On double Pythagorean-hodograph curves of degree seven

Knez, Marjetka (author), Praprotnik, Selena (author)

This paper presents a comprehensive and constructive analysis of degree 7 double Pythagorean-hodograph (DPH) curves for the three distinct structural classes. A unified framework is introduced for their construction, particularly useful for interpolation tasks involving prescribed boundary data. The approach explicitly identifies the degrees of freedom available in each class and distinguishes between helical and non-helical curve types. Furthermore, the structure of a specific rational curve in the complex plane that via the normalized Hopf map generates the tangent indicatrix is revealed for all degree 7 DPH curves. This confirms known results for helical curves and extends the interpretation to non-helical cases. The practical applicability of the derived curves is demonstrated through a numerical interpolation example, which also validates the stated number of degrees of freedom.

• Language: English
• Keywords: double Pythagorean-hodograph curves, helical/non-helical curves, quaternions, Hopf map, Frenet frame, interpolation
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics; FGG - Faculty of Civil and Geodetic Engineering
• Publication status: Published
• Publication version: Version of Record
• Publication date: 01.04.2026
• Year: 2026
• Number of pages: 20 str.
• Numbering: Vol. 126, article no. 102532
• PID: 20.500.12556/RUL-181708
• UDC: 519.6
• ISSN on article: 0167-8396
• DOI: 10.1016/j.cagd.2026.102532
• COBISS.SI-ID: 275061507
• Publication date in RUL: 14.04.2026

Record is a part of a journal

• Title: Computer Aided Geometric Design
• Shortened title: Comput. aided geom. des.
• Publisher: Elsevier
• ISSN: 0167-8396
• COBISS.SI-ID: 25266176

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: krivulje s pitagorejskim hodografom, kvaternioni, interpolacija