User:Michael Retriever/Ellipse perimeters
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This is a comparative test that attempts to verify the efficiency of various formulas (infinite binomial expansions) for the perimeter of an ellipse; to see whether they need more or less terms within their infinite expansions to reach a satisfactory result. The charts have been done with MS Excel.
Contents |
[edit] Formulas
- 1st
![z=(1+f')^{-1}\left[1+\left({1\over 2}\right)^2f'^2+\left({1\over 3^2}\right)\left({1\cdot 3\over 2\cdot 4}\right)^2f'^4+\left({1\over 5^2}\right)\left({1\cdot 3\cdot 5\over 2\cdot 4\cdot 6}\right)^2f'^6+\dots\right]\,](../../../math/b/2/9/b299a9b7f04fb8903b5c209cb630a87f.png)
- Used within the charts with the form:
![z=\left[{1\over 1+f'}+\left({1\over 2}\right)^2{f'^2\over 1+f'}+\left({1\over 3^2}\right)\left({1\cdot 3\over 2\cdot 4}\right)^2{f'^4\over 1+f'}+\left({1\over 5^2}\right)\left({1\cdot 3\cdot 5\over 2\cdot 4\cdot 6}\right)^2{f'^6\over 1+f'}+\dots\right]\,](../../../math/d/b/d/dbd12009bf936bb87b224540747d4d7a.png)
- 2nd
![z=\left[1 - \left({1\over 2}\right)^2e^2 - \left({1\over 3}\right)\left({1\cdot 3\over 2\cdot 4}\right)^2e^4 - \left({1\over 5}\right)\left({1\cdot 3\cdot 5\over 2\cdot 4\cdot 6}\right)^2e^6 - \dots\right]\,](../../../math/1/3/0/130a4c53d994bdbb31fceafcf81f0192.png)
- 3rd (website link)
![z=\left[1 + \left({1 \over 2}\right)^2 n - \left({1 \over 2\cdot 4}\right)^2 3n^2 + \left({1 \cdot 3 \over 2\cdot 4\cdot 6}\right)^2 5n^3 - \left({1 \cdot 3 \cdot 5\over 2\cdot 4\cdot 6\cdot 8}\right)^2 7n^4 + \dots\right]\,](../../../math/c/7/5/c75300c7fad7eb65eee34c6b7b727ce6.png)
- Used within the charts with the form:
![z=\left[\left(\frac{b}{a}\right) + \left({1 \over 2}\right)^2 \left(\frac{b}{a}\right) n- \left({1\over 3}\right)\left({1\cdot 3\over 2\cdot 4}\right)^2 \left(\frac{b}{a}\right) n^2 + \left({1\over 5}\right)\left({1\cdot 3\cdot 5\over 2\cdot 4\cdot 6}\right)^2 \left(\frac{b}{a}\right) n^3 - \dots\right]\,](../../../math/2/b/4/2b45b5c34be621458eff0b820fb52939.png)
- Equations
- The various elements of the formulas can be expressed through elementary equations or through trigonometric equations. For more information you can visit eccentricity and angular eccentricity.
[edit] Polygonal paths
Check explanation for the polygonal path. The polygonal path 1 measures the distance between two points in the ellipse's quarter of a perimeter every 0.01 degrees. The polygonal path 2 measures the distance between two points in the ellipse's quarter of a perimeter every 0.001 degrees.
[edit] Comparative test
[edit] Conclusion
As we can see, the first formula is the most efficient one.
