# A New Approximation for the Perimeter of an Ellipse

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## Abstract

**:**

## 1. Introduction

#### 1.1. The Perimeter of the Ellipse

#### 1.2. Ramanujan’s Approximations

#### 1.3. Symbolic Regression and Learning from Data

#### 1.4. Structure of the Paper

## 2. Materials and Methods

## 3. Results

#### 3.1. A New Asymptotic Bound Leads to a More Accurate Approximation for High Eccentricity

#### 3.2. Use of a Highly Accurate Padé Approximation Formula for Low Eccentricities

## 4. Comparative Results with Sýkora’s Curated Collection of Approximations

#### 4.1. Keplerian Equations

#### 4.2. Keplerian Padè Equations

#### 4.3. Exact Extremes (No Crossing) Equations

#### 4.4. Combined Padè Equations with Exact Extremes (No Crossing)

#### 4.5. Exact Extremes and Crossing Equations

#### 4.6. Algebraic Equations

#### 4.7. All S-Class Equations

#### 4.8. A Detailed Comparison of Results with the Best Performers in the S-Class of Sýkora’s Collection

## 5. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Sykora, S. Approximations of Ellipse Perimeters and of the Complete Elliptic Integral E(x). Review of known formulae. In Stan’s Library; Extra Byte: Castano Primo, Italy, 2005. [Google Scholar] [CrossRef]
- Nemes, G. A historic comment on ellipse perimeter approximations. In Stan’s Library; Extra Byte: Castano Primo, Italy, 2005. [Google Scholar] [CrossRef]
- Barnard, R.W.; Pearce, K.; Schovanec, L. Inequalities for the Perimeter of an Ellipse. J. Math. Anal. Appl.
**2001**, 260, 295–306. [Google Scholar] [CrossRef] - Tomkys, H. Formula for the Perimeter of an Ellipse. Nature
**1902**, 65, 563. [Google Scholar] [CrossRef] - Muir, T. Formula for the Perimeter of an Ellipse. Nature
**1902**, 66, 174–175. [Google Scholar] [CrossRef] - Rogers, R. Perimeter of an Ellipse. Nature
**1920**, 105, 8. [Google Scholar] [CrossRef] - Ramanujan, S. Modular equations and approximations to π. Q. J. Math. Oxf.
**1914**, 45, 350–372. [Google Scholar] - Villarino, M.B. A note on the accuracy of Ramanujan’s approximative formula for the perimeter of an ellipse. J. Inequal. Pure Appl. Math.
**2006**, 7, 21. [Google Scholar] - Wang, Y.; Yao, Q.; Kwok, J.T.; Ni, L.M. Generalizing from a few examples: A survey on few-shot learning. ACM Comput. Surv.
**2021**, 53, 63. [Google Scholar] [CrossRef] - Lu, J.; Gong, P.; Ye, J.; Zhang, C. A survey on machine learning from few samples. Pattern Recognit.
**2023**, 139, 109480. [Google Scholar] [CrossRef] - TuringBot. Documentation for TuringBot: Symbolic Regression Software. 2024. Available online: https://turingbotsoftware.com/documentation.html (accessed on 12 September 2024).
- Villarino, M.B. Ramanujan’s inverse elliptic arc approximation. Ramanujan J.
**2014**, 34, 157–161. [Google Scholar] [CrossRef] - Grosser, M. The Discovery of Neptune; Harvard University Press: Cambridge, MA, USA, 1962. [Google Scholar]
- Lyttleton, R.A. A short method for the discovery of Neptune. Mon. Not. R. Astron. Soc.
**1958**, 118, 551–559. [Google Scholar] [CrossRef] - Michaud, E.J.; Liu, Z.; Tegmark, M. Precision Machine Learning. Entropy
**2023**, 25, 175. [Google Scholar] [CrossRef] - Sun, H.; Moscato, P. A Memetic Algorithm for Symbolic Regression. In Proceedings of the 2019 IEEE Congress on Evolutionary Computation (CEC), Wellington, New Zealand, 10–13 June 2019; pp. 2167–2174. [Google Scholar] [CrossRef]
- Moscato, P.; Ciezak, A.; Noman, N. Dynamic depth for analytic continued fraction regression. In Proceedings of the 2023 Annual Conference on Genetic and Evolutionary Computation, Lisbon, Portugal, 15–19 July 2023; pp. 520–528. [Google Scholar] [CrossRef]

**Figure 1.**Relative absolute error plot for the values in Table 1 that for a are between 1 and 80 while keeping $b=1$. We compare the approximation of the upper bound model ${m}_{u}$ (Equation (5)) and our proposed model ${m}_{MC}$ (Equation (12)). We observe a significant difference in the performance of ${m}_{MC}$, particularly for higher values of a.

