# Developable Ruled Surfaces Generated by the Curvature Axis of a Curve

^{1}

^{2}

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## Abstract

**:**

“If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.”

—John von Neumann

## 1. Introduction

## 2. Preliminary Concepts and Definitions

**Theorem**

**1.**

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

- $\mathbf{d}$ represents a line at the point $\alpha \left({s}_{0}\right)+\frac{1}{\kappa \left({s}_{0}\right)}\mathbf{n}\left({s}_{0}\right)$.
- $\alpha \left({s}_{0}\right)$ is a position vector associated with a point on a curve at a particular parameter value ${s}_{0}$.
- $\frac{1}{\kappa \left({s}_{0}\right)}\mathbf{n}\left({s}_{0}\right)$ is the normal component, where $\kappa \left({s}_{0}\right)$ represents the curvature of the curve at the point ${s}_{0}$, and $\mathbf{n}\left({s}_{0}\right)$ is the unit normal vector at that point. This component ensures that $\mathbf{d}$ is displaced from the curve in the direction of its normal.

**Definition**

**5.**

**Remark**

**1.**

## 3. Construction of the Developable Ruled Surface

**Theorem**

**2.**

**Proof.**

## 4. Finding the Curve $\mathbf{\alpha}$

#### 4.1. Generalized Cylinder-Developable Surfaces

#### 4.2. Conical-Developable Surfaces

#### 4.3. Tangent-Developable Surface

#### 4.4. Comparisons

**Corollary**

**1.**

**Example**

**1.**

## 5. Singularities

- A tangent-developable surface is considered a cuspidal edge surface along $\alpha \left(t\right)$ if $\tau \left(t\right)\ne 0$.
- The tangent-developable surface of a space curve possesses a cuspidal cross-cap singularity at $\alpha \left({t}_{0}\right)$ if $\tau \left({t}_{0}\right)=0$ and ${\tau}^{\prime}\left({t}_{0}\right)\ne 0$.

**Theorem**

**3.**

**Example**

**2**

**Example**

**3**

**Example**

**4**

## 6. Conclusions and Recommendations for Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CAGD | computer-aided geometric design |

CAD | computer-aided design |

ODE | ordinary differential equation |

AI | artificial intelligence |

## Appendix A. Ruled Surfaces in Art

**Figure A1.**Barbara Hepworth’s sculptures related to ruled surfaces (copyright for personal or educational use only from https://barbarahepworth.org.uk/, accessed on 22 October 2023).

