Effects of the Numerical Values of the Parameters in the Gielis Equation on Its Geometries

The Lamé curve is an extension of ellipse, the latter being a special case. Dr. Johan Gielis further extended in polar coordinate system by introducing additional parameters (n1, n2, n3; m): rφ=1Acosm4φn2+1Bsinm4φn3−1/n1, which can be applied to model natural geometries. Here, r radius corresponding angle φ; A, B, n1, n2 and n3 are estimated; m positive real number that determines angles curve. Most prior studies on equation focused mainly its applications. However, also generate large shapes rotationally symmetric axisymmetric when A = B n3, interrelated with parameter m, n1 determining curves. In this paper, we prove relationship between rotational symmetry axial from theoretical point view condition n3. We set take negative numbers rather than only taking numbers, then classify curves based extremal properties r(φ) at φ 0, π/m different intervals, analyze how precisely affect