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

Tracing an ellipse with a pencil and a string of fixed length around two fixed pins illustrates a path where the combined distance to the pins remains constant.This defines the geometry of an ellipse, while its shape depends on its eccentricity, e&#8212;the ratio of the focal distance to the semi-major axis length.When e is zero, the ellipse is a perfect circle. As e increases, the ellipse stretches along the major axis.This shape appears in orbital paths such as that of Halley&#8217;s Comet, which has an eccentricity of 0.967 and the Sun at one focus.The major axis of the orbit measures approximately 35.88 astronomical units&#8212;a standard unit of measurement in astronomy.To express Halley&#8217;s orbit in standard form, the ellipse is centered at the origin, with its major axis aligned along the x-axis.Multiplying the semi-major axis by e gives the focal distance.The semi-minor axis length is then calculated using the Pythagorean theorem, which relates it to the semi-major axis and the focal distance, completing the overall shape of the ellipse.These values are substituted into the standard form to model the orbit.

excentriciteit van een ellips

Een ellips is een fundamentele kegelsnede die wordt gedefinieerd door de constante som van de afstanden van een willekeurig punt op de kromme tot twee vaste punten, de brandpunten. Deze geometrische eigenschap kan fysiek worden gedemonstreerd met een potlood, een touwtje en twee pinnen: door het touwtje aan beide uiteinden te verankeren en het strak te houden met het potlood, kan men de omtrek van een ellips tekenen.De vorm en omvang van de ellips worden bepaald door de excentriciteit e, gedefinieerd als de verhouding van de afstand tussen het middelpunt en een brandpunt (c) tot de lengte van de halve lange as (a). Wanneer e = 0, is de ellips een cirkel. Naarmate e naar 1 toeneemt, rekt de ellips uit langs de lange as.Deze eigenschap verklaart het elliptische karakter van hemelbanen. De komeet Halley beweegt zich bijvoorbeeld langs een zeer excentrische elliptische baan met e = 0,967. De lange as van die baan meet ongeveer 35,88 astronomische eenheden (AE), een gebruikelijke astronomische eenheid die overeenkomt met de gemiddelde afstand tussen de aarde en de zon. Dit betekent dat de halve lange as a gelijk is aan 17,95 AE. De afstand van het middelpunt van de ellips tot elk brandpunt (de brandpuntsafstand) is a &#215; e. Voor de komeet Halley resulteert dit in<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1027/59;' src='https://cloudfront.jove.com/files/ftp_upload/41036/41036_Equation_1_editor_41036.svg' alt='41036_Equation_1.svg' /></figure>De halve korte as b kan worden berekend met behulp van de relatie<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:1605/143;' src='https://cloudfront.jove.com/files/ftp_upload/41036/41036_Equation_2_editor_41036.svg' alt='41036_Equation_2.svg' /></figure>Wanneer men de ellips in standaardvorm plaatst, gecentreerd in de oorsprong en uitgelijnd langs de x-as, luidt de baanvergelijking:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:731/200;' src='https://cloudfront.jove.com/files/ftp_upload/41036/41036_Equation_3_editor_41036.svg' alt='41036_Equation_3.svg' /></figure>Volgens dit model bedraagt de periheliumafstand (dichtstbijzijnde afstand tot de zon) a &#8722; c &#8776; 0,59 AE, en de apheliumafstand (verste punt) a + c &#8776; 35,29 AE. Deze aanzienlijke variatie illustreert de uitgesproken excentriciteit van Halley&#8217;s baan en het nut van de elliptische geometrie in de hemelmechanica.

Excentriciteit van een ellips

Een ellips is een fundamentele kegelsnede die wordt gedefinieerd door de constante som van de afstanden van een willekeurig punt op de kromme tot twee ...

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Eccentricity of an Ellipse

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

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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...

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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.

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Ellipse Shape and Orbits Explained

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