The Circumference and Area of an Ellipse

The methods for calculating the area and circumference of an ellipse may seem similar, but in reality, their levels of difficulty are worlds apart.

The area enclosed by an ellipse is S = \pi ab, where a and b are the semi-major and semi-minor axes, respectively. This can be derived solely from the standard equation of the ellipse.

Currently, no general closed-form formula for the circumference of an ellipse has been found. To solve it precisely, one must use the following infinite series: C = 2\pi a \left[ 1 - \left(\frac{1}{2}\right)^2 \left(\frac{c}{a}\right)^2 - \left(\frac{1 \cdot 3}{2 \cdot 4}\right)^2 \frac{c^4}{3a^4} - \left(\frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6}\right)^2 \frac{c^6}{5a^6} - \dots \right] Which can be written as: C = 2\pi a \sum_{n=0}^{\infty} \left\{ - \left[ \prod_{m=1}^n \left( \frac{2m-1}{2m} \right) \right]^2 \frac{c^{2n}}{a^{2n}(2n - 1)} \right\}

The distance c is called the linear eccentricity of the ellipse, which is equal to the distance from the center to either focus.

Of course, if you find these complex, do not be discouraged, as the mathematician Ramanujan provided a relatively simple and highly accurate approximation formula: C \approx \pi \left[ 3(a+b) - \sqrt{(3a+b)(a+3b)} \right]

There is also another highly accurate approximation formula (which is said to be sufficient even for calculating planetary orbits): C = \pi (a+b) \left[ 1 + \frac{3 \left( \frac{a-b}{a+b} \right)^2}{10 + \sqrt{4 - 3 \left( \frac{a-b}{a+b} \right)^2}} \right] \cdot \left[ 1 + \left( \frac{22}{7\pi} - 1 \right) \left( \frac{a-b}{a} \right)^{33.697} \right]

Below are some reference values for the circumference of an ellipse:

When reposting, please include the original article link: https://kexue.fm/archives/47

For more detailed information on reposting, please refer to: Scientific Space FAQ