Planar Typical Bézier Curves Made Simple
Recently, He et al. derived several remarkable properties of the so-called typical Bézier curves, a subset of constrained Bézier curves introduced by Mineur et al. In particular, He et al. proved that such curves display at most one curvature extremum, give an explicit formula of the parameter at th...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-11-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/23/3017 |
| _version_ | 1827674897322606592 |
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| author | Javier Sánchez-Reyes |
| author_facet | Javier Sánchez-Reyes |
| author_sort | Javier Sánchez-Reyes |
| collection | DOAJ |
| description | Recently, He et al. derived several remarkable properties of the so-called typical Bézier curves, a subset of constrained Bézier curves introduced by Mineur et al. In particular, He et al. proved that such curves display at most one curvature extremum, give an explicit formula of the parameter at the extremum, and show that subdividing a curve at this point furnishes two new typical curves. We recall that typical curves amount to segments of a special family of sinusoidal spirals, curves already studied by Maclaurin in the early 18th century and whose properties are well-known. These sinusoidal spirals display only one curvature extremum (i.e., vertex), whose parameter is simply that corresponding to the axis of symmetry. Subdividing a segment at an arbitrary point, not necessarily the vertex, always yields two segments of the same spiral, hence two typical curves. |
| first_indexed | 2024-03-10T04:48:45Z |
| format | Article |
| id | doaj.art-cb2c6a46cd1d404d97da404c587ee0b5 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T04:48:45Z |
| publishDate | 2021-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-cb2c6a46cd1d404d97da404c587ee0b52023-11-23T02:44:47ZengMDPI AGMathematics2227-73902021-11-01923301710.3390/math9233017Planar Typical Bézier Curves Made SimpleJavier Sánchez-Reyes0IMACI, ETS Ingeniería Industrial Ciudad Real, Universidad de Castilla-La Mancha, 13071 Ciudad Real, SpainRecently, He et al. derived several remarkable properties of the so-called typical Bézier curves, a subset of constrained Bézier curves introduced by Mineur et al. In particular, He et al. proved that such curves display at most one curvature extremum, give an explicit formula of the parameter at the extremum, and show that subdividing a curve at this point furnishes two new typical curves. We recall that typical curves amount to segments of a special family of sinusoidal spirals, curves already studied by Maclaurin in the early 18th century and whose properties are well-known. These sinusoidal spirals display only one curvature extremum (i.e., vertex), whose parameter is simply that corresponding to the axis of symmetry. Subdividing a segment at an arbitrary point, not necessarily the vertex, always yields two segments of the same spiral, hence two typical curves.https://www.mdpi.com/2227-7390/9/23/3017curvature extremumsubdivisionsinusoidal spiralTschirnhausen cubictypical Bézier curvevertex |
| spellingShingle | Javier Sánchez-Reyes Planar Typical Bézier Curves Made Simple Mathematics curvature extremum subdivision sinusoidal spiral Tschirnhausen cubic typical Bézier curve vertex |
| title | Planar Typical Bézier Curves Made Simple |
| title_full | Planar Typical Bézier Curves Made Simple |
| title_fullStr | Planar Typical Bézier Curves Made Simple |
| title_full_unstemmed | Planar Typical Bézier Curves Made Simple |
| title_short | Planar Typical Bézier Curves Made Simple |
| title_sort | planar typical bezier curves made simple |
| topic | curvature extremum subdivision sinusoidal spiral Tschirnhausen cubic typical Bézier curve vertex |
| url | https://www.mdpi.com/2227-7390/9/23/3017 |
| work_keys_str_mv | AT javiersanchezreyes planartypicalbeziercurvesmadesimple |