Fig.1   Cardioid

    As described in the beginning of the page of "Hart Curves II" (here in ), the equation of the Cardioid as shown in Fig.1 in this page is written in the angular coordinates as

                     ,                        (1)
where the variables and indicate the moving radius and the phase angle respectively, and the definition region of is .     The usual conversion formula into the orthogonal coordinates is described as

                     .                        (2)

    In the present time, we merely treat the variables both of and as general intermediate variables, and then, we will consider the following equations which are deformed from Eqs.(1) and (2) with the use of some exponential functions.

                     ,                        (3)
where , and a, b, p and q are the constants.
    When the intermediate variable varies in the range of , the curves which Eq.(3) display, and its painted one are shown in the following figures.

    As a reference, when the region of is expanded over the full range, Eq.(3) gives Fig.3.

Fig.3