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<p>In this paper, we obtain all the locally convex plane curves with proportional curvatures (Euclidean and affine). The analysis is performed from a special second order differential equation that, in appropriated coordinates, and using the support function the problem is reduced to the properties of a periodic linear nonhomogeneous second differential equation of the type <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(y''+y=a(t), \; a(t+2\pi )=a(t)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>y</mi> <mrow> <mo>′</mo> <mo>′</mo> </mrow> </msup> <mo>+</mo> <mi>y</mi> <mo>=</mo> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mspace width="0.277778em" /> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Plane Curves with Proportional Affine and Euclidean Curvatures

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年3月16日

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  • Ronaldo A. Garcia,
  • Armengol Gasull

摘要

In this paper, we obtain all the locally convex plane curves with proportional curvatures (Euclidean and affine). The analysis is performed from a special second order differential equation that, in appropriated coordinates, and using the support function the problem is reduced to the properties of a periodic linear nonhomogeneous second differential equation of the type \(y''+y=a(t), \; a(t+2\pi )=a(t)\) y ′ ′ + y = a ( t ) , a ( t + 2 π ) = a ( t ) .