Degenerations of Pascal lines
Abstract Let $${\mathcal {K}}$$ denote a nonsingular conic in the projective plane. Pascal’s theorem says that, given six distinct points A, B, C, D, E, F on $${\mathcal {K}}$$, the three intersection points $$AE \cap BF, AD \cap CF, BD \cap CE$$ are collinear. The line containing them is called the...
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| Autor*in: |
Chipalkatti, Jaydeep [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2022 |
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© The Managing Editors 2022 |
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| Übergeordnetes Werk: |
Enthalten in: Beiträge zur Algebra und Geometrie - Springer Berlin Heidelberg, 1971, 64(2022), 3 vom: 22. Juli, Seite 761-781 |
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| Übergeordnetes Werk: |
volume:64 ; year:2022 ; number:3 ; day:22 ; month:07 ; pages:761-781 |
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10.1007/s13366-022-00655-x |
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| 520 | |a Abstract Let $${\mathcal {K}}$$ denote a nonsingular conic in the projective plane. Pascal’s theorem says that, given six distinct points A, B, C, D, E, F on $${\mathcal {K}}$$, the three intersection points $$AE \cap BF, AD \cap CF, BD \cap CE$$ are collinear. The line containing them is called the Pascal line of the sextuple. However, this construction may fail when some of the six points come together. In this paper, we find the indeterminacy locus where the Pascal line is not well-defined and then use blow-ups along polydiagonals to define it. We analyse the geometry of Pascals in these degenerate cases. Finally we offer some remarks about the indeterminacy of other geometric elements in Pascal’s hexagrammum mysticum. | ||
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10.1007/s13366-022-00655-x doi (DE-627)OLC2144476438 (DE-He213)s13366-022-00655-x-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn 31.00 bkl Chipalkatti, Jaydeep verfasserin aut Degenerations of Pascal lines 2022 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Managing Editors 2022 Abstract Let $${\mathcal {K}}$$ denote a nonsingular conic in the projective plane. Pascal’s theorem says that, given six distinct points A, B, C, D, E, F on $${\mathcal {K}}$$, the three intersection points $$AE \cap BF, AD \cap CF, BD \cap CE$$ are collinear. The line containing them is called the Pascal line of the sextuple. However, this construction may fail when some of the six points come together. In this paper, we find the indeterminacy locus where the Pascal line is not well-defined and then use blow-ups along polydiagonals to define it. We analyse the geometry of Pascals in these degenerate cases. Finally we offer some remarks about the indeterminacy of other geometric elements in Pascal’s hexagrammum mysticum. Pascal’s theorem Pascal lines Hexagrammum Mysticum Da Silva, Sergio aut Enthalten in Beiträge zur Algebra und Geometrie Springer Berlin Heidelberg, 1971 64(2022), 3 vom: 22. Juli, Seite 761-781 (DE-627)129565830 (DE-600)223551-1 (DE-576)015035735 0138-4821 nnns volume:64 year:2022 number:3 day:22 month:07 pages:761-781 https://doi.org/10.1007/s13366-022-00655-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_267 GBV_ILN_2018 GBV_ILN_4310 GBV_ILN_4318 31.00 VZ AR 64 2022 3 22 07 761-781 |
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10.1007/s13366-022-00655-x doi (DE-627)OLC2144476438 (DE-He213)s13366-022-00655-x-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn 31.00 bkl Chipalkatti, Jaydeep verfasserin aut Degenerations of Pascal lines 2022 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Managing Editors 2022 Abstract Let $${\mathcal {K}}$$ denote a nonsingular conic in the projective plane. Pascal’s theorem says that, given six distinct points A, B, C, D, E, F on $${\mathcal {K}}$$, the three intersection points $$AE \cap BF, AD \cap CF, BD \cap CE$$ are collinear. The line containing them is called the Pascal line of the sextuple. However, this construction may fail when some of the six points come together. In this paper, we find the indeterminacy locus where the Pascal line is not well-defined and then use blow-ups along polydiagonals to define it. We analyse the geometry of Pascals in these degenerate cases. Finally we offer some remarks about the indeterminacy of other geometric elements in Pascal’s hexagrammum mysticum. Pascal’s theorem Pascal lines Hexagrammum Mysticum Da Silva, Sergio aut Enthalten in Beiträge zur Algebra und Geometrie Springer Berlin Heidelberg, 1971 64(2022), 3 vom: 22. Juli, Seite 761-781 (DE-627)129565830 (DE-600)223551-1 (DE-576)015035735 0138-4821 nnns volume:64 year:2022 number:3 day:22 month:07 pages:761-781 https://doi.org/10.1007/s13366-022-00655-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_267 GBV_ILN_2018 GBV_ILN_4310 GBV_ILN_4318 31.00 VZ AR 64 2022 3 22 07 761-781 |
