# On Geometric Interpretations of Euler’s Substitutions

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## Abstract

**:**

## 1. Introduction

## 2. Three Classical Euler’s Substitutions

#### 2.1. First Euler Substitution

#### 2.2. Second Euler Substitution

#### 2.3. Third Euler Substitution

#### 2.4. Original Euler’s Approach

## 3. Geometric Interpretation

#### 3.1. Elliptic Case: $a<0$

#### 3.2. Parabolic Case: $a=0$

#### 3.3. Hyperbolic Case: $a>0$

#### 3.4. Rational Parameterization: Standard Approach

**Corollary**

**1.**

## 4. New Insights from the Geometric Interpretation

#### 4.1. Fourth Euler’s Substitution

#### 4.2. Simplifying Euler’s First Substitution

#### 4.3. Euler’s First Substitution as a Limit of the Generic Case

## 5. Euler’s Substitutions versus Trigonometric Substitutions

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Geometric interpretation of the second Euler substitution in the case $a<0$ and $c>0$. The point P is parameterized by the slope t of the line ${P}_{0}P$.

**Figure 2.**Geometric interpretation of the second Euler substitution in the case $a>0$ and $c>0$. The point P is parameterized by the slope t of the line ${P}_{0}P$.

**Figure 3.**Geometric interpretation of the third Euler substitution in the case $a<0$ and $\Delta >0$. The point P is parameterized by the slope t of the line ${P}_{0}P$.

**Figure 4.**Geometric interpretation of the third Euler substitution in the case $a>0$. The point P is parameterized by the slope t of the line ${P}_{0}P$.

**Figure 5.**Geometric interpretation of the first Euler substitution. The points P and ${P}_{1}$ are parameterized by intersections t and ${t}_{1}$, respectively, of the y-axis with the line parallel to one of the asymptotes of the hyperbola ${y}^{2}=a{x}^{2}+bx+c$.

**Figure 6.**Characteristic points on the graph of an ellipse: intersections with the coordinate axes (provided that they exist) and extremes (minimum ${M}_{1}$ and maximum ${M}_{2}$).

**Figure 7.**Characteristic points on the graphs of hyperbolas (two hyperbolas with the same $\left|a\right|$ are presented): intersections with the coordinate axes (${V}_{1}$, ${V}_{2}$, ${R}_{1}$, ${R}_{2}$) and extremes (${M}_{1}$, ${M}_{2}$).

**Figure 8.**Geometric interpretation of the fourth Euler substitution in the case $a>0$. The point P is parameterized by the slope t of the line ${P}_{0}P$, where ${P}_{0}={M}_{1}$.

**Figure 9.**Geometric interpretation of the fourth Euler substitution in the case $a<0$. The point P is parameterized by the slope t of the line ${P}_{0}P$, where ${P}_{0}={M}_{1}$.

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**MDPI and ACS Style**

Cieśliński, J.L.; Jurgielewicz, M.
On Geometric Interpretations of Euler’s Substitutions. *Symmetry* **2023**, *15*, 1932.
https://doi.org/10.3390/sym15101932

**AMA Style**

Cieśliński JL, Jurgielewicz M.
On Geometric Interpretations of Euler’s Substitutions. *Symmetry*. 2023; 15(10):1932.
https://doi.org/10.3390/sym15101932

**Chicago/Turabian Style**

Cieśliński, Jan L., and Maciej Jurgielewicz.
2023. "On Geometric Interpretations of Euler’s Substitutions" *Symmetry* 15, no. 10: 1932.
https://doi.org/10.3390/sym15101932