关于曲线族产生曲面的理论证明的一点补充——多项式的结式在几何上的一个应用

在空间解析几何中,我们把柱面,锥面和旋转曲面看成是动曲线的轨迹,来建立这几类特殊曲面的方程,例如,把柱面看成是与空间定曲线(准线)相交且和定方向(母线方向)平行的动直线的轨迹。建立柱面方程的步骤是先写出满足条件的动直线的方程——含有参数的直线族方程,然后从     

用准线和焦点推导双曲线方程

《全日制十年制学校高中课本》第二册 (以下简称《课本》) 双曲线的方程是用右面一套准线和焦点推得的。教师在讲完这一内容时,学生往往会提问:“用其它的准线和焦点能否推出双曲线的方程?”为了对这个问题进行全面讨论,本文将用左、右两套准线和焦点分各种情况对双曲线两支曲线的方程进行推导,以求得对双曲线方程的进一步认识。取《课本》P179页例3:求圆锥曲线(到焦点和到准线的距离的比是е的点的轨迹) 的极坐标方程 (取е>1,即为双曲线方程)     

一个丢番图方程及其它的整数解

研究丢番图方程x~y+y~z+z~x=0的可解性,并求该方程的所有整数解.本文利用初等方法及整数的整除性质研究这一问题,获得了彻底解决.即就是证明了方程x~y+y~z+z~x=0有且仅有六组整数解(x,y,z)=(-2,1,1),(1,-2,1),(1,1,-2),(1,-1,-2),(-1,-2,1),(-2,1,-1)     

一题三观寻根引源

2009年高考重庆卷第20题： 已知以原点O为中心的椭圆的一条准线方程为y=3/4∫3,     

新题征展(54)

A题组新编 1.反比例函数y=k/x(k＞0)的图象是双曲线,则其渐近线方程是＿＿;对称轴方程是＿＿,顶点坐标是＿＿;离心率是＿＿;焦点坐标是＿＿;准线方程是——.     

关于本原商高数的新猜想

设a,b,c是满足a=m²-n2-n²,b=2mn,c=m2,b=2mn,c=m²+n2+n²的正整数,其中m,n是适合m>n,gcd(m,n)=1,2|mn的正整数.运用初等数论方法讨论了方程c2的正整数,其中m,n是适合m>n,gcd(m,n)=1,2|mn的正整数.运用初等数论方法讨论了方程c^x+bx+b^y=ay=a^z的正整数解(x,y,z).证明(m,n)≡(0,1),(0,5),(1,2),(2,3),(3,4),(4,1),(4,5),(5,6),(6,7)或(7,0)(mod8)时,方程无解.上述结果部分地解决了有关本原商高数的一个新猜想.     

一道高考解析几何题的推广

题目 (07重庆,22)如图1,中心在原点O的椭圆的右焦点为F(3,0),右准线Z的方程为:x=12.……     

关于Kenichiro的一个丢番图方程问题

研究了Kenichiro提出的轮换对称形式的丢番图方程,即方程ab bc ca=ya+b+c3的求解问题,利用素数整除的一些性质,证明了该方程仅有平凡解a=b=c以及非平凡解(a,b,c)=(k,k,4k),(k,4k,k),(4k,k,k)(k∈N),从而完全解决了这个方程.     

抛物线参数方程的应用

本文将介绍抛物线的一种参数方程的一些应用。为此,先说明抛物线参数方程中参数的意义。为方便起见,我们将抛物线的标准方程写成: y~2=4ax 这里a为抛物线的焦点到它的准线的距离的一半,所以a>0。设抛物线y~2=4ax的这种参数方程为:     

与圆锥曲线准线有关的一个性质及其证明

笔者在研究2006年湖北省高考试题(理科)第20题时,通过在准线l:x=4上取两个不同的点P,分别求出对应的直线MN的方程,发现这两条不同的直线均过焦点F(1,0),于是引起了笔者的极大兴趣,通过探讨,笔者得到如下与圆锥曲线的准线有关的三个结论.     

旋转抛物面的新定义及其性质

从点的轨迹的角度,将R<'3>中的旋转抛物面定义为:R<'3>中到一定点与到一定平面(点不在平面上)距离相等的点的轨迹.同时引入旋转抛物面的焦点、准平面、准线等概念,并在此基础上证明关于旋转抛物面的焦点弦、准平面、顶点、对称轴、切平面之间的若干重要性质.     

