Eccentricity

Their characteristics are classified according to their shapes, which are determined by an intriguing element known as eccentricity. The eccentricity of circles is zero, but the eccentricity of parabolas is one. The eccentricities of ellipses and hyperbolas vary. The distance between the centres of two cylindrical objects, one of which surrounds the other, as between an eccentric and the shaft on which it is placed, is typically represented by the letter e (don’t confuse this with Euler’s number “e,” they are completely different). The ratio of the distance between a point on a conic and a focus and the distance between the point and the directrix is a constant.

What is Eccentricity? 

The ratio of the distance from any point on the conic section to the focus to the perpendicular distance from that point to the nearest directrix is the eccentricity of conic sections. The eccentricity of a conic section is the distance from any point on the curve to its focus minus the distance from the same point to its directrix = a constant for every conic section. Eccentricity, represented by e, is the name given to this constant value. The roundness of a curved shape is determined by its eccentricity. As the eccentricity rises, the curvatures decrease. Any conic section can be defined as a collection of points with a fixed ratio of distances to a point (the focus) and a line (the directrix). The eccentricity, abbreviated as e, is the name given to this ratio. The distance between the centre of an ellipse or hyperbola and either of its two foci is symbolized by the letter c (or occasionally f or e). The ratio of the linear eccentricity to the semimajor axis an is known as eccentricity. 

The eccentricity of any conic section can be calculated using the following formula:

Eccentricity is defined as e = c/a.

Where,

c = the distance between the centre and the focus

a = the distance between the vertex and the centre

The following is the quadratic version of the general equation for any conic section: 

Ax2 + Bxy + Cy2 + Dx + Ey + F = Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

A conic section is any curve that is created by the intersection of a plane with a right circular cone in geometrical terms. Depending on the angle of the plane with respect to the cone, the intersection is a circle, an ellipse, a hyperbola, or a parabola; otherwise, it is a circle.  When the plane passes through simply the apex (creating a single point) or between the apex and another point on the cone, special (degenerate) examples of intersection arise. Conics are plane curves that are the pathways (loci) of a point travelling in such a way that the ratio of its distance from a given point (the focus) to its distance from a fixed line (the directrix) is a constant, which is known as the eccentricity of the curve. The curve is a circle if the eccentricity is zero; a parabola if equal to one; an ellipse if less than one; and a hyperbola if larger than one. Lets take a look at eccentricity uses.

Eccentricity equation for Circle 

When it comes to plane geometry, a circle is defined as a collection of points on a plane that are all equally far from the “centre,” which is a fixed point on the plane surface. It is the distance between the circle’s centre and one of its points that is referred to as the “radius.” When the circle’s centre is located at the origin, determining the equation of a circle is straightforward. 

Parabola eccentricity

It is defined as the set of points P with distances from a fixed point F (focus) in the plane equal to their distances from a fixed-line l (directrix) in the plane that form a parabola. With this in mind, the distance between a fixed point and a fixed line in a plane has a constant ratio equal to the distance between the two points in the plane. 

As a consequence, the eccentricity of the parabola is equal to one (e = 1). 

The eccentricity of a parabola is given as 1 and the basic equation is expressed as x2 = 4ay.

Ellipse’s eccentricity

A collection of points in a plane where the sum of distances between two fixed points is constant is defined as the collection of points in a plane. A more technical way of putting it is that the distance between two fixed points in a plane is proportionately less than the distance between two fixed lines on a planar surface. 

A result of this is that the eccentricity of the ellipse is less than one, which is indicated by the symbol e1. 

An ellipse’s general equation is written as:(x-h)2a2+(y-k)2b2=1

The eccentricity formula is as follows:e = c/a.

The lengths of the semi-major and semi-minor axes, respectively, are a and b for an ellipse.

Hyperbola Eccentricity

A hyperbola is defined as the set of all points in a plane whose distances from two fixed points differ by a constant amount. In other words, the distance from a plane’s fixed point is proportionally bigger than the distance from a plane’s fixed line.

Conclusion 

The curvatures of conic sections are determined by the eccentricity of the sections. 

The eccentricity of a circle is zero, but the eccentricity of a parabola is one. 

The various eccentricities of ellipses and parabolas are computed using the formula e = c/a, where a and b are the semi-axes of a hyperbola, and c is the semi-axes of an ellipse in the case of a hyperbola. 

Eccentricity of Line = Infinity (i.e.) e =1.

Eccentricity of Circle = 0 (i.e.) e =0.

Eccentricity of Ellipse = Between 0 and 1 (i.e.) 0 <e <1.

Eccentricity of Parabola = 1(i.e.) e =1.

Eccentricity of Hyperbola = Greater than 1(i.e.) e > 1