# JP2000090072A - Method for obtaining function values of circle and sphere with arbitrary accuracy and method for calculating numerical stream for arbitrary number of digits - Google Patents

Method for obtaining function values of circle and sphere with arbitrary accuracy and method for calculating numerical stream for arbitrary number of digits

## Info

Publication number
JP2000090072A
JP2000090072A JP29597698A JP29597698A JP2000090072A JP 2000090072 A JP2000090072 A JP 2000090072A JP 29597698 A JP29597698 A JP 29597698A JP 29597698 A JP29597698 A JP 29597698A JP 2000090072 A JP2000090072 A JP 2000090072A
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Japan
Prior art keywords
length
approximate
calculation
notation
ratio
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Pending
Application number
JP29597698A
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Japanese (ja)
Inventor
Hisato Sugawara

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Individual
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Individual
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Priority to JP29597698A priority Critical patent/JP2000090072A/en
Publication of JP2000090072A publication Critical patent/JP2000090072A/en
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• Complex Calculations (AREA)

## Abstract

PROBLEM TO BE SOLVED: To easily operate the numerical value of a digit for an arbitrary length by improving the accuracy of various calculations by obtaining a number π, the tangential rate and inverse tangential rate of chord and circular arc and spherical rate W. SOLUTION: While defining the midpoint of a line segment of extension 2*R as a center, concerning an equilateral polygon composed of the aggregate of 2'N pieces of isosceles triangles having two side lengths R sharing this point, initial values are made into number n=2, area E1=0, height h1 to bottom side = 0, bottom side length L1=2*R, full peripheral side length of regular polygon = Lλ*2=4*R, for n=1 time, the values are made into number 2'=(n+1), area En+1=R*Ln/4, bottom side length Ln+1=((R-hn)'2+(Ln/2)'2(1/2), height hn+1 to the bottom side = 2*En+1/Ln+1, full peripheral side length of regular polygon = Ln+λ*2'(n+1) and approximate number π = full peripheral side length 2/R, the calculation is repeated for N times corresponding to the required accuracy rate (approximation expression) 1±1/2'(2*N+1), and the full peripheral side length and approximate number of the regular polygon are obtained.

## Description

【発明の詳細な説明】DETAILED DESCRIPTION OF THE INVENTION

【０００１】[0001]

【産業上の利用分野】高精度の計算を行う全分野と、円
と角度と球について高精度の計算を行う全分野
[Industrial applications] All fields that perform high-precision calculations, and all fields that perform high-precision calculations on circles, angles, and spheres

【０００２】[0002]

【従来の技術】近年の電子式演算装置の一般社会への普

2. Description of the Related Art Due to the recent popularization of electronic arithmetic devices in the general society and the improvement in device capabilities, complicated procedures and arithmetic processes in various types of advance which were not possible in the past with an arithmetic method in terms of time. Became faster and possible.

【０００３】[0003]

【発明が解決しようとする課題】従来の角度と円と球に

が真値と一致するかが不明で、三角関数値と逆三角関数

に、各種の図形の代表点が、不明確で、重心が、針上均

を係数とする２＊Ｗ＊円周率＾２＊半径＾２である事が

In the conventional calculations relating to angles, circles, and spheres, the meaning of the calculations is unknown, it is unknown whether the power series operation matches the true value, and the trigonometric function value and the inverse trigonometric function are used. The range of the error was unknown because it was impossible to obtain the value, angle value, and pi with any precision. Therefore, the fact that the representative points of various figures are unclear, the center of gravity is the needle equilibrium point, and the definition of the centroid is unclear is not widely spread, confusion occurs frequently, and various formulas are used. In many fields, there were many errors, and the improvement of accuracy was neglected in many fields. The following formula is described in the standard BASIC language. The surface area of the sphere is 1/4
Spherical ratio W (folding ratio, aperture ratio) (2 / number of neighbors of pi)
Since it was unclear that 2 * W * pi ＾ 2 * radius ＾ 2 with the coefficient as, the understanding of the meaning of the relationship between phenomena was unclear. [Definition of reception surface] A continuous collection of tangent lines without thickness. It was difficult to publicize the solution, and it was not understood that the invention / invention was the same as the problem solution.

【０００４】[0004]

【課題を解決するための手段】請求項１，２，３の方法
により、円の全円周、一部の円弧、弦を正多角形の部分
として辺の１／２分割、又は２倍の辺長関係で、計算回

４折込みでの、面積率（球面率）を求めて、半径と表面

According to the method of claims 1, 2 and 3, the whole circumference of a circle, a part of an arc, and a chord are divided into half of a side or a double of a side as a regular polygon. By reducing the number of calculations in relation to the side length, the length ratio of the pi and the radius of the arc and chord with arbitrary precision is obtained. According to the method of claim 4,
The surface area of the sphere and the side area of the circumscribed cylinder are 1 /
Calculate the area ratio (spherical ratio) at 4 folds and clarify the relationship between radius and surface area. The method according to claim 5, 6, or 7, enables a numerical operation of an arbitrary digit length between arbitrary storage media,
Improve accuracy.