## References

- Study, E. Geometrie der Dynamen. In Druck und Verlag von BG Teubner; Mathematiker Deutschland Publisher: Leibzig, German, 1903. [Google Scholar]
- Pottmann, H.; Wallner, J. Computational Line Geometry; Springer: Berlin/Heidelberg, Germany, 2001. [Google Scholar]
- Tas, F.; Gürsoy, O. On the Line Congruences. Int. Electron. J. Geom.
**2018**, 11, 47–53. [Google Scholar] [CrossRef] - Taş, F.; İlarslan, K. A new approach to design the ruled surface. Int. J. Geom. Methods Mod. Phys.
**2019**, 16, 1950093. [Google Scholar] [CrossRef] - Bo, P.; Fan, H.; Bartoň, M. Efficient 5-axis CNC trochoidal flank milling of 3D cavities using custom-shaped cutting tools. Comput. Aided Des.
**2022**, 151, 103334. [Google Scholar] [CrossRef] - Chu, C.H.; Wang, C.C.; Tsai, C.R. Computer aided geometric design of strip using developable Bézier patches. Comput. Ind.
**2008**, 59, 601–611. [Google Scholar] [CrossRef] - Peternell, M. Developable surface fitting to point clouds. Comput. Aided Geom. Des.
**2004**, 21, 785–803. [Google Scholar] [CrossRef] - Li, C.Y.; Wang, R.H.; Zhu, C.G. An approach for designing a developable surface through a given line of curvature. Comput. Aided Des.
**2013**, 45, 621–627. [Google Scholar] [CrossRef] - Agoston, M.K.; Agoston, M.K. Computer Graphics and Geometric Modeling; Springer: Berlin/Heidelberg, Germany, 2005; Volume 1. [Google Scholar]
- Fernández-Jambrina, L. B-spline control nets for developable surfaces. Comput. Aided Geom. Des.
**2007**, 24, 189–199. [Google Scholar] [CrossRef] - Lang, J.; Röschel, O. Developable (1, n)-Bézier surfaces. Comput. Aided Geom. Des.
**1992**, 9, 291–298. [Google Scholar] [CrossRef] - Ravani, B.; Ku, T.S. Bertrand offsets of ruled and developable surfaces. Comput. Aided Des.
**1991**, 23, 145–152. [Google Scholar] [CrossRef] - Ziatdinov, R.; Nabiyev, R.; Kim, H.; Lim, S.H. The Concept of a Dew Collection Device Based on the Mathematical Model of Sliding Liquid Drops on an Inclined Solid Surface. In IOP Conference Series: Earth and Environmental Science; IOP Publishing: Bristol, UK, 2019; Volume 272, p. 022091. [Google Scholar]
- Zhang, Y.; Zheng, J. An Overview of Developable Surfaces in Geometric Modeling. Recent Patents Eng.
**2022**, 16, 87–103. [Google Scholar] [CrossRef] - Izumiya, S.; Takeuchi, N. New special curves and developable surfaces. Turk. J. Math.
**2004**, 28, 153–164. [Google Scholar] - Lawrence, S. Developable surfaces: Their history and application. Nexus Netw. J.
**2011**, 13, 701–714. [Google Scholar] [CrossRef] - Pottmann, H.; Wallner, J. Approximation algorithms for developable surfaces. Comput. Aided Geom. Des.
**1999**, 16, 539–556. [Google Scholar] [CrossRef] - Tang, C.; Bo, P.; Wallner, J.; Pottmann, H. Interactive design of developable surfaces. Acm Trans. Graph. (TOG)
**2016**, 35, 1–12. [Google Scholar] [CrossRef] - Liu, Y.; Pottmann, H.; Wallner, J.; Yang, Y.L.; Wang, W. Geometric modeling with conical meshes and developable surfaces. In ACM SIGGRAPH 2006 Papers; ACM: New York, NY, USA, 2006; pp. 681–689. [Google Scholar]
- Glaeser, G.; Gruber, F. Developable surfaces in contemporary architecture. J. Math. Arts
**2007**, 1, 59–71. [Google Scholar] [CrossRef] - Ishikawa, G. Singularities of developable surfaces. In London Mathematical Society Lecture Note Series; Cambridge University Press: Cambridge, UK, 1999; pp. 403–418. [Google Scholar]
- Chalfant, J.S.; Maekawa, T. Design for manufacturing using B-spline developable surfaces. J. Ship Res.
**1998**, 42, 207–215. [Google Scholar] [CrossRef] - Ali, A.T.; Aziz, H.S.A.; Sorour, A.H. Ruled surfaces generated by some special curves in Euclidean 3-Space. J. Egypt. Math. Soc.
**2013**, 21, 285–294. [Google Scholar] [CrossRef] - Izumiya, S.; Romero Fuster, M.C.; Ruas, M.A.S.; Tari, F. Differential Geometry from a Singularity Theory Viewpoint; World Scientific Publishing Co., Pte. Ltd.: Hackensack, NJ, USA, 2016. [Google Scholar]
- Lipschutz, M.M. Schaum’s Outline of Theory and Problems of Differential Geometry; McGraw-Hill: New York, NY, USA, 1969. [Google Scholar]
- Struik, D.J. Lectures on Classical Differential Geometry; Addison-Wesley Publishing Company, Inc.: Boston, MA, USA, 1961. [Google Scholar]
- Somasundaram, D. Differential Geometry: A First Course; Alpha Science Int’l Ltd.: Oxford, UK, 2005. [Google Scholar]
- Umehara, M.; Yamada, K. Differential Geometry of CURVES and Surfaces; Dover Publications: Mineola, NY, USA, 2016. [Google Scholar]
- Abbena, E.; Salamon, S.; Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica; CRC Press: Boca Raton, FL, USA, 2017. [Google Scholar]
- Hacisalihoglu, H.H. Diferensiyel Geometri; Inonu University Yayinlari: Malatya, Turkey, 1983. [Google Scholar]
- Goldstein, H.; Poole, C.P.; Safko, J.L. Classical Mechanics, 3rd ed.; Addison Wesley: Boca Raton, FL, USA, 2001; ISBN 0201657023. [Google Scholar]
- Cassiday, G.L.; Fowles, G.R. Analytical Mechanics; Saunders College: Rochester, NY, USA, 1993. [Google Scholar]
- Strogatz, S.H. Nonlinear Dynamics and Chaos with Student Solutions Manual: With Applications to Physics, Biology, Chemistry, and Engineering; CRC Press: Boca Raton, FL, USA, 2018. [Google Scholar]
- Liu, H.; Liu, Y.; Jung, S.D. Ruled invariants and total classification of non-developable ruled surfaces. J. Geom.
**2022**, 113, 21. [Google Scholar] [CrossRef] - Emap. Rhino. Available online: https://docs.mcneel.com/rhino/5/help/en-us/commands/emap.htm (accessed on 14 November 2023).
- Yoshida, N.; Saito, T. Interactive aesthetic curve segments. Vis. Comput.
**2006**, 22, 896–905. [Google Scholar] [CrossRef] - Yoshida, N.; Fukuda, R.; Saito, T. Log-aesthetic space curve segments. In Proceedings of the 2009 SIAM/ACM Joint Conference on Geometric and Physical Modeling, San Francisco, CA, USA, 4–8 October 2009; pp. 35–46. [Google Scholar]
- Miura, K.T.; Suzuki, S.; Gobithaasan, R.U.; Usuki, S. A new log-aesthetic space curve based on similarity geometry. Comput. Aided Des. Appl.
**2019**, 16, 79–88. [Google Scholar] [CrossRef] - Ziatdinov, R. Family of superspirals with completely monotonic curvature given in terms of Gauss hypergeometric function. Comput. Aided Geom. Des.
**2012**, 29, 510–518. [Google Scholar] [CrossRef] - Nishikawa, S.; Suzuki, A.; Maekawa, T.; Takizawa, K. Geometric modeling of umbrella surfaces based on piecewise bilinear surfaces. Proc. Des. Syst. Conf.
**2021**, 31, 3308. [Google Scholar] [CrossRef]