**Figure 2.**Relative absolute error plot with log scale on the axes. We show the results for our upper bound model ${m}_{u}$ (Equation (5)), our best model ${m}_{MC}$ (Equation (12)), and the other models listed in Table 1. The equations were produced using the samples from Table 1, which were then re-evaluated on 10,000 uniform values of a in the log space between $a=1$ and $a=1\times {10}^{9}$ with $b=1$. We observe strong performance from ${m}_{MC}$ in both extremities of a relative to the other models.

**Figure 3.**Comparison of the Keplerian and Keplerian Padé equation classes in Sýkora’s analysis in subfigures (

**a**) and (

**b**), respectively. The proposed model ${m}_{MC}$ (Equation (12)) appears in black in each subfigure, and the axes plot the log of the relative absolute error across the log space between a values between 1 and $1\times {10}^{9}$ with $b=1$.

**Figure 4.**Comparison of the Padé equation classes exact extremes no-crossing and exact extremes in Sýkora’s analysis in subfigures (

**a**) and (

**b**), respectively. The proposed model ${m}_{MC}$ (Equation (12)) appears in black in each subfigure, and the axes plot the log of the relative absolute error across the log space between a values between 1 and $1\times {10}^{9}$ with $b=1$.

**Figure 5.**Comparison of the exact extremes crossing and algebraic equation classes in Sýkora’s analysis given in subfigures (

**a**) and (

**b**), respectively. The proposed model ${m}_{MC}$ (Equation (12)) appears in black in each subfigure, and the axes plot the log of the relative absolute error across the log space between a values between 1 and $1\times {10}^{9}$ with $b=1$.

**Figure 6.**Relative absolute error plot with log scale on the axes, comparing our equation ${m}_{MC}$ (Equation (12)) with models from Sýkora’s S-class. The equations are evaluated on 10,000 uniform values of a in the log space between a = 1 and $a=1\times {10}^{9}$.

**Table 1.**The table shows the absolute relative error for each approximation of an ellipse for different eccentricities within the training samples, where a varies but $b=1$ is constant. The smallest errors are in bold within each row.

a | ${\mathit{m}}_{\mathit{R}}$ Equation (2) Err | ${\mathit{m}}_{\mathit{u}}$ Equation (5) Err | ${\mathit{m}}_{\mathit{Ru}}$ Equation (8) Err | ${\mathit{m}}_{\mathit{p}}$ Equation (9) Err | ${\mathit{m}}_{\mathit{MC}}$ Equation (12) Err |
---|---|---|---|---|---|

1.05 | 1.707$\times {10}^{-21}$ | 2.793$\times {10}^{-07}$ | 1.707$\times {10}^{-21}$ | $\mathbf{1}.\mathbf{743}\times {\mathbf{10}}^{-\mathbf{28}}$ | $\mathbf{1}.\mathbf{743}\times {\mathbf{10}}^{-\mathbf{28}}$ |

1.15 | 6.296$\times {10}^{-17}$ | 2.284$\times {10}^{-06}$ | 6.296$\times {10}^{-17}$ | $\mathbf{4}.\mathbf{317}\times {\mathbf{10}}^{-\mathbf{22}}$ | $\mathbf{4}.\mathbf{317}\times {\mathbf{10}}^{-\mathbf{22}}$ |

1.25 | 6.679$\times {10}^{-15}$ | 5.784$\times {10}^{-06}$ | 6.679$\times {10}^{-15}$ | $\mathbf{2}.\mathbf{959}\times {\mathbf{10}}^{-\mathbf{19}}$ | $\mathbf{2}.\mathbf{959}\times {\mathbf{10}}^{-\mathbf{19}}$ |

