Someyama, Norihiro On the outlines of plane curves of the form \((ax)^{\alpha} + (by)^{\alpha} = r^{\alpha}\) with \(\alpha > 0\). (English) Zbl 1448.53008 J. Ramanujan Soc. Math. Math. Sci. 7, No. 2, 99-108 (2020). Summary: 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 \(\alpha\). MSC: 53A04 Curves in Euclidean and related spaces 14H50 Plane and space curves 26B10 Implicit function theorems, Jacobians, transformations with several variables Keywords:plane curve; implicit function; orthogonal representation; polar representation; differential calculus; area; integral PDFBibTeX XMLCite \textit{N. Someyama}, J. Ramanujan Soc. Math. Math. Sci. 7, No. 2, 99--108 (2020; Zbl 1448.53008) Full Text: Link References: [1] Brieskorn, E. and Kn¨orrer, H. (trans. Stillwell, J.), Plane Algebraic Curves, Birkhaeuser (2012). · Zbl 1254.14002 [2] Friedman, A., Advanced Calculus, Dover Publications (2007). · Zbl 0225.26002 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.