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<p>We introduce a parametric family of plane curves, which we call <i>supercardioids</i>, obtained by applying the Gielis transformation to the classical cardioid. This construction yields a continuous deformation governed by parameters controlling symmetry and sharpness, producing a geometrically rich class of curves. We investigate fundamental properties of supercardioids, including symmetry, extremal structure, and global geometric properties. In particular, we establish explicit bounds for the area enclosed by these curves and show that it remains uniformly bounded by a universal constant multiple of the area of the classical cardioid. Furthermore, we analyze the asymptotic behaviour with respect to the deformation parameter, identifying a transition from smooth perturbations of the cardioid to non-smooth, piecewise-defined limiting shapes. These results demonstrate that classical plane curves can generate structurally rich families of geometric objects under nonlinear transformations, providing a tractable framework that combines elementary trigonometric constructions with nontrivial analytic features.</p>

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On supercardioids: geometry and area bounds

  • 期刊论文

  • 发表日期：

    2026年7月9日

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  • Luděk Spíchal

摘要

We introduce a parametric family of plane curves, which we call supercardioids, obtained by applying the Gielis transformation to the classical cardioid. This construction yields a continuous deformation governed by parameters controlling symmetry and sharpness, producing a geometrically rich class of curves. We investigate fundamental properties of supercardioids, including symmetry, extremal structure, and global geometric properties. In particular, we establish explicit bounds for the area enclosed by these curves and show that it remains uniformly bounded by a universal constant multiple of the area of the classical cardioid. Furthermore, we analyze the asymptotic behaviour with respect to the deformation parameter, identifying a transition from smooth perturbations of the cardioid to non-smooth, piecewise-defined limiting shapes. These results demonstrate that classical plane curves can generate structurally rich families of geometric objects under nonlinear transformations, providing a tractable framework that combines elementary trigonometric constructions with nontrivial analytic features.