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平面内常见曲线有:线段的垂直平分线,角平分线,圆及圆锥曲线等.他们的定义分别如下:(1)线段的垂直平分线是平面内到两定点的距离相等的点的轨迹.(2)角平分线是平面内到角两边距离相等的点的轨迹.(3)圆是平面内到一定点的距离等于定长的点的轨迹. 相似文献
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问题1 平面内到定点F的距离比到定直线l的距离大(小)d的轨迹一定是以定点F为焦点的抛物线吗?
北师大版新课标教材<数学(选修2-1)>(以下简称新教材)第73页练习2第4题:
平面上动点M到点F(3,0)的距离比M到y轴的距离大3,求动点M满足的方程.…… 相似文献
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1841年,D elaunay获得如下定理:如果在一平面上沿定直线滚动一条二次圆锥直线,然后将其焦点的轨迹绕定直线旋转,则所得到的曲面具有常数平均曲率,反之,所有旋转常数平均曲率曲面(除球面外)都有如此构造.本文将以上的D elaunay定理推广到Lorentz-M inkow sk i空间Rn1 1中类空的Sm型旋转W超曲面. 相似文献
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<正>轨迹意识是平面解析几何中的一种重要行为意识,也是平面解析几何中的重要思想方法.除在解析几何中熟练应用外,在解三角形、平面向量以及立体几何等其他场合,也经常借助轨迹意识来解决相应的数学问题,直观形象.1 解析几何中的轨迹意识解析几何中的轨迹问题,其实质就是由曲线上的动点变化规律,按照一个条件的变化引起其他相关新动点的变化情况,利用对图形结构的理解、探索与联想,构建“形”与“数”之间的联系,进而探究新动点的轨迹. 相似文献
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椭圆、双曲线第一定义 :平面上到两个定点F1,F2 距离之和等于常数 ( >|F1F2 | )的动点的轨迹叫椭圆 ,两距离之差的绝对值等于常数 ( <|F1F2 | )的动点的轨迹叫双曲线 .圆锥曲线第二定义 :平面上到定点的距离与到定直线的距离的比等于常数e的动点的轨迹叫… ,换言之 :平面上到定点F的距离与定直线l的距离的e倍相等的点的轨迹叫… .在创新思想指导下 ,将第一、第二定义剪辑后再嫁接 ,提出开放的新问题 :若动点M到定点F的距离与M到定直线l的距离的e倍的和 (或差的绝对值 )等于常数 ,动点M的轨迹是什么呢 ?以定直线l为x轴 ,过定点F且与l垂… 相似文献
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椭圆、双曲线、抛物线统称为圆锥曲线.它们表示到定点F和定直线l的距离的比是一个常数e的点M的轨迹.当01时,点M的轨迹是双曲线;当e=1时,点M的轨迹是抛物线.其中定点F叫做焦点;定直线l叫做准线;定比e叫做离心率.这样的 相似文献
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题目在平面直角坐标系xOy中,点P到点F(3,0)的距离的4倍与它到直线x=2的距离的3倍之和记为d.当点P运动时,d恒等于点P的横坐标与18之和.
(Ⅰ)求点P的轨迹C;
(Ⅱ)设过点F的直线l与轨迹C相交于M,N两点,求线段MN长度的最大值.…… 相似文献
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V. K. Srinivasan 《International Journal of Mathematical Education in Science & Technology》2013,44(5):791-800
Given a conic section, the locus of a moving point in the plane of the conic section such that the two tangent lines drawn to the conic section from the moving point are all mutually perpendicular is a curve. In the case of an ellipse and hyperbola this curve is a circle referred to as the director circle. In the case of the parabola this curve coincides with the directrix of the parabola. The last section is devoted to the graphical illustrations of director circles for circles, parabolas, ellipses and hyperbolas using the built-in Maple V software of Scientific Work Place 3.0. 相似文献
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Alexey V. Borisov Alexander A. Kilin Ivan S. Mamaev 《Regular and Chaotic Dynamics》2018,23(3):339-354
In this paper, equations of motion for the problem of a ball rolling without slipping on a rotating hyperbolic paraboloid are obtained. Integrals of motions and an invariant measure are found. A detailed linear stability analysis of the ball’s rotations at the saddle point of the hyperbolic paraboloid is made. A three-dimensional Poincaré map generated by the phase flow of the problem is numerically investigated and the existence of a region of bounded trajectories in a neighborhood of the saddle point of the paraboloid is demonstrated. It is shown that a similar problem of a ball rolling on a rotating paraboloid, considered within the framework of the rubber model, can be reduced to a Hamiltonian system which includes the Brower problem as a particular case. 相似文献
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We consider strongly monotone continuous planar vector fields with a finite number of fixed points. The fixed points fall into three classes, attractors, repellers and saddles. Naturally, the relative positions of the fixed points must obey a set of restrictions imposed by monotonicity. The study of these restrictions is the main goal of the paper. With any given vector field, we associate a matrix describing the arrangement of the fixed points on the plane. We then use these matrices to formulate simple necessary and sufficient conditions which allow one to determine whether a finite set of attractors, repellers and saddles at given positions on the plane can be realized as the fixed point set of a strongly monotone vector field. 相似文献
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Sang Won Bae Chunseok Lee Hee-Kap Ahn Sunghee Choi Kyung-Yong Chwa 《Computational Geometry》2009,42(9):903-912
