ON THE OUTLINES OF PLANE CURVES OF THE FORM $(ax)^{\alpha}+(by)^{\alpha}=r^{\alpha}$ WITH $\alpha>0$

Print ISSN: 2319-1023 | Online ISSN: 2582-5461

Abstract

We consider plane curves of the form $(ax)^{\alpha}+(by)^{\alpha}=r^{\alpha}$ defined on the first quadrant of $\mathbb R^2$, where $\alpha>0$ and $a,b,r>0$. We summarize the outlines of them by using elementary differential calculus. We will in this note understand that they are classified into three types of curves, convex, straight and concave, depending on $\alpha$.

Keywords and Phrases

Plane curve, implicit function, orthogonal representation, polar representation, differential calculus, area, integral.

A.M.S. subject classification

14H50, 26B10.

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