﻿ Equation of Curve Expressing Character S

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Equation of Curve Expressing Character S

Nobuo YAMAMOTO

Equation of curve expressing character S is tried to be composed with the use of the exponential function.

 Fig.1 Fig.2

As seen in Fig.1, it is supposed that a point P moves on a line AB as an intermediate variable varies in the range of .     As the orthogonal coordinates of the point P is put with , the equation expressing such coordinates is assumed as the following with the use of the constants a, b and p.
.                      (1)
where Eq.(1) gives the curve shown in Fig.2.
In the next, vector which makes point P a cardinal point is assumed as the following with the use of the constants c, k and q.
,                      (2)
where i represents the imaginary unit.
Then, the coordinates is expressed as the following equations, and the point Q may display a curve which represents character S as the intermediate variable varies in the range of

,                      (3)
and

,                      (4)
where .
When the intermediate variable varies in the range of , the curves which Eqs.(3) and (4) display are shown in the following figures.

 Fig.3 Fig.4 Fig.5

 Fig.6 Fig.7

 Fig.8 /CENTER> Fig.9 Fig.10

 In purpose to calculate the numerical coordinates data of a curve expressing character S as shown in Figs.3 to 10, a C++ program originated from Eqs.(3) and (4) is given by .    By executing the either C++ program, a text file named "S_curve.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 expressing character S with the use of a graph wizard attached on the Excel file.

It is tried to express the shape of the question mark (?) by some equation in the same manner as described above.

(Method 1)

The following two equations are treated corresponding to Eqs.(3) and (4).

,                      (5)
and

,                      (6)
where .
When the intermediate variable varies in the range of , the curves which Eqs.(5) and (6) display are shown in the following figures.

 Fig.11 Fig.12

 In purpose to calculate the numerical coordinates data of a curve expressing the question mark (?) as shown in Figs.11 and 12, a C++ program originated from Eqs.(5) and (6) is given by .

(Method 2)

The following two equations are treated corresponding to Eqs.(3) and (4).

,                      (7)
and

,                      (8)
where .
When the intermediate variable varies in the range of , the curves which Eqs.(7) and (8) display are shown in the following figures.

 Fig.13 Fig.14

 In purpose to calculate the numerical coordinates data of a curve expressing the question mark (?) as shown in Figs.13 and 14, a C++ program originated from Eqs.(7) and (8) is given by .

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Updated: 2010.02.21, edited by N. Yamamoto.
Revised on Mar. 16, 2015, Jul. 24, 2016, May 04, 2020, Feb. 02, 2021, May 09, 2021 and Mar. 30, 2022.