We study the problems of computing two non-convex enclosing shapes with the minimum area; the L-shape and the rectilinear convex hull. Given a set of n points in the plane, we find an L-shape enclosing the points or a rectilinear convex hull of the point set with minimum area over all orientations. We show that the minimum enclosing shapes for fixed orientations change combinatorially at most O(n) times while rotating the coordinate system. Based on this, we propose efficient algorithms that compute both shapes with the minimum area over all orientations. The algorithms provide an efficient way of maintaining the set of extremal points, or the staircase, while rotating the coordinate system, and compute both minimum enclosing shapes in O(n2) time and O(n) space. We also show that the time complexity of maintaining the staircase can be improved if we use more space. 相似文献
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Tomasz Szemberg Halszka Tutaj-Gasinska 《Proceedings of the American Mathematical Society》2002,130(9):2515-2524
We study linear series on a projective plane blown up in a bunch of general points. Such series arise from plane curves of fixed degree with assigned fat base points. We give conditions (expressed as inequalities involving the number of points and the degree of the plane curves) on these series to be base point free, i.e. to define a morphism to a projective space. We also provide conditions for the morphism to be a higher order embedding. In the discussion of the optimality of obtained results we relate them to the Nagata Conjecture expressed in the language of Seshadri constants and we give a lower bound on these invariants. 相似文献
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《Topology and its Applications》1988,29(3):213-217
Though fixed point free homeomorphisms of the plane would appear to exhibit the simplest dynamical behavior, we show that the minimal sets can be quite complex. Every homeomorphism which is conjugate to a translation must have a closed invariant line. However we construct an orientation preserving fixed point free homeomorphism of the plane which admits no closed invariant line. We verify that no such line exists by considering the ‘fundamental regions” of our example. Fundamental regions, studied first by Stephen Andrea, are equivalence classes of points in the plane associated with a given homeomorphism. Two points are said to be in the same equivalence class if they can be connected by an arc which diverges to infinity under both the forward and backward iterates of the homeomorphism. Our example contains no invariant fundamental regions. 相似文献
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《复变函数与椭圆型方程》2012,57(1):69-81
We consider the generalization of linear fractional transformations of the plane to $ {\shadC}^n $ . Analogs of the one-variable theory are developed including fixed point sets and points of symmetry. The domains in $ {\shadC}^n $ that are images of the ball under these transformations are found. Finally, we see some examples where classical fixed point results follow from this theory in a natural way. 相似文献
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We consider the nilpotent left-invariant sub-Riemannian structure on the Engel group. This structure gives a fundamental local approximation of a generic rank 2 sub-Riemannian structure on a 4-manifold near a generic point (in particular, of the kinematic models of a car with a trailer). On the other hand, this is the simplest sub-Riemannian structure of step three. We describe the global structure of the cut locus (the set of points where geodesics lose their global optimality), the Maxwell set (the set of points that admit more than one minimizer), and the intersection of the cut locus with the caustic (the set of conjugate points along all geodesics). The group of symmetries of the cut locus is described: it is generated by a one-parameter group of dilations R+ and a discrete group of reflections Z2 × Z2 × Z2. The cut locus admits a stratification with 6 three-dimensional strata, 12 two-dimensional strata, and 2 one-dimensional strata. Three-dimensional strata of the cut locus are Maxwell strata of multiplicity 2 (for each point there are 2 minimizers). Two-dimensional strata of the cut locus consist of conjugate points. Finally, one-dimensional strata are Maxwell strata of infinite multiplicity, they consist of conjugate points as well. Projections of sub-Riemannian geodesics to the 2-dimensional plane of the distribution are Euler elasticae. For each point of the cut locus, we describe the Euler elasticae corresponding to minimizers coming to this point. Finally, we describe the structure of the optimal synthesis, i. e., the set of minimizers for each terminal point in the Engel group. 相似文献
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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. 相似文献