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The area of Hügelschäffer curves via Taylor series

Maja Petrovic, Branko Malesevic

Abstract

In this paper, we give new Taylor approximative formulae for the area of the egg-shaped parts of Hügelschäffer curves. Based on a parametrization of the Hügelschäffer curve, a formula for the area of the egg-shaped part of such a curve is derived via elliptic integrals of the first and second kind. Furthermore, new approximative formulae for calculating this area derived from standard and double Taylor approximations are given. A representation of the value $\frac{1}π$ was also obtained using an appropriate series.

The area of Hügelschäffer curves via Taylor series

Abstract

In this paper, we give new Taylor approximative formulae for the area of the egg-shaped parts of Hügelschäffer curves. Based on a parametrization of the Hügelschäffer curve, a formula for the area of the egg-shaped part of such a curve is derived via elliptic integrals of the first and second kind. Furthermore, new approximative formulae for calculating this area derived from standard and double Taylor approximations are given. A representation of the value was also obtained using an appropriate series.
Paper Structure (6 sections, 3 theorems, 76 equations, 3 figures)

This paper contains 6 sections, 3 theorems, 76 equations, 3 figures.

Table of Contents

  1. Hügelschäffer curve $\mathcal{F}_{\hbox{\scriptsize \boldmath $q$},\, egg}$
  2. The area of curve $\mathcal{F}_{\hbox{\scriptsize \boldmath $q$},\, egg}$
  3. Taylor approximations of elliptic integrals
  4. Taylor approximations for the area of Hügelschäffer curve $\mathcal{F}_{\hbox{\scriptsize \boldmath $q$}, \, egg}$
  5. Approximative formulae for the area of curve $\mathcal{F}_{\hbox{\scriptsize \boldmath $q$}, \, egg}$
  6. Conclusion

Key Result

Theorem 2.1

For the area $\mathcal{A}_{\hbox{\scriptsize \boldmath$q$},\,egg}$ of curve $\mathcal{F}_{\hbox{\scriptsize \boldmath $q$},\, egg}$ it holds that$:$ and where $a$, $b$, $w$ are Hügelschäffer curve parameters and $k = \dfrac{\hbox{\boldmath $q$}^2 w}{a}$.

Figures (3)

  • Figure 1: A Hügelschäffer curve $\mathcal{F} = \mathcal{F}_{egg} \cup \mathcal{F}_{hyp}$; Source: © First Author
  • Figure 2: Hügelschäffer's construction of an egg curve $\mathcal{F}_{\hbox{\scriptsize \boldmath $q$},\, egg}$; Source: © First Author
  • Figure 3: A comparison of the areas of a parabola, Hügelschäffer egg curve $\mathcal{F}_{\hbox{\scriptsize \boldmath $q$},\,egg}$, and an ellipse; Source: © First Author

Theorems & Definitions (9)

  • Theorem 2.1
  • Remark 2.2
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.4: Theorem WD
  • Remark 4.1
  • Theorem 5.1
  • Remark 5.2