Journal of Zhejiang University

ENGINEERING Information Technology & Electronic Engineering

Accepted manuscript available online (unedited version)

Pseudo-evolute curves and caustic surfaces

Author(s): Hande Nur DALKILIÇ, Yusuf YAYLI
Affiliation(s): Department of Mathematics, Graduate School of Natural and Applied Sciences, Ankara University, Ankara 06000, Türkiye; more
Corresponding email(s): hncetin@ankara.edu.tr, yayli@science.ankara.edu.tr
Key Words: Pseudo-evolute curves; Caustic surfaces; Developable surfaces; Rectifying caustic developable surfaces; Osculating caustic developable surfaces; Normal caustic developable surfaces

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Hande Nur DALKILIÇ, Yusuf YAYLI. Pseudo-evolute curves and caustic surfaces[J]. Frontiers of Information Technology & Electronic Engineering,in press.https://doi.org/10.1631/FITEE.2400930

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Abstract
Chinese Summary
Academic Network
Reviewer Comment

Abstract: In this study, osculating caustic developable surfaces and rectifying caustic developable surfaces were obtained by considering space curves and curves on surfaces as base curves and changing the direction of the light source reflected by the mirror surface. It was proved that pseudo-evolute curves represent the striction curves (regression edges) of these surfaces. For developable surfaces based on curves on surfaces, it was observed that osculating caustic developable surfaces are equivalent to rectifying caustic developable surfaces if the curve is geodesic. Additionally, when the base curve was taken over any surface, the caustic surfaces were characterized as flat or normal approximation surfaces, depending on the direction of the light source.

伪渐屈线与焦散面

Hande Nur DALKILIÇ¹,Yusuf YAYLI²
¹安卡拉大学自然与应用科学研究生院数学系，土耳其安卡拉省，06000
²安卡拉大学理学院数学系，土耳其安卡拉省，06000
摘要：本文通过将空间曲线和曲面上的曲线视为基线，并改变镜面反射光源方向，获得密切焦散可展曲面和从切焦散可展曲面。证明伪渐屈线代表这些曲面的腰曲线（回归边）。对基于曲面上曲线的可展曲面，观察到如果曲线是测地线，则密切焦散可展曲面等价于从切焦散可展曲面。此外，当基曲线取自任意曲面时，根据光源方向的不同，焦散曲面可被表征为平坦型或法向逼近型曲面。

关键词组：伪渐屈线；焦散面；可展曲面；从切焦散可展曲面；密切焦散可展曲面；法向焦散可展曲面

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Reference

[1]Fuchs D, 2013. Evolutes and involutes of spatial curves. Am Math Mon, 120(3):217-231.

[2]Fuchs D, Izmestiev I, Raffaelli M, et al., 2024. Differential geometry of space curves: forgotten chapters. Math Intell, 46(1):9-21.

[3]Hananoi S, Izumiya S, 2017. Normal developable surfaces of surfaces along curves. Proc R Soc Edinb Sect A Math, 147(1):177-203.

[4]Hoffmann M, Juhász I, Troll E, 2022. Caustics of developable surfaces. Front Inform Technol Electron Eng, 23(3):479-487.

[5]Izumiya S, Otani S, 2015. Flat approximations of surfaces along curves. Demonstr Math, 48(2):217-241.

[6]Izumiya S, Takeuchi N, 2004. New special curves and developable surfaces. Turk J Math, 28(2):153-164.

[7]Köse B, Yayli Y, 2023. Approximations of parallel surfaces along curves. Int Electr J Geom, 16(2):715-726.

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Full Text: <2894>

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CLC number: O182;O435

On-line Access: 2025-10-13

Received: 2024-10-20

Revision Accepted: 2025-01-24

Crosschecked: 2025-10-13

Cited: 0

Clicked: 1937

Citations: Bibtex RefMan EndNote GB/T7714

ORCID:

Hande Nur DALKILIÇ

https://orcid.org/0009-0003-9875-781X

Yusuf YAYLI

https://orcid.org/0000-0003-4398-3855

伪渐屈线与焦散面

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