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<p>We present the list of unavoidable local phenomena (transitions) occurring on the configuration of the flecnodal and parabolic curves of evolving smooth surfaces in <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10883_2024_9697_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {R}^3\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> (or <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10883_2024_9697_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {R}\mathrm P^3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">R</mi> <msup> <mi mathvariant="normal">P</mi> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>). We also present the list of transitions occurring on the curve of inflections of the solutions of evolving implicit differential equations (IDE). Our results are based on the properties of the contours of surfaces (in a contact 3-space) for projections all whose fibres are Legendrian.</p>

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Evolving Surfaces and Evolving Implicit Differential Equations Via Contact Geometry and Singularities

  • 期刊论文

  • 发表日期：

    2025年7月8日

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  • Ricardo Uribe-Vargas

摘要

We present the list of unavoidable local phenomena (transitions) occurring on the configuration of the flecnodal and parabolic curves of evolving smooth surfaces in \(\mathbb {R}^3\) R 3 (or \(\mathbb {R}\mathrm P^3\) R P 3 ). We also present the list of transitions occurring on the curve of inflections of the solutions of evolving implicit differential equations (IDE). Our results are based on the properties of the contours of surfaces (in a contact 3-space) for projections all whose fibres are Legendrian.