Name: ellipse shape and orbits explained
Uploaded: 2025-11-26T05:44:07.000Z
Duration: 1 min 27 s
Channel: JoVE

用铅笔和一根固定长度的细线绕两个固定的图钉描绘椭圆，可以说明一条到两个图钉的距离之和保持不变的路径。这定义了椭圆的几何特征，而其形状则取决于偏心率 e，即焦距与半长轴长度之比。当离心率 e 为零时，椭圆是一个完美的圆。随着 e 增大，椭圆沿长轴方向拉伸。这种形状出现在轨道路径中，例如哈雷彗星的轨道，其偏心率为 0.967，太阳位于其中一个焦点上。轨道的长轴约为35.88个天文单位——这是天文学中的一种标准测量单位。为了以标准形式表示哈雷轨道，椭圆以原点为中心，其长轴沿 x 轴对齐。将半长轴乘以 e 可得到焦距。然后利用勾股定理计算短半轴长度，该定理将其与长半轴和焦距相关联，从而确定椭圆的整体形状。将这些值代入标准形式以模拟轨道。

椭圆的偏心率

Eccentricity of an Ellipse

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Education

教育

JoVE Core

Math Fundamentals

数学基础

Unit

单位

Analytic Geometry

解析几何

KEY TERMS AND CONCEPTS

关键术语与概念

ellipse shape and orbits explained

An ellipse is a key conic section with two foci, or fixed points inside the shape. For any point on the ellipse, the sum of the distances to the two foci stays the same. A pencil, string, and two pins can show this shape in a simple way. If the string is held tight while the pencil moves, it traces an ellipse.

The size and shape of an ellipse depend on its eccentricity, written as e. Eccentricity is the ratio of the distance from the center to a focus, c, to the semi-major axis length, a. When e = 0, the ellipse is a circle. As e gets closer to 1, the ellipse stretches more along its major axis.

This idea helps explain the paths of celestial objects. Elliptical orbits are common in astronomy, and Halley’s Comet is a well-known example. Its orbit has a high eccentricity of 0.967, so it is very elongated. The major axis of the orbit is about 35.88 astronomical units, or AU, which is a unit equal to the average distance between Earth and the Sun.

From that value, the semi-major axis is 17.95 AU. The focal distance is found by multiplying a by e, so c = a × e. The semi-minor axis, b, can also be calculated from the ellipse relationships. When the ellipse is written in standard form, centered at the origin and aligned along the x-axis, it matches the orbital model used for the comet.

That model also gives the comet’s distance from the Sun at different points in its orbit. The perihelion, or closest approach to the Sun, is about 0.59 AU. The aphelion, or farthest point, is about 35.29 AU. These large changes in distance show why Halley’s Comet is a strong example of a highly eccentric ellipse in celestial mechanics.

Ellipse Shape and Orbits Explained

inclination of a line

Slope Angle and Line Direction

The slope angle shows how a line tilts on the Cartesian plane. It gives the line's direction by linking geometry with slope, which tells how steeply the line rises or falls. A line has no thickness, but its orientation still matters. This angle is measured counterclockwise from the positive x-axis. For nonhorizontal lines, the inclination angle ranges from 0 to π radians. That range helps describe the line's tilt in a clear geometric way. When two lines intersect, each line has its own...

parabolas

Parabolas: Focus, Directrix, and Symmetry

A parabola is a conic section with a clear geometric rule. It forms when a plane cuts a double-napped cone parallel to the cone’s slant height. The result is an open curve used in coordinate geometry and defined by a fixed point and a fixed line. That fixed point is the focus, and the fixed line is the directrix. Every point on a parabola stays the same distance from the focus as from the directrix. In mathematics, this set of points is called the locus of points with equal distance from both.<br...

reflective property of parabolas

Parabola Focus and Signal Reflection

A parabola is a U-shaped conic section with an important reflective property. It forms when a plane cuts a double-napped cone in a direction parallel to one side of the cone. In geometry, a parabola is also defined as the set of points that are equally far from a fixed point, called the focus, and a fixed line, called the directrix. This curve has a useful way of handling light, sound, and other parallel rays. Incoming rays that travel parallel to the axis of symmetry reflect toward one point.

ellipses

Ellipse Geometry: Foci, Axes, and Equations

An ellipse is a closed curve in geometry with two foci, a major axis, and a minor axis. It is formed when a right circular cone is cut by an inclined plane that does not pass through the cone’s base. The shape is symmetric, which makes its structure easy to study in analytic geometry. One key way to define an ellipse is by distance. For any point on the curve, the sum of the distances to the two fixed points, called the foci, stays constant. That constant sum is equal to the length of the...

eccentricity of an ellipse

An ellipse is a key conic section with two foci, or fixed points inside the shape. For any point on the ellipse, the sum of the distances to the two foci stays the same. A pencil, string, and two pins can show this shape in a simple way. If the string is held tight while the pencil moves, it traces an ellipse. The size and shape of an ellipse depend on its eccentricity, written as e. Eccentricity is the ratio of the distance from the center to a focus, c, to the semi-major axis length, a. When...

hyperbolas

Hyperbolas: Foci, Vertices, and Asymptotes

Hyperbolas are conic sections with two separate branches. They form when a plane cuts through a double-napped cone at a steeper angle than the cone’s slope, so the plane passes through both nappes. The two curves are mirror images and open away from each other along the transverse axis. The closest points on the branches to the center are called vertices. The distance from the center to a vertex is labeled a. The axis perpendicular to the transverse axis is the conjugate axis, and it is linked...

geometry of hyperbolas

Hyperbola Properties and Asymptotes

A hyperbola is a curve defined by two fixed points called foci. For any point on the curve, the absolute difference in its distances to the foci stays constant. This distance rule gives the hyperbola its shape and separates it from other conic sections. The standard equation shows two branches that extend forever. Each branch moves closer to two asymptotes, which are lines that guide the curve as it grows. The parameters a and b control key features of the graph. The value a measures the...

polar equations of conics

Conic Sections in Polar Form

Conic sections can be written in polar form using a focus and a directrix. The focus is a fixed point, and the directrix is a fixed line. A point on the conic is defined by the ratio of its distance to the focus and its distance to the directrix. That ratio is called the eccentricity. This setup gives one framework for ellipses, parabolas, and hyperbolas. When the focus is placed at the origin of the polar coordinate system, one polar equation can represent any conic section. That makes the...

key terms and concepts

Learn how ellipse shape changes with eccentricity and how that math models Halley’s Comet, from foci and axes to perihelion and aphelion.

Ellipse Shape and Orbits Explained | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Ellipse Shape and Orbits Explained

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