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Equation to Display a Shape of Rectangle with the Use of Exponential Function


      There is an equation as below.
      ,                      (1)
where a, b and c are arbitrary real numbers.
  We have found that the above equation displays the envelope of a rectangle in the case that the absolute values of b and c take sufficiently large (over about ten).

  If we rewrite Eq.(1), the following equation is led to
      ,                      (2)
where the defined region is given as
      .                      (3)

      In the case of a=1, if we calculate Eq.(2) as varying the several values of and with the use of computer, the each rectangle is drawn as shown in Fig.1 to 8.      The each length of longitudinal and horizontal sides of a rectangle is about and respectively.

Fig.1 a=1, b=100, c=50
Fig.2 a=1, b=100, c=100

Fig.3 a=1, b=10, c=10
Fig.4 a=1, b=3, c=3

Fig.5 a=1, b=0, c=0
Fig.6 a=1, b=-100, c=100

Fig.7 a=1, b=-100, c=-100
Fig.8 a=1, b=100, c=-100

      In purpose to calculate the numerical coordinates data of a rectangle as shown in Figs.1 to 8, a C++ program originated from Eq. (2) is given by C++_program_of_rectangle
     By executing this C++ program, a text file named "rectangle.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 figure of rectangle with the use of a graph wizard attached on the Excel file.

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Updated: 2008.6.23, edited by N. Yamamoto.
Revised on Mar. 16, 2015, May 05, 2020, May 17, 2020, Feb. 01, 2021, May 09, 2021 and Mar. 30, 2022.