1.35 | 1.268$\times {10}^{-13}$ | 1.037$\times {10}^{-05}$ | 1.268$\times {10}^{-13}$ | $\mathbf{1}.\mathbf{825}\times {\mathbf{10}}^{-\mathbf{17}}$ | $\mathbf{1}.\mathbf{825}\times {\mathbf{10}}^{-\mathbf{17}}$ |

1.05 | 1.707$\times {10}^{-21}$ | 2.793$\times {10}^{-07}$ | 1.707$\times {10}^{-21}$ | $\mathbf{1}.\mathbf{743}\times {\mathbf{10}}^{-\mathbf{28}}$ | $\mathbf{1}.\mathbf{743}\times {\mathbf{10}}^{-\mathbf{28}}$ |

1.15 | 6.296$\times {10}^{-17}$ | 2.284$\times {10}^{-06}$ | 6.296$\times {10}^{-17}$ | $\mathbf{4}.\mathbf{317}\times {\mathbf{10}}^{-\mathbf{22}}$ | $\mathbf{4}.\mathbf{317}\times {\mathbf{10}}^{-\mathbf{22}}$ |

1.25 | 6.679$\times {10}^{-15}$ | 5.784$\times {10}^{-06}$ | 6.679$\times {10}^{-15}$ | $\mathbf{2}.\mathbf{959}\times {\mathbf{10}}^{-\mathbf{19}}$ | $\mathbf{2}.\mathbf{959}\times {\mathbf{10}}^{-\mathbf{19}}$ |

1.35 | 1.268$\times {10}^{-13}$ | 1.037$\times {10}^{-05}$ | 1.268$\times {10}^{-13}$ | $\mathbf{1}.\mathbf{825}\times {\mathbf{10}}^{-\mathbf{17}}$ | $\mathbf{1}.\mathbf{825}\times {\mathbf{10}}^{-\mathbf{17}}$ |

1.45 | 1.049$\times {10}^{-12}$ | 1.574$\times {10}^{-05}$ | 1.049$\times {10}^{-12}$ | 3.516$\times {10}^{-16}$ | $\mathbf{3}.\mathbf{516}\times {\mathbf{10}}^{-\mathbf{16}}$ |

1.55 | 5.328$\times {10}^{-12}$ | 2.166$\times {10}^{-05}$ | 5.328$\times {10}^{-12}$ | 3.422$\times {10}^{-15}$ | $\mathbf{3}.\mathbf{422}\times {\mathbf{10}}^{-\mathbf{15}}$ |

1.65 | 1.966$\times {10}^{-11}$ | 2.794$\times {10}^{-05}$ | 1.966$\times {10}^{-11}$ | 2.130$\times {10}^{-14}$ | $\mathbf{2}.\mathbf{130}\times {\mathbf{10}}^{-\mathbf{14}}$ |

1.75 | 5.799$\times {10}^{-11}$ | 3.445$\times {10}^{-05}$ | 5.796$\times {10}^{-11}$ | 9.684$\times {10}^{-14}$ | $\mathbf{9}.\mathbf{683}\times {\mathbf{10}}^{-\mathbf{14}}$ |

1.85 | 1.450$\times {10}^{-10}$ | 4.109$\times {10}^{-05}$ | 1.448$\times {10}^{-10}$ | 3.493$\times {10}^{-13}$ | $\mathbf{3}.\mathbf{492}\times {\mathbf{10}}^{-\mathbf{13}}$ |

1.95 | 3.193$\times {10}^{-10}$ | 4.778$\times {10}^{-05}$ | 3.186$\times {10}^{-10}$ | 1.056$\times {10}^{-12}$ | $\mathbf{1}.\mathbf{055}\times {\mathbf{10}}^{-\mathbf{12}}$ |

2 | 4.560$\times {10}^{-10}$ | 5.112$\times {10}^{-05}$ | 4.546$\times {10}^{-10}$ | 1.738$\times {10}^{-12}$ | $\mathbf{1}.\mathbf{737}\times {\mathbf{10}}^{-\mathbf{12}}$ |

3 | 3.298$\times {10}^{-08}$ | 1.122$\times {10}^{-04}$ | 3.107$\times {10}^{-08}$ | 6.960$\times {10}^{-10}$ | $\mathbf{6}.\mathbf{687}\times {\mathbf{10}}^{-\mathbf{10}}$ |

