Gielis Formula

An observed difficulty between the design, simulation, and construction of a patch antenna is the differences between the design data, the simulation, and the construction of the antenna. An antenna design considered acceptable presents a difference that can vary between 1 and 5% of the values obtained considering the operating frequencies, bandwidth, and the applied technology. To reduce differences by implementing equations for automatic generation of radiating element shapes. A polar equation used for generating shapes observed in nature, for generating shapes for patch antennas is the Gielis formula^12,13.

According to Refs.^12,13 circular shapes, squares, ellipses and rectangles are members of the group of superellipses which have limited symmetry as a disadvantage. Using the polar coordinate, by substituting of x=rcos(θ) and y=rsin(θ), and entering the m/4 angle argument, introduces a specific rotational symmetry. The arguments ni and m, to the set of real numbers, and a and b are nonzero real numbers. Gielis formula can be used by multiplying by a function, forming a generic equation that generates a large class of super, and sub-forms, including the super and the sub-circle as a special case, similar structures to the generated by L-system and by fractals, Euclidean and non-Euclidean forms, such as those observed in nature. Gielis formula is giver by: