On perturbation of aesthetic curves

Alina Ramona Baias; Ioana Crǎciun; Alina Ramona Baias; Ioana Crǎciun

doi:10.3934/math.2023883

AIMS Mathematics

2023, Volume 8, Issue 7: 17272-17283. doi: 10.3934/math.2023883

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Research article

On perturbation of aesthetic curves

Alina Ramona Baias ^{1
,
,},
Ioana Crǎciun ²

1.
Department of Mathematics, Technical University of Cluj-Napoca, 28 Memorandumului Street, 400114, Cluj-Napoca, Romania
2.
Department of Automotive Engineering and Transports, Technical University of Cluj-Napoca, 28 Memorandumului Street, 400114, Cluj-Napoca, Romania

Received: 17 January 2023 Revised: 10 April 2023 Accepted: 17 April 2023 Published: 18 May 2023
MSC : 53A04, 34D20

The notion of aesthetic curve plays an essential role in the generation of aesthetic forms. In order to quantify the aesthetics in art and science some aesthetic standards were introduced but, the beauty is perceived by people in various ways. Bearing this idea in mind, we introduce in this paper the notion of an approximate aesthetic curve by perturbing the equation that defines it. Moreover, we study the relation between the solutions of the perturbed equation and the solutions of the exact equation. For the equation of neutral curves, some stability results in Ulam sense are obtained. We prove that the equation of aesthetic curves by perturbations to an equation for convergent curves is Ulam stable, while by perturbation to an equation for divergent curves, it is not Ulam stable.
- aesthetic curve,
- approximate aesthetic curve,
- Ulam stability
Citation: Alina Ramona Baias, Ioana Crǎciun. On perturbation of aesthetic curves[J]. AIMS Mathematics, 2023, 8(7): 17272-17283. doi: 10.3934/math.2023883

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Abstract

The notion of aesthetic curve plays an essential role in the generation of aesthetic forms. In order to quantify the aesthetics in art and science some aesthetic standards were introduced but, the beauty is perceived by people in various ways. Bearing this idea in mind, we introduce in this paper the notion of an approximate aesthetic curve by perturbing the equation that defines it. Moreover, we study the relation between the solutions of the perturbed equation and the solutions of the exact equation. For the equation of neutral curves, some stability results in Ulam sense are obtained. We prove that the equation of aesthetic curves by perturbations to an equation for convergent curves is Ulam stable, while by perturbation to an equation for divergent curves, it is not Ulam stable.

References

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