4 | 2.499$\times {10}^{-07}$ | 1.584$\times {10}^{-04}$ | 2.009$\times {10}^{-07}$ | 1.180$\times {10}^{-08}$ | $\mathbf{9}.\mathbf{571}\times {\mathbf{10}}^{-\mathbf{09}}$ |

5 | 8.514$\times {10}^{-07}$ | 1.923$\times {10}^{-04}$ | 5.339$\times {10}^{-07}$ | 6.511$\times {10}^{-08}$ | $\mathbf{3}.\mathbf{926}\times {\mathbf{10}}^{-\mathbf{08}}$ |

6 | 1.965$\times {10}^{-06}$ | 2.175$\times {10}^{-04}$ | 8.884$\times {10}^{-07}$ | 2.081$\times {10}^{-07}$ | $\mathbf{8}.\mathbf{627}\times {\mathbf{10}}^{-\mathbf{08}}$ |

7 | 3.629$\times {10}^{-06}$ | 2.363$\times {10}^{-04}$ | 1.080$\times {10}^{-06}$ | 4.867$\times {10}^{-07}$ | $\mathbf{1}.\mathbf{325}\times {\mathbf{10}}^{-\mathbf{07}}$ |

8 | 5.821$\times {10}^{-06}$ | 2.506$\times {10}^{-04}$ | 9.776$\times {10}^{-07}$ | 9.345$\times {10}^{-07}$ | $\mathbf{1}.\mathbf{600}\times {\mathbf{10}}^{-\mathbf{07}}$ |

9 | 8.487$\times {10}^{-06}$ | 2.614$\times {10}^{-04}$ | 5.271$\times {10}^{-07}$ | 1.570$\times {10}^{-06}$ | $\mathbf{1}.\mathbf{572}\times {\mathbf{10}}^{-\mathbf{07}}$ |

10 | 1.156$\times {10}^{-05}$ | 2.695$\times {10}^{-04}$ | 2.684$\times {10}^{-07}$ | 2.398$\times {10}^{-06}$ | $\mathbf{1}.\mathbf{181}\times {\mathbf{10}}^{-\mathbf{07}}$ |

11 | 1.497$\times {10}^{-05}$ | 2.756$\times {10}^{-04}$ | 1.371$\times {10}^{-06}$ | 3.415$\times {10}^{-06}$ | $\mathbf{4}.\mathbf{146}\times {\mathbf{10}}^{-\mathbf{08}}$ |

12 | 1.865$\times {10}^{-05}$ | 2.800$\times {10}^{-04}$ | 2.728$\times {10}^{-06}$ | 4.610$\times {10}^{-06}$ | $\mathbf{7}.\mathbf{110}\times {\mathbf{10}}^{-\mathbf{08}}$ |

13 | 2.254$\times {10}^{-05}$ | 2.831$\times {10}^{-04}$ | 4.281$\times {10}^{-06}$ | 5.968$\times {10}^{-06}$ | $\mathbf{2}.\mathbf{153}\times {\mathbf{10}}^{-\mathbf{07}}$ |

14 | 2.660$\times {10}^{-05}$ | 2.852$\times {10}^{-04}$ | 5.977$\times {10}^{-06}$ | 7.471$\times {10}^{-06}$ | $\mathbf{3}.\mathbf{853}\times {\mathbf{10}}^{-\mathbf{07}}$ |

15 | 3.078$\times {10}^{-05}$ | 2.864$\times {10}^{-04}$ | 7.768$\times {10}^{-06}$ | 9.104$\times {10}^{-06}$ | $\mathbf{5}.\mathbf{744}\times {\mathbf{10}}^{-\mathbf{07}}$ |

16 | 3.504$\times {10}^{-05}$ | 2.869$\times {10}^{-04}$ | 9.615$\times {10}^{-06}$ | 1.085$\times {10}^{-05}$ | $\mathbf{7}.\mathbf{758}\times {\mathbf{10}}^{-\mathbf{07}}$ |

17 | 3.935$\times {10}^{-05}$ | 2.869$\times {10}^{-04}$ | 1.148$\times {10}^{-05}$ | 1.269$\times {10}^{-05}$ | $\mathbf{9}.\mathbf{830}\times {\mathbf{10}}^{-\mathbf{07}}$ |

