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On the outlines of plane curves of the form \((ax)^{\alpha} + (by)^{\alpha} = r^{\alpha}\) with \(\alpha > 0\). (English) Zbl 1448.53008

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
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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
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