Plane Curves with Proportional Affine and Euclidean Curvatures

Abstract 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 $$y''+y=a(t), \; a(t+2\pi )=a(t)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>y</mml:mi> <mml:mrow> <mml:mo>′</mml:mo> <mml:mo>′</mml:mo> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>y</mml:mi> <mml:mo>=</mml:mo> <mml:mi>a</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mspace/> <mml:mi>a</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>a</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> .