18 | 4.370$\times {10}^{-05}$ | 2.864$\times {10}^{-04}$ | 1.335$\times {10}^{-05}$ | 1.461$\times {10}^{-05}$ | $\mathbf{1}.\mathbf{190}\times {\mathbf{10}}^{-\mathbf{06}}$ |

19 | 4.805$\times {10}^{-05}$ | 2.855$\times {10}^{-04}$ | 1.519$\times {10}^{-05}$ | 1.660$\times {10}^{-05}$ | $\mathbf{1}.\mathbf{393}\times {\mathbf{10}}^{-\mathbf{06}}$ |

20 | 5.239$\times {10}^{-05}$ | 2.843$\times {10}^{-04}$ | 1.699$\times {10}^{-05}$ | 1.864$\times {10}^{-05}$ | $\mathbf{1}.\mathbf{586}\times {\mathbf{10}}^{-\mathbf{06}}$ |

21 | 5.671$\times {10}^{-05}$ | 2.829$\times {10}^{-04}$ | 1.874$\times {10}^{-05}$ | 2.072$\times {10}^{-05}$ | $\mathbf{1}.\mathbf{767}\times {\mathbf{10}}^{-\mathbf{06}}$ |

22 | 6.100$\times {10}^{-05}$ | 2.812$\times {10}^{-04}$ | 2.044$\times {10}^{-05}$ | 2.284$\times {10}^{-05}$ | $\mathbf{1}.\mathbf{934}\times {\mathbf{10}}^{-\mathbf{06}}$ |

23 | 6.524$\times {10}^{-05}$ | 2.794$\times {10}^{-04}$ | 2.206$\times {10}^{-05}$ | 2.499$\times {10}^{-05}$ | $\mathbf{2}.\mathbf{085}\times {\mathbf{10}}^{-\mathbf{06}}$ |

24 | 6.943$\times {10}^{-05}$ | 2.775$\times {10}^{-04}$ | 2.362$\times {10}^{-05}$ | 2.715$\times {10}^{-05}$ | $\mathbf{2}.\mathbf{220}\times {\mathbf{10}}^{-\mathbf{06}}$ |

25 | 7.356$\times {10}^{-05}$ | 2.754$\times {10}^{-04}$ | 2.511$\times {10}^{-05}$ | 2.932$\times {10}^{-05}$ | $\mathbf{2}.\mathbf{337}\times {\mathbf{10}}^{-\mathbf{06}}$ |

26 | 7.763$\times {10}^{-05}$ | 2.733$\times {10}^{-04}$ | 2.652$\times {10}^{-05}$ | 3.150$\times {10}^{-05}$ | $\mathbf{2}.\mathbf{436}\times {\mathbf{10}}^{-\mathbf{06}}$ |

27 | 8.164$\times {10}^{-05}$ | 2.710$\times {10}^{-04}$ | 2.787$\times {10}^{-05}$ | 3.367$\times {10}^{-05}$ | $\mathbf{2}.\mathbf{518}\times {\mathbf{10}}^{-\mathbf{06}}$ |

28 | 8.558$\times {10}^{-05}$ | 2.688$\times {10}^{-04}$ | 2.914$\times {10}^{-05}$ | 3.585$\times {10}^{-05}$ | $\mathbf{2}.\mathbf{584}\times {\mathbf{10}}^{-\mathbf{06}}$ |

29 | 8.945$\times {10}^{-05}$ | 2.665$\times {10}^{-04}$ | 3.035$\times {10}^{-05}$ | 3.801$\times {10}^{-05}$ | $\mathbf{2}.\mathbf{634}\times {\mathbf{10}}^{-\mathbf{06}}$ |

30 | 9.325$\times {10}^{-05}$ | 2.641$\times {10}^{-04}$ | 3.149$\times {10}^{-05}$ | 4.016$\times {10}^{-05}$ | $\mathbf{2}.\mathbf{669}\times {\mathbf{10}}^{-\mathbf{06}}$ |

