Effects of the numerical values of the parameters in the Gielis equation on its geometries

The Lame curve is an extension of an ellipse, the latter being a special case. Dr. Johan Gielis further extended the Lame curve in the polar coordinate system by introducing additional parameters (n1, n2, n3; m): r(o)=(||1Acos(m4o)||n2+||1Bsin(m4o)||n3)-1/n1, which can be applied to model natural geometries. Here, r is the polar radius corresponding to the polar angle o; A, B, n1, n2 and n3 are parameters to be estimated; m is the positive real number that determines the number of angles of the Gielis curve. Most prior studies on the Gielis equation focused mainly on its applications. However, the Gielis equation can also generate a large number of shapes that are rotationally symmetric and axisymmetric when A = B and n2 = n3, interrelated with the parameter m, with the parameters n1 and n2 determining the shapes of the curves. In this paper, we prove the relationship between m and the rotational symmetry and axial symmetry of the Gielis curve from a theoretical point of view with the condition A = B, n2 = n3. We also set n1 and n2 to take negative real numbers rather than only taking positive real numbers, then classify the curves based on extremal properties of r(o) at o = 0, PI/m when n1 and n2 are in different intervals, and analyze how n1, n2 precisely affect the shapes of Gielis curves.