Eccentricity of Curves

Any curved shape’s eccentricity, regardless of size, defines its shape. A circle, an ellipse, a parabola, and a hyperbola are formed when a plane intersects with a double-napped cone.

Their characteristics are classified based on their shapes, which are determined by an intriguing factor known as eccentricity. The eccentricity of circles is zero, while the eccentricity of parabolas is one. The eccentricities of ellipses and hyperbolas vary. Let us go over how to calculate the eccentricities of conic sections in greater detail.

Eccentricity of Conic Sections – Hyperbola, Ellipse, Parabola, Circle

There are various conics, such as a parabola, ellipse, hyperbola, and circle. The eccentricity of a conic section can be defined as the distance between any two points divided by the perpendicular distance between those two points and the nearest directrix. 

A hyperbola is defined as a smooth curve in a plane that is defined by its geometric properties or by equations for which it is the solution set. A hyperbola is generally composed of two connected components or branches that are mirror images of one another and resemble two infinite bows.

Parabola

In mathematics, a parabola is a plane curve that is mirror-symmetrical and roughly U-shaped. It corresponds to several seemingly disparate mathematical descriptions, all of which define the same curves.

Circle Conic Section

The circle, in terms of conic section, is the intersection of a plane perpendicular to the cone’s axis. 

Ellipse Conic Section

A plane curve that surrounds two focal points and has a constant sum of the two distances to the focal points for all points on the curve is called an ellipse.

For any conic, the eccentricity value is constant. When the eccentricities are large, the curves are small. As a result, we conclude that as the eccentricities of these conic sections increase, so do their curvatures.

  • A circle’s eccentricity is equal to zero.
  • An ellipse’s eccentricity lies between 0 and 1.
  • A parabola’s eccentricity is equal to 1.
  • A hyperbola’s eccentricity is greater than one.
  • A line’s eccentricity equals infinity.

Eccentricity Formula 

We all know that the planets revolve in an elliptical orbit around the Earth. The eccentricity of Earth’s orbit (e = 0.0167) is less than that of Mars (e = 0.0935). The shape looks less like a circle as the eccentricity value moves away from zero. A parabola has one focus and one directrix, whereas an ellipse and a hyperbola have two foci and two directrixes. Their eccentricity formulas are given in terms of their semimajor axis(a) and semiminor axis(b) for an ellipse, and a = semi-transverse axis and b = semi-conjugate axis for a hyperbola. 

The eccentricity formula is given by, 

Eccentricity equals Distance to the focus/Distance to the directrix 

Eccentricity (e) = c/a

Where, c equals the distance between the centre and the focus

A equals the distance between the centre and the vertex.

Eccentricity in Solar System

The gravitational pull of our solar system’s two largest gas giant planets, Jupiter and Saturn, causes the shape of Earth’s orbit to shift from nearly circular to slightly elliptical over time. Eccentricity is a measure of how far the Earth’s orbit deviates from a perfect circle.

History of Ellipse

Menaechmus was the first to investigate the ellipse. The ellipse was first described by Euclid, and Apollonius gave it its current name. 

In 1602, Kepler stated that he thought Mars’ orbit was oval, but later discovered that it was an ellipse with the sun at one focus. In fact, it was Kepler who coined the term “focus” and published his findings in 1609. Planetary orbits have a low eccentricity (i.e. they are close to circles). Mars has an eccentricity of 1/11 and the Earth has an eccentricity of 1/60.

In 1705, Halley demonstrated that the comet named after him moved in an elliptical orbit around the sun.

The eccentricity and hyperbola concept can be learnt in an interesting way at cuemath.com

 

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