﻿ 双曲平面上非蜕化圆锥截线的聚焦性 The Focus Properties of Nonsingular Conic Curves in Hyperbolic Plane

Pure Mathematics
Vol.07 No.04(2017), Article ID:21446,15 pages
10.12677/PM.2017.74043

The Focus Properties of Nonsingular Conic Curves in Hyperbolic Plane

Hai He, Youning Wang

School of Mathematical Science, Beijing Normal University, Beijing

Received: Jul. 1st, 2017; accepted: Jul. 15th, 2017; published: Jul. 21st, 2017

ABSTRACT

In the plane geometry, as we all know, the light through a focus of the ellipse is reflected by ellipse to focus on another; similarly the light through the hyperbola one focus, after reflection, reversely focuses on another; and for the parabola it is true that the light from focus is reflected into parallel light. This paper proved that the nonsingular conic curves on the hyperbolic plane have corresponding focus properties; moreover, there are some curves having focusing properties, which are not conic curves.

Keywords:Hyperbolic Plane, Conic Curves, Focus, Beltrami-Klein Coordinate System

1. 引言

2. 双曲平面的Beltrami-Klein坐标系和点的几何不变性

3. 圆锥截线的聚焦性

3.1. 椭圆和双曲线的聚焦性

(3.1.1)

(3.1.2)

，方程的解为。若，方程的解为

，根据引理1，由常微分方程理论知，方程(3.1.2)的两个解总是存在的，注意到微分方程的齐次性，假设其积分曲线方程形如

(3.1.3)

(3.1.4)

(3.1.5)

3.2. 测地圆和极限圆的一点聚焦性

(3.2.1)

(3.2.2)

(3.2.3)

(3.2.4)

，则由(3.2.3)式有

(3.2.5)

(3.2.6)

(3.2.7)

(3.2.8)

3.3. 抛物线的聚焦性

(3.3.1)

(3.3.2)

(3.3.3)

(3.3.4)

(3.3.5)

(3.3.6)

(3.3.7)

(3.3.8)

(3.3.9)

(3.3.10)

(3.3.11)

(3.3.12)

(3.3.13)

(3.3.14)

(3.3.15)

3.4. 极限曲线的聚焦性

(3.4.1)

(3.4.2)

(3.4.3)

(3.4.4)

(3.4.5)

(3.4.6)

(3.4.7)

，分别考察极限椭圆和极限双曲线的焦点坐标，二者共有一个焦点，极限椭圆的另一个焦点坐标为

The Focus Properties of Nonsingular Conic Curves in Hyperbolic Plane[J]. 理论数学, 2017, 07(04): 334-348. http://dx.doi.org/10.12677/PM.2017.74043

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