**Figure 1.**Applications of ruled surfaces: (

**a**) AI-enhanced image of a dew collector based on a ruled surface (helicoid) [13]. (

**b**) The grid shell of the Shukhov Tower in Moscow, whose sections are doubly ruled. The image has been adapted from Wikipedia.

**Figure 2.**AI image generated from a Wikipedia photo of hyperbolic cooling towers (the surface can be doubly ruled) at Didcot Power Station, UK.

**Figure 5.**A surface of revolution with a cusp singularity created by rotating a cuspidal NURBS curve around an axis. Zebra stripes on the outer and inner parts of the surface behave in a specific way around the singularity.

**Figure 6.**Environmental map with a circular pattern applied to the surface with a singularity. White circles tend to be elongated and increase in number around a single point, forming a flower-like pattern.

**Figure 7.**Singular developable surfaces: cuspidal-edge (

**left**), swallowtail (

**center**), and cuspidal cross-cap (

**right**).

**Figure 8.**Cuspidal cross-cap swallowtail developable surface generated by the curvature axis of the singular space curve $({t}^{2},{t}^{3},{t}^{4})$.

**Figure 9.**Cuspidal cross-cap swallowtail developable surface visualisation using checker textures, where self-intersection and singularity point are not clearly visible. The example shows that other surface quality methods such as zebra lines or Gaussian curvature visualisation can be used, or an environmental map with a circular pattern can be applied.

**Figure 10.**Singular developable surfaces associated with the 3D asteroid curve (blue). From

**left**to

**right**: different values of $\lambda $.

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**MDPI and ACS Style**

Taş, F.; Ziatdinov, R.
Developable Ruled Surfaces Generated by the Curvature Axis of a Curve. *Axioms* **2023**, *12*, 1090.
https://doi.org/10.3390/axioms12121090

**AMA Style**

Taş F, Ziatdinov R.
Developable Ruled Surfaces Generated by the Curvature Axis of a Curve. *Axioms*. 2023; 12(12):1090.
https://doi.org/10.3390/axioms12121090

**Chicago/Turabian Style**

Taş, Ferhat, and Rushan Ziatdinov.
2023. "Developable Ruled Surfaces Generated by the Curvature Axis of a Curve" *Axioms* 12, no. 12: 1090.
https://doi.org/10.3390/axioms12121090