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10.1007/s13366-022-00655-x doi (DE-627)OLC2144476438 (DE-He213)s13366-022-00655-x-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn 31.00 bkl Chipalkatti, Jaydeep verfasserin aut Degenerations of Pascal lines 2022 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Managing Editors 2022 Abstract Let $${\mathcal {K}}$$ denote a nonsingular conic in the projective plane. Pascal’s theorem says that, given six distinct points A, B, C, D, E, F on $${\mathcal {K}}$$, the three intersection points $$AE \cap BF, AD \cap CF, BD \cap CE$$ are collinear. The line containing them is called the Pascal line of the sextuple. However, this construction may fail when some of the six points come together. In this paper, we find the indeterminacy locus where the Pascal line is not well-defined and then use blow-ups along polydiagonals to define it. We analyse the geometry of Pascals in these degenerate cases. Finally we offer some remarks about the indeterminacy of other geometric elements in Pascal’s hexagrammum mysticum. Pascal’s theorem Pascal lines Hexagrammum Mysticum Da Silva, Sergio aut Enthalten in Beiträge zur Algebra und Geometrie Springer Berlin Heidelberg, 1971 64(2022), 3 vom: 22. Juli, Seite 761-781 (DE-627)129565830 (DE-600)223551-1 (DE-576)015035735 0138-4821 nnns volume:64 year:2022 number:3 day:22 month:07 pages:761-781 https://doi.org/10.1007/s13366-022-00655-x lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_267 GBV_ILN_2018 GBV_ILN_4310 GBV_ILN_4318 31.00 VZ AR 64 2022 3 22 07 761-781 |
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Abstract Let $${\mathcal {K}}$$ denote a nonsingular conic in the projective plane. Pascal’s theorem says that, given six distinct points A, B, C, D, E, F on $${\mathcal {K}}$$, the three intersection points $$AE \cap BF, AD \cap CF, BD \cap CE$$ are collinear. The line containing them is called the Pascal line of the sextuple. However, this construction may fail when some of the six points come together. In this paper, we find the indeterminacy locus where the Pascal line is not well-defined and then use blow-ups along polydiagonals to define it. We analyse the geometry of Pascals in these degenerate cases. Finally we offer some remarks about the indeterminacy of other geometric elements in Pascal’s hexagrammum mysticum. © The Managing Editors 2022 |
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Abstract Let $${\mathcal {K}}$$ denote a nonsingular conic in the projective plane. Pascal’s theorem says that, given six distinct points A, B, C, D, E, F on $${\mathcal {K}}$$, the three intersection points $$AE \cap BF, AD \cap CF, BD \cap CE$$ are collinear. The line containing them is called the Pascal line of the sextuple. However, this construction may fail when some of the six points come together. In this paper, we find the indeterminacy locus where the Pascal line is not well-defined and then use blow-ups along polydiagonals to define it. We analyse the geometry of Pascals in these degenerate cases. Finally we offer some remarks about the indeterminacy of other geometric elements in Pascal’s hexagrammum mysticum. © The Managing Editors 2022 |
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Abstract Let $${\mathcal {K}}$$ denote a nonsingular conic in the projective plane. Pascal’s theorem says that, given six distinct points A, B, C, D, E, F on $${\mathcal {K}}$$, the three intersection points $$AE \cap BF, AD \cap CF, BD \cap CE$$ are collinear. The line containing them is called the Pascal line of the sextuple. However, this construction may fail when some of the six points come together. In this paper, we find the indeterminacy locus where the Pascal line is not well-defined and then use blow-ups along polydiagonals to define it. We analyse the geometry of Pascals in these degenerate cases. Finally we offer some remarks about the indeterminacy of other geometric elements in Pascal’s hexagrammum mysticum. © The Managing Editors 2022 |
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Degenerations of Pascal lines |
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https://doi.org/10.1007/s13366-022-00655-x |
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Da Silva, Sergio |
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10.1007/s13366-022-00655-x |
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2024-07-03T22:30:34.784Z |
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