圆锥部分

Modern Astrodynamics Pub Date : 2021-01-12 DOI:10.1142/9789814295871_0005

David Pierce

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引用次数: 0

摘要

直线是最短的路径。1. 蚂蚁Z又渴又累。他想在回家的路上去看看附近的一条直溪。为他找到最短的路径。(给定两个点A,B和一条不经过A和B的直线l，在l上找到一个点X，使得AX + XB是可能的最小值。)蚂蚁Z位于一个立方体的顶点。他想要到达对顶点。在盒子的表面上为他找到最短的部分。3.求蚂蚁Z的最短路径:(a)从a点到B点都在圆柱形罐的一边;(b)从a点到蛋筒上的b点;(c)从球上的a点到B点。4. 长方形台球桌上有两个球，红球和白球。你要击打红球，使它先击中桌子的两面墙，然后再击中白球。你应该如何指挥红球?椭圆。固定两个点A和B，并使数字A > AB。平面上所有点X使AX + XB = A的集合称为椭圆。点A和点B称为椭圆的焦点(焦点的复数形式)。椭圆各焦点之间的距离称为焦距。如果两个焦点重合，椭圆就变成圆。椭圆的中心是连接其焦点的线段的中点。直径是穿过中心的弦。所有行星的轨道都是以太阳为焦点的椭圆。5. 一个非圆的椭圆有两条对称轴:一条是AB线，另一条是AB线的垂直平分线。椭圆内部由这两条线切割的线段称为长轴和短轴。6. 设a为长轴，b为短轴，c为焦距。证明c = a - b。椭圆的切线是与椭圆正好相交于一点的直线。对于椭圆上的任意点X，存在一条穿过X与椭圆相切的唯一直线。7. 设X为椭圆Γ的一点，t为Γ在X处的切线，设a和B为Γ的两个焦点。证明AX和t的夹角等于BX和t的夹角。

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CONIC SECTIONS

A straight line is the shortest path. 1. Ant Z is thirsty and tired. He wants to visit a straight stream nearby on his way home. Find the shortest path for him. (Given two points A,B and a line l not passing through A and B. Find a point X on l such that AX + XB is the minimal possible.) 2. Ant Z is sitting at a vertex of a cubic box. He wants to get to the opposite vertex. Find the shortest part for him on the surface of the box. 3. Find the shortest path for Ant Z: (a) from a point A to a point B both on a side of a cylindrical can; (b) from a point A to a point B on an ice-cream cone; (c) from a point A to a point B on a ball. 4. There are two balls, red and white, on the rectangular billiard table. You want to strike the red ball so that it hits two walls of the table first and then hits the white ball. How should you direct the red ball? Ellipse. Fix two points A and B and the number a > AB. The set of all points X on the plane such that AX + XB = a is called an ellipse. The points A and B are called the foci (plural of focus) of an ellipse. The distance between the foci of an ellipse is called the focal distance. In case when two foci coincide an ellipse becomes a circle. The center of an ellipse is the midpoint of the segment joining its foci. A diameter is a chord passing through the center. All planets’ orbits are ellipses with the sun at a focus. 5. An ellipse which is not a circle has two axes of symmetry: one is the line AB and another one is the perpendicular bisector to AB. The segments inside the ellipse cut by those lines are called the major axis and the minor axis. 6. Let a be the major axis, b be the minor axis and c be the focal distance. Show that c = a − b. A tangent line to an ellipse is a line that intersects the ellipse at exactly one point. For any point X on an ellipse there exists a unique line through X tangent to the ellipse. 7. Let X be a point of an ellipse Γ, t be the tangent line to Γ at X. Let A and B be two foci of Γ. Show that the angle between AX and t equals the angle between BX and t.

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Modern Astrodynamics

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