The eccentricity of a curved shape determines its shape no matter what the size of the shape is. If a plane intersects with a double-napped cone, then four curves are formed and they are circle, ellipse, parabola, and hyperbola. The features of such curves are determined by a factor known as eccentricity. A circle has zero eccentricity, a parabola has unit eccentricity, whereas ellipses and hyperbolas have varying eccentricity.
The ratio of the distance from a particular point on the conic section to the focus to the perpendicular distance from the same point to the nearest directrix is known as the eccentricity of the conic sections. The eccentricity is a constant value and it is denoted by ‘e’. The roundness of a curved shape is determined by the eccentricity of the shape. As the eccentricity of the shape increases, the curvature decreases. From this, we can conclude that the eccentricity of a shape is inversely related to its curvature.
The eccentricity of the earth’s orbit is, e = 0.0167, while the eccentricity of the orbit of Mars is, e = 0.0935. We can see that the eccentricity of the earth’s orbit is less than that of Mars. As the value of the eccentricity starts moving away from 0, the shape appears to be round or circle. An ellipse and hyperbola have two foci as well as two directrices, whereas a parabola has only one focus and one directrix. Let us learn about the formula that we use to find out the eccentricity of a shape.
Eccentricity = Distance to the focus/Distance to the directrix
i.e., e = c/a
the conditions given are,
e = eccentricity value, c = distance from a particular point on the conic section to the focus, a = distance from a particular point on the conic section to its directrix.
Eccentricity of Ellipse
A set of all the points in a plane where the sum of distances from two fixed points, also known as foci, in the plane are found to be constant is known as an ellipse. The ratio of the distance from its center to one of the foci and one of the vertices is known as the eccentricity of an ellipse.
Eccentricity of Circle
We all know that the set of all the points in a plane that are equidistant from the center point of the plane is known as a circle. In a circle both the foci coincide with its center. Since, both the foci are at the same point, the distance from the center to either of its foci is zero. Thus, the eccentricity makes the circle round in shape and the value of the eccentricity of a circle is always 0.
Eccentricity of Parabola
The set of all the points in a plane that are equidistant from the directrix and the focus is known as a parabola. When the value of the eccentricity e = 1, the locus of the moving point forms the shape of a parabola. Therefore, the U shape to the parabola curve is formed. We can conclude that the eccentricity of a parabola is 1.
The locus of the points in a plane, where the sum of the distances from two fixed points is a constant value, is known as an ellipse. Those two fixed points are known as the foci of the ellipse.
Properties of an Ellipse
- When a plane intersects a cone at the angle of its base is known as an ellipse.
- Every ellipse has two focal points, i.e., foci. The sum of the distances from a particular to either of the foci is constant.
- The eccentricity value is always less than 1.
- Every ellipse has a center and a minor and major axis.
To know more about the eccentricity of other shapes, visit Cuemath.