有关Euler函数(n)的方程的正整数解

设(n)是Euler函数.主要研究了方程(xy)=3((x)+(y))的可解性问题,利用初等的方法给出了这一方程的所有的35组正整数解.对于任意素数k>3,(x,y)=(3k,4k),(4k,3k)是方程(xy)=k((x)+(y))的2个正整数解.证明了更为一般的结论:对于任意奇数k>3,当gcd(k,3)=1时,(x,y)=(3k,4k),(4k,3k)是方程(xy)=k((x)+(y))的2个正整数解.     

An Efficient and Accurate Spectral Method for Acoustic Scattering in Elliptic Domains

An efficient and accurate method for solving the two-dimensional Helmhokz equation in domains exterior to elongated obstacles is developed in this paper. The method is based on the so called transformed field expansion (TFE) coupled with a spectral-Galerkin solver for elliptical domain using Mathieu functions. Numerical results are presented to show the accuracy and stability of the proposed method.     

Pillai's conjecture revisited

We prove a generalization of an old conjecture of Pillai (now a theorem of Stroeker and Tijdeman) to the effect that the Diophantine equation 3^x−2^y=c has, for |c|>13, at most one solution in positive integers x and y. In fact, we show that if N and c are positive integers with N?2, then the equation |(N+1)^x−N^y|=c has at most one solution in positive integers x and y, unless (N,c)∈{(2,1),(2,5),(2,7),(2,13),(2,23),(3,13)}. Our proof uses the hypergeometric method of Thue and Siegel and avoids application of lower bounds for linear forms in logarithms of algebraic numbers.     

On the Parameterization of Rational Ringed Surfaces and Rational Canal Surfaces

Ringed surfaces and canal surfaces are surfaces that contain a one-parameter family of circles. Ringed surfaces can be described by a radius function, a directrix curve and vector field along the directrix curve, which specifies the normals of the planes that contain the circles. In particular, the class of ringed surfaces includes canal surfaces, which can be obtained as the envelopes of a one-parameter family of spheres. Consequently, canal surfaces can be described by a spine curve and a radius function. We present parameterization algorithms for rational ringed surfaces and rational canal surfaces. It is shown that these algorithms may generate any rational parameterization of a ringed (or canal) surface with the property that one family of parameter lines consists of circles. These algorithms are used to obtain rational parameterizations for Darboux cyclides and to construct blends between pairs of canal surfaces and pairs of ringed surfaces.     

从射影观点看焦点

用射影几何知识讨论欧氏平面上二次曲线焦点与准线的性质.     

二次曲线局部与整体的关系

用射影几何知识讨论欧氏平面上二次曲线局部与整体的关系,讨论如何通过二次曲线的一些已知点与切线判断它的类型,作出它的对称轴,渐近线,焦点与准线.     

广义Ramanujan-Nagell方程x~2+D~m=p~n的解的注记

设a为偶数,p为素数,D=3a~2+1,p=4a~2+1。本文指出了乐茂华文献中的错误,并利用两个对数的线性型上界估计的P-adic形式以及广义Fermat方程的解的一些新结论,证明了方程x~2+D~m=p~n仅有两组正整数解(x,m,n)=(0,1,1),(8a~3+ 3a,1,3)．     

Intersection of a ruled surface with a free-form surface

This paper presents a simple method for computing the intersection curve of a ruled surface and a free-form surface. The basic idea is to reduce the problem of surface intersection to the one of projecting an appropriate curve such as a directrix of the ruled surface, along its indicatrix curve (direction vector field of its generating lines), onto the free-form surface; the projection curve is just the intersection curve. With techniques in classical differential geometry, we derive the differential equations of the intersection curve in the cases of parametrically and implicitly defined free-form surfaces. The intersection curve naturally inherits the parameter of the chosen directrix. Moreover, it is independent of the base surface geometry and its parameterization, and is obtained by numerically solving the initial-value problem for a system of first-order ordinary differential equations in the parametric domain associated to the surface representation for parametric case or in 3D space for implicit case. Some experimental examples are also given to demonstrate that the presented method is effective and potentially useful in computer aided design and computer graphics. An erratum to this article can be found at     

Higher Singularities and the Twistor Fibration π: CP3 → S4

We use the Klein correspondences to write down an explicit relationship between two holomorphic curves, namely the directrix curve and the twistor lift, associated to a superminimal map from a Riemann surface to the 4-sphere. We also explain how the singularities of the corresponding osculating curves are related and show that they occur at the branch points or the umbilics of the corresponding superminimal map.