along with a determinable substitutable group at the C3 position is highly valuable, as it allows the oxime ether to act as a convenient precursor for a variety of useful products, playing key components in many drugs. And with a substituted nitrate group, which is an O-nucleophile, the oxime ether has the potential to become an NO-donor, and hence become a possible intermediate in a wide array of NO donor drugs. Co(NO)3 was used in the synthesis of the cyclic oxime ether, directly from a phenyl substituted bis(oxy)enamine intermediate, producing an entirely new compound: α-hydroxyoxime nitrates, the oxime ether being substituted with a nitrate group. This new reaction of the synthesis of α-hydroxyoxime nitrates was further studied for optimization purposes, in order to open a new class of NO donor precursors. In addition, other nucleophiles were also explored in this class of reactions, forming important bonds such as C-N and C-S bonds, with key structures for other types of synthesis intermediates and precursors. Different metal nitrates, or various other nucleophiles in place of the nitrates, were used in reaction with bis(oxy)enamine, and the yield and structure of the final products were determined by NMR spectra. Successful optimization of the synthesis of α-hydroxyoxime nitrates has been achieved, where the conditions for optimum synthesis involve using Cr(NO3)3•9H2O which achieved a high yield of 76%, dissolved in THF with the bis(oxy)enamine starting compound. It has been determined that the metal in the salt affects the reaction pathway, as the nature of the metal cation affects its efficiency to cleave the N-O bond in the starting compound (with d-block elements being the best performing), and H+ ions can promote the reaction as well. Also, the reaction proceeds with different types of bis(oxy)enamines, meaning the substrate scope can be expanded to give a variety of products. The reaction can also proceed to form other products with different nucleophiles other than the nitrate group, where the C-N and C-S bonds were successfully formed in the reactions from bis(oxy)enamine to oxime ether. Thus, this class of reaction in converting the bis(oxy)enamine to a cyclic oxime ether has potentially opened a new class of NO donor compounds, and further possesses the potential to form a wide variety of products to be used in other important synthesis procedures.
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