40 | 1.274$\times {10}^{-04}$ | 2.407$\times {10}^{-04}$ | 3.976$\times {10}^{-05}$ | 6.060$\times {10}^{-05}$ | $\mathbf{2}.\mathbf{398}\times {\mathbf{10}}^{-\mathbf{06}}$ |

50 | 1.554$\times {10}^{-04}$ | 2.194$\times {10}^{-04}$ | 4.395$\times {10}^{-05}$ | 7.852$\times {10}^{-05}$ | $\mathbf{1}.\mathbf{560}\times {\mathbf{10}}^{-\mathbf{06}}$ |

60 | 1.783$\times {10}^{-04}$ | 2.009$\times {10}^{-04}$ | 4.578$\times {10}^{-05}$ | 9.393$\times {10}^{-05}$ | $\mathbf{6}.\mathbf{352}\times {\mathbf{10}}^{-\mathbf{07}}$ |

70 | 1.974$\times {10}^{-04}$ | 1.851$\times {10}^{-04}$ | 4.626$\times {10}^{-05}$ | 1.072$\times {10}^{-04}$ | $\mathbf{1}.\mathbf{905}\times {\mathbf{10}}^{-\mathbf{07}}$ |

80 | 2.134$\times {10}^{-04}$ | 1.714$\times {10}^{-04}$ | 4.595$\times {10}^{-05}$ | 1.186$\times {10}^{-04}$ | $\mathbf{8}.\mathbf{676}\times {\mathbf{10}}^{-\mathbf{07}}$ |

90 | 2.271$\times {10}^{-04}$ | 1.596$\times {10}^{-04}$ | 4.521$\times {10}^{-05}$ | 1.285$\times {10}^{-04}$ | $\mathbf{1}.\mathbf{399}\times {\mathbf{10}}^{-\mathbf{06}}$ |

100 | 2.390$\times {10}^{-04}$ | 1.493$\times {10}^{-04}$ | 4.421$\times {10}^{-05}$ | 1.372$\times {10}^{-04}$ | $\mathbf{1}.\mathbf{805}\times {\mathbf{10}}^{-\mathbf{06}}$ |

500 | 3.572$\times {10}^{-04}$ | 4.225$\times {10}^{-05}$ | 1.780$\times {10}^{-05}$ | 2.295$\times {10}^{-04}$ | $\mathbf{1}.\mathbf{409}\times {\mathbf{10}}^{-\mathbf{06}}$ |

1000 | 3.784$\times {10}^{-04}$ | 2.249$\times {10}^{-05}$ | 1.004$\times {10}^{-05}$ | 2.470$\times {10}^{-04}$ | $\mathbf{6}.\mathbf{122}\times {\mathbf{10}}^{-\mathbf{07}}$ |

10,000 | 3.998$\times {10}^{-04}$ | 2.422$\times {10}^{-06}$ | 1.158$\times {10}^{-06}$ | 2.649$\times {10}^{-04}$ | $\mathbf{1}.\mathbf{644}\times {\mathbf{10}}^{-\mathbf{08}}$ |

100,000 | 4.023$\times {10}^{-04}$ | 2.453$\times {10}^{-08}$ | 1.577$\times {10}^{-08}$ | 2.671$\times {10}^{-04}$ | $\mathbf{5}.\mathbf{054}\times {\mathbf{10}}^{-\mathbf{15}}$ |

1,000,000,000 | 4.023$\times {10}^{-04}$ | 2.454$\times {10}^{-11}$ | 3.957$\times {10}^{-09}$ | 2.671$\times {10}^{-04}$ | $\mathbf{3}.\mathbf{974}\times {\mathbf{10}}^{-\mathbf{15}}$ |

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**MDPI and ACS Style**

Moscato, P.; Ciezak, A.
A New Approximation for the Perimeter of an Ellipse. *Algorithms* **2024**, *17*, 464.
https://doi.org/10.3390/a17100464

**AMA Style**

Moscato P, Ciezak A.
A New Approximation for the Perimeter of an Ellipse. *Algorithms*. 2024; 17(10):464.
https://doi.org/10.3390/a17100464

**Chicago/Turabian Style**

Moscato, Pablo, and Andrew Ciezak.
2024. "A New Approximation for the Perimeter of an Ellipse" *Algorithms* 17, no. 10: 464.
https://doi.org/10.3390/a17100464