Pseudo-evolute curves and caustic surfaces

Hande Nur DALKILIÇ; Yusuf YAYLI

doi:10.1631/FITEE.2400930

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Pseudo-evolute curves and caustic surfaces

Regular Papers | Updated：2025-11-05

- Pseudo-evolute curves and caustic surfaces
- 伪渐屈线与焦散面
- Frontiers of Information Technology & Electronic Engineering Vol. 26, Issue 9, Pages: 1733-1741(2025)
- Affiliations：
  
  1.Department of Mathematics, Graduate School of Natural and Applied Sciences, Ankara University, Ankara 06000, Türkiye
  2.Department of Mathematics, Faculty of Science, Ankara University, Ankara 06000, Türkiye
- Author bio：
  
  ‡ Corresponding author
  yayli@science.ankara.edu.tr
- Funds：
- DOI：10.1631/FITEE.2400930
  CLC： O182;O435
- Received：20 October 2024，
  
  Revised：2025-01-24，
  
  Published：2025-09
- Accepted：
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Hande Nur DALKILIÇ, Yusuf YAYLI. Pseudo-evolute curves and caustic surfaces[J]. Frontiers of Information Technology & Electronic Engineering, 2025, 26(9): 1733-1741.
DOI：

Hande Nur DALKILIÇ, Yusuf YAYLI. Pseudo-evolute curves and caustic surfaces[J]. Frontiers of Information Technology & Electronic Engineering, 2025, 26(9): 1733-1741. DOI： 10.1631/FITEE.2400930.

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。

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.

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references

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