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Equations of Convex and Concave Circles with the Use of Exponential Equation

Nobuo YAMAMOTO

There is a set of three equations described as below. ,                            (1) ,                                     (2)
and ,                     (3)
where the values of a, b and c are arbitrary real numbers in Eqs.(1) and (2), and where, in Eq.(3), the values of b and c are also arbitrary real numbers except for the value of a which is restricted by the following equation. .                                              (4)

In the case of , it is shown in the foregoing calculations that both of Eq.(1) and Eq.(3) are led to Eq.(2) by means of MacLaurin series expansion. .                                 (5) .                               (6) .                                        (7)

As such the led Eq.(7) may be regarded to be an independent of Eqs.(1) and (3), we can remove the coefficient number "2" from the right-hand side of Eq.(7) freely.        Thus, the constants a, b and c in Eq.(2) are newly defined as arbitrary real numbers.

In the case of b=c=1, if Eqs.(1) to (3) are numerically calculated and graphically displayed in the various values of a with the use of a computer, the various closed-curves related to a are obtained as shown in Figs.1 to 10.
Eq.(1), Eq.(2) and Eq.(3) give convex, true and concave circles respectively.        Moreover, Eq,(1) nearly gives rectangles under sufficiently large values of a, and Eq.(3) nearly gives an asteroid under the minimum value of a which is restricted in the defined region as given in Eq.(4).          When a takes sufficiently large value, Eq,(1) gives rectangles which have different sizes between longitudinal and horizontal sides as varying the values of b and c in the same manner as in another way in this button .        When a=10, two cases of b=1 and c=2, and b=2 and c=1 are displayed in the followings.  In purpose to calculate the numerical coordinates data of convex circles as shown in Figs.1 to 4 and Figs 11 to 12, a C++ program originated from Eq. (1) is given by .      In purpose to calculate the numerical coordinates data of a circle as shown in Fig.5, a C++ program originated from Eq. (2) is given by .      In purpose to calculate the numerical coordinates data of concave circles as shown in Figs.6 to 10, a C++ program originated from Eq. (3) is given by .      By executing these C++ programs, a common text file named "distorted_circle.txt" or "circle.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 convex, true or concave circle with the use of a graph wizard attached on the Excel file.
[Reference]

General Method

Generally, the convex, true and concave circles or ellipses are given by the following equation. ,                                         (8)
Two examples of the cases of b=1 and c=2, and b=2 and c=1 are shown in Fig.13 and Fig.14 respectively.  _ returning to the page which opened just before with 'return' of browser _

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Updated: 2008.7.02, edited by N. Yamamoto.
Revised on Aug.30, 2013, Dec. 09, 2013, Mar. 16, 2015, May 05, 2020, Jan. 31, 2021, May 15, 2021 and Mar. 25, 2022.