| Title: | On $L^2$ approximation by spatial Pythagorean-hodograph curves |
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| Authors: | ID Farouki, Rida T. (Author)
ID Knez, Marjetka (Author)
ID Vitrih, Vito (Author)
ID Žagar, Emil (Author) |
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| Files: |  PDF - Presentation file, download (5,65 MB)
MD5: 85C798ACBB919FC508EBFF78668BC879
 URL - Source URL, visit https://www.sciencedirect.com/science/article/pii/S0377042726006291
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | Two methods for the $L^2$ approximation of smooth space curves by spatial Pythagorean-hodograph (PH) curves are considered. The first method employs direct approximation in ${\mathbb R}^3$, which results in a non-linear system of equations that may be solved by a numerical iteration or optimization scheme. The second method performs the optimization in the quaternion preimage space of PH curves, which results in a linear system of equations. The preimage of a curve in ${\mathbb R}^3$ is a surface in the quaternion space, and an appropriate locus on this surface must be identified for the given space curve. This is accomplished by observing that PH curves equipped with a rational rotation-minimizing frame (RMF) have preimages that are geodesic loci on the surface, and computing a discretized approximation to the RMF on the given curve. The methods are also extended to the approximation of spatial B-spline curves. Detailed algorithm descriptions are provided for both methods, and several computed examples illustrate their performance. |
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| Keywords: | $L^2$ approximation, quaternions, Pythagorean-hodograph curves, B-splines, preimage |
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| Publication version: | Version of Record |
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| Publication date: | 01.02.2027 |
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| Year of publishing: | 2027 |
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| Number of pages: | 16 str. |
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| Numbering: | Vol. 491, article no. 117987 |
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| PID: | 20.500.12556/DiRROS-31390  |
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| UDC: | 519.6 |
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| ISSN on article: | 0377-0427 |
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| DOI: | 10.1016/j.cam.2026.117987 |
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| COBISS.SI-ID: | 286529283 |
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| Note: |
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| Publication date in DiRROS: | 31.07.2026 |
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| Views: | 90 |
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| Downloads: | 82 |
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