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Equation of Spade Curve


    Although the head shape of a spade seems like the reverse shape of a heart curve, it is very difficult to find the equation expressing the head of a spade because the margin of the choice of a spade curve is much more narrow than that in the case of a heart curve.
    In this time, we find only a heart curve, which may be adopted as the shape of a spade after the top and the bottom of the heart curve were made reverse, among a lot of the obtained heart curves.
    By the way, the shape of a spade is indicated in this site.

    If the top and the bottom of the heart curve which is displayed in the third section of the page of Heart Curves III (here in Heart Curves III) are made inverse, relatively good shape of a spade yields.     Then, if only the sign of Eq.(6) in the page of heart curve III is changed while that of Eq.(5) is not changed, the following two equations are rewritten.

                     ,                        (1)

                     ,                        (2)
where .
    When the intermediate variable varies in the range of , two examples of curves displaying Eqs.(1) and (2) and these painted figures are shown in the followings.



     In purpose to calculate the numerical coordinates data of a spade curve as shown in Fig.1, a C++ program originated from Eqs.(1) and (2) is given by C++ program for the calculation of spade curve.      By executing the C++ program, a text file named "spade.txt" including the calculated coordinates data is produced.      Each interval of these data is divided by 'comma'.      After moving these calculated data into an Excel file, we obtain a spade curve with the use of a graph wizard attached on the Excel file.

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Updated: 2010.03.17, edited by N. Yamamoto.
Revised on Mar. 16, 2015, May 01, 2020, May 04, 2020, Feb. 02, 2021, May 09, 2021 and Mar. 30, 2022.