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Equation of Apple Shaped Curve


When a Cardioid is deformed into another figure with introducing some exponential functions,
an apple shaped curve is found accidentally.

Fig.1   Cardioid

    As described in the beginning of the page of "Hart Curves II" (here in heart curves II), 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.


     In purpose to calculate the numerical coordinates data of an apple shaped curve as shown in Fig.2, a C++ program originated from Eq.(3) is given by C++ program for the calculation of apple curve.      By executing the C++ program, a text file named "apple.txt" including the calculated data is produced.      Each interval of these data is divided by 'comma'.      After moving these calculated data into an Excel file, we obtain a curve with the use of a graph wizard attached on the Excel file.

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


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Updated: 2010.03.29, edited by N. Yamamoto.
Revised on Mar. 16, 2015, May 05, 2020, Jan. 30, 2021, May 15, 2021 and Mar. 22, 2022.