# On Approximate Aesthetic Curves

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Aesthetic Curves

**Definition**

**1.**

**Definition**

**2.**

- Convergent: if the LCG gradient is positive;
- Divergent: if the LCG gradient is negative;
- Neutral: if the LCG gradient is zero.

**Theorem**

**1.**

- $\left(C\right)$is convergent if and only if$\left|k\right|$is strictly decreasing;
- $\left(C\right)$is divergent if and only if$\left|k\right|$is strictly increasing;
- $\left(C\right)$is neutral if and only if$k\left(s\right)={c}_{2}{e}^{{c}_{1}s},\text{}0\le s\le S$, and c
_{1}, c_{2}are real constants.

**Proof.**

## 3. Approximate Neutral Curves

**Definition**

**3.**

**Theorem**

**2.**

**Proof.**

**Remark**

**1.**

**Theorem**

**3.**

**Proof.**

**Application**

**1.**

**Application**

**2.**

**Application**

**3.**

## 4. Approximate Aesthetic Curves

**Definition**

**4.**

**Theorem**

**4.**

_{1}, C

_{2}are real constants. Moreover,${C}_{1}=\frac{k\left(0\right)}{{k}^{\prime}\left(0\right)},\text{}{C}_{2}=k\left(0\right).$

**Proof.**

^{2}− 4ac.

**Theorem**

**5.**

- (i)
- If ${\delta}_{1}>0,\text{}{\delta}_{2}0$ we have$${\left|\frac{\epsilon +\sqrt{{\delta}_{2}}}{\epsilon -\sqrt{{\delta}_{2}}}\cdot \frac{as+b-\epsilon +\sqrt{{\delta}_{2}}}{as+b-\epsilon -\sqrt{{\delta}_{2}}}\right|}^{\frac{1}{\sqrt{{\delta}_{2}}}}\le \left|\frac{k\left(s\right)}{k\left(0\right)}\right|\le {\left|\frac{\epsilon -\sqrt{{\delta}_{1}}}{\epsilon +\sqrt{{\delta}_{1}}}\cdot \frac{as+b+\epsilon +\sqrt{{\delta}_{1}}}{as+b+\epsilon -\sqrt{{\delta}_{1}}}\right|}^{\frac{1}{\sqrt{{\delta}_{1}}}},\text{}s\in \left[0,S\right].$$
- (ii)
- If ${\delta}_{1}={\delta}_{2}=0,$ then$${e}^{\frac{-2as}{\left(b-\epsilon \right)\left(as+b-\epsilon \right)}}\le \left|\frac{k\left(s\right)}{k\left(0\right)}\right|\le {e}^{\frac{-2as}{\left(b+\epsilon \right)\left(as+b+\epsilon \right)}},\text{}s\in \left[0,S\right].$$
- (iii)
- If ${\delta}_{1}<0,\text{}{\delta}_{2}0$$${e}^{\frac{2}{\sqrt{-{\delta}_{2}}}\left(\mathrm{arctan}\frac{b-\epsilon}{\sqrt{-{\delta}_{2}}}-\mathrm{arctan}\frac{as+b-\epsilon}{\sqrt{-{\delta}_{2}}}\right)}\le \left|\frac{k\left(s\right)}{k\left(0\right)}\right|\le {e}^{\frac{2}{\sqrt{-{\delta}_{1}}}\left(\mathrm{arctan}\frac{b+\epsilon}{\sqrt{-{\delta}_{1}}}-\mathrm{arctan}\frac{as+b+\epsilon}{\sqrt{-{\delta}_{1}}}\right)},\text{}s\in \left[0,S\right].$$

**Proof.**

**Remark**

**2.**

**Application**

**4.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Crăciun, I.; Popa, D.; Serdean, F.; Tudose, L.
On Approximate Aesthetic Curves. *Symmetry* **2020**, *12*, 1394.
https://doi.org/10.3390/sym12091394

**AMA Style**

Crăciun I, Popa D, Serdean F, Tudose L.
On Approximate Aesthetic Curves. *Symmetry*. 2020; 12(9):1394.
https://doi.org/10.3390/sym12091394

**Chicago/Turabian Style**

Crăciun, Ioana, Dorian Popa, Florina Serdean, and Lucian Tudose.
2020. "On Approximate Aesthetic Curves" *Symmetry* 12, no. 9: 1394.
https://doi.org/10.3390/sym12091394