On Self-Intersections of Cubic Bézier Curves
Cubic Bézier curves are widely used in computer graphics and geometric modeling, favored for their intuitive design and ease of implementation. However, self-intersections within these curves can pose significant challenges in both geometric modeling and analysis. This paper presents a comprehensive...
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-03-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/12/6/882 |
| _version_ | 1797240114269650944 |
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| author | Ying-Ying Yu Xin Li Ye Ji |
| author_facet | Ying-Ying Yu Xin Li Ye Ji |
| author_sort | Ying-Ying Yu |
| collection | DOAJ |
| description | Cubic Bézier curves are widely used in computer graphics and geometric modeling, favored for their intuitive design and ease of implementation. However, self-intersections within these curves can pose significant challenges in both geometric modeling and analysis. This paper presents a comprehensive approach to detecting and computing self-intersections of cubic Bézier curves. We introduce an efficient algorithm that leverages both the geometric properties of Bézier curves and numerical methods to accurately identify intersection points. The self-intersection problem of cubic Bézier curves is firstly transformed into a quadratic problem by eliminating trivial solutions. Subsequently, this quadratic system is converted into a linear system that may be easily analyzed and solved. Finally, the parameter values corresponding to the self-intersection points are computed through the solution of the linear system. The proposed method is designed to be robust and computationally efficient, making it suitable for real-time applications. |
| first_indexed | 2024-04-24T18:02:17Z |
| format | Article |
| id | doaj.art-230becf1e84e4eda8c8a8631e18da8fc |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-04-24T18:02:17Z |
| publishDate | 2024-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-230becf1e84e4eda8c8a8631e18da8fc2024-03-27T13:53:13ZengMDPI AGMathematics2227-73902024-03-0112688210.3390/math12060882On Self-Intersections of Cubic Bézier CurvesYing-Ying Yu0Xin Li1Ye Ji2School of Mathematics, Liaoning Normal University, Dalian 116029, ChinaSchool of Mathematical Sciences, Dalian University of Technology, Dalian 116024, ChinaDelft Institute of Applied Mathematics, Delft University of Technology, 2628 CD Delft, The NetherlandsCubic Bézier curves are widely used in computer graphics and geometric modeling, favored for their intuitive design and ease of implementation. However, self-intersections within these curves can pose significant challenges in both geometric modeling and analysis. This paper presents a comprehensive approach to detecting and computing self-intersections of cubic Bézier curves. We introduce an efficient algorithm that leverages both the geometric properties of Bézier curves and numerical methods to accurately identify intersection points. The self-intersection problem of cubic Bézier curves is firstly transformed into a quadratic problem by eliminating trivial solutions. Subsequently, this quadratic system is converted into a linear system that may be easily analyzed and solved. Finally, the parameter values corresponding to the self-intersection points are computed through the solution of the linear system. The proposed method is designed to be robust and computationally efficient, making it suitable for real-time applications.https://www.mdpi.com/2227-7390/12/6/882geometric modelingBézier curvesself-intersections |
| spellingShingle | Ying-Ying Yu Xin Li Ye Ji On Self-Intersections of Cubic Bézier Curves Mathematics geometric modeling Bézier curves self-intersections |
| title | On Self-Intersections of Cubic Bézier Curves |
| title_full | On Self-Intersections of Cubic Bézier Curves |
| title_fullStr | On Self-Intersections of Cubic Bézier Curves |
| title_full_unstemmed | On Self-Intersections of Cubic Bézier Curves |
| title_short | On Self-Intersections of Cubic Bézier Curves |
| title_sort | on self intersections of cubic bezier curves |
| topic | geometric modeling Bézier curves self-intersections |
| url | https://www.mdpi.com/2227-7390/12/6/882 |
| work_keys_str_mv | AT yingyingyu onselfintersectionsofcubicbeziercurves AT xinli onselfintersectionsofcubicbeziercurves AT yeji onselfintersectionsofcubicbeziercurves |