Ellipse Calculator

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Lets first understand, what is an ellipse and how we can find different properties of an ellipse?

Ellipse

An ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the fixed point is a constant. The fixed point is called focal points of the ellipse.

Circle is the special type of ellipse in which the two focal points are the same.

The elongation of an ellipse is measured by its eccentricity e, a number ranging from [0,1).

An angled cross section of a cylinder is also an ellipse.

Equation of an ellipse

The equation of a standard ellipse centered at the origin (0,0) with major-axis '2a' and minor-axis '2b' is

$$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$$

If a > b, then the foci of the ellipse will be at C \((\pm c,0)\) for \(c=\sqrt{a^2-b^2}\)

The standard parametric equation of the ellipse is

$$(x,y)=(a\hspace{0.1cm}cos\theta ,b\hspace{0.1cm}sin\theta)$$

where, \(0\leq \theta \leq 2 \pi \).

An ellipse may also be defined in terms of one focal point and a line outside the ellipse called the directrix, for all points on the ellipse, the ratio between the distance to the focus and the distance to the directrix is a constant. This constant ratio is the eccentricity of the ellipse, given by

$$e=\dfrac{c}{a}=\sqrt{1-\dfrac{b^2}{a^2}}$$

Parameters of an ellipse

Principal axes of an ellipse

Generally, the semi-major and semi-minor axes of the ellipse, are denoted a and b, respectively, where \(a\geq b>0\), then the equation of the ellipse is

$$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$$

Linear eccentricity of an ellipse

This is the distance from the center of the ellipse to its focus, \(c=\sqrt{a^2-b^2}\).

Eccentricity of an ellipse

The eccentricity of an ellipse can be expressed as

$$e=\dfrac{c}{a}=\sqrt{1-\dfrac{b^2}{a^2}}$$

Latus rectum of an ellipse

The length of the chord through one focus, perpendicular to the major axis of the ellipse, is called the latus rectum of the ellipse, given by

$$l=\dfrac{2b^2}{a}=2a(1-e^2)$$

Area of an ellipse

Area enclosed by the ellipse is given by,

$$A=\pi ab$$

Circumference of an ellipse

Circumference of an ellipse is given by

$$C\approx \pi \left(3*(a+b)-\sqrt{(3*a+b)*(a+3*b)}\right)$$

For calculating different properties of the ellipse using above ellipse calculator, you have to just write the values in the given input boxes and press the calculate button, you will get the result.

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