【０００５】[0005]

【作用】請求項１，２，３の方法により、従来のベキ級

によって分割数２＾Ｎに相当するＮ回の演算回数で、請

＊Ｎ＋１）、請求項２の方法でサインを［近似式］１±
１／２＾（２＊Ｎ＋５）、請求項３の方法で逆サインを
［近似式］１±１／２＾（２＊Ｎ＋３）の各精度率で各

であることを認識し、隣数の意味を確認する．請求項
５、６、７の方法により、任意の記憶媒体間での、任意

According to the first, second, and third methods, the number of operations corresponding to the number of divisions of 2 ＾ N can be reduced from the conventional method of calculating the power series to the ratio by a half or two times. In the method of claim 1, the pi is set to [approximate expression] 1 ± 1/2 ± (2
* N + 1), the sign is calculated by the method of claim 2 [approximate expression] 1 ±
The inverse sine is calculated by the method of claim 3 at an accuracy rate of 1 ± 1/2 (2 * N + 3). According to a fourth aspect of the present invention, the meaning of the number of neighbors is confirmed by recognizing that the sphere ratio W at the 1/4 folding of the contact surface is the number of neighbors of 2 / pi, by the minute piece division method. According to the method of claims 5, 6, and 7, the operation of a numerical string of an arbitrary length between arbitrary storage media is almost independent of the performance of the processing device, and the accuracy is improved.

【０００６】[0006]

【発明を実施するための最良の形態】請求項５、６、７
の方法は、電子演算装置のファイル＝コンペアにより、

BEST MODE FOR CARRYING OUT THE INVENTION Claims 5, 6, and 7
In the method of the above, the file of the electronic computing device = compare,
A file of the intermediate arithmetic storage is constructed from a comparison between a plurality of files, and the processing is sequentially performed. The final result is a normal decimal notation.

【０００７】[0007]

【実施例１】請求項１の方法で、多角形辺長、近似円周

＝２５で、求めた例を図５に示す．
Embodiment 1 According to the method of claim 1, the polygon side length, the approximate circumference length and the pi are determined by the element relationships shown in FIG. 1 and FIG.
FIG. 5 shows an example of the calculation at = 25.

【０００８】[0008]

【実施例２】請求項２の方法で、円弧に正接する弦長率
を、図２に示す要素関係で、Ｎ＝５、１０、１５、２
０、２１、２２、２３、２４、３０で、求めた例を図６
に示す．
[Embodiment 2] In the method according to the second aspect, the chord length ratio tangent to the arc is determined by the element relation shown in FIG.
Examples obtained at 0, 21, 22, 23, 24 and 30 are shown in FIG.
Shown in

【０００９】[0009]

【実施例３】請求項３の方法で、弦に正接する円弧長率
を、図３に示す要素関係で、Ｎ＝２５で、求めた例を図
７に示す．
Embodiment 3 FIG. 7 shows an example in which the arc length ratio tangent to the chord is obtained by the method of claim 3 at N = 25 in the element relation shown in FIG.

【００１０】[0010]

【実施例４】請求項４の方法で、応接面の１／４折込み
での、球面率Ｗを、図４に示す要素関係で、図８でＮ＝
２、５、６、１０、２０、図９でＮ＝２００、図１０で
Ｎ＝１０，０００、図１１でＮ＝８０，０００、図１２
でＮ＝４００，０００、図１３でＮ＝８００，０００、

[Fourth Embodiment] In the method of the fourth embodiment, the sphere ratio W at the quarter-folding of the contact surface is calculated by using the element relation shown in FIG.
2, 5, 6, 10, 20, N = 200 in FIG. 9, N = 10,000 in FIG. 10, N = 80,000 in FIG. 11, FIG.
N = 400,000 and N = 800,000 in FIG.
FIG. 14 shows an example in which N = 10,000,000 is obtained.

【００１１】[0011]

【実施例５】請求項１２３４の方法で、構成した、表計

の実行実例を物件提出で、示す．
[Embodiment 5] A program disclosed in a spreadsheet program 123 constructed according to the method of claim 1234. WJ1 and its execution example are shown in the property submission.

【００１２】[0012]

【発明の効果】任意精度の、三角関数値と逆三角関数値
と円弧長に相当する角度値と円周率を任意の精度で求め
られ、各種の計算精度が向上し、各種の不明確点を、確

According to the present invention, it is possible to obtain an arbitrary value of a trigonometric function value, an inverse trigonometric function value, an angle value corresponding to an arc length and a pi with an arbitrary accuracy, thereby improving various calculation accuracy and various unclear points. Can be confirmed.

【００１３】[0013]

【図面の簡単な説明】[Brief description of the drawings]

【図１の１】請求項１の方法の各要素の関係を多角形の

FIG. 1 shows the relationship between the elements of the method of claim 1 as a part of a polygon.

【図１の２】請求項１の方法のＥｎとＥｎ＋１の位置関

FIG. 1 shows the positional relationship between En and En + 1 in the method of claim 1 as a part of a polygon.

【図２】請求項２の方法で、Ｒ＝１とした場合の各要素
の関係を示す．
FIG. 2 shows the relationship among elements when R = 1 in the method of claim 2;

【図３】請求項３の方法で、Ｒ＝１とした場合の各要素
の関係を示す．
FIG. 3 shows a relationship between respective elements when R = 1 in the method of claim 3;

【図４】請求項４の方法の各要素の関係を示す．FIG. 4 shows the relationship between elements of the method of claim 4;

【図５】請求項１の方法で、多角形総辺長と、近似円周

FIG. 5 shows an example in which the total side length of the polygon and the approximate pi are determined at N = 25 by the method of claim 1;

【図６】請求項２の方法で、円弧に正接する弦の近似長

２４、３０で、求めた例を示す．
FIG. 6 is a graph showing a method of calculating the approximate length ratio of a chord tangent to an arc by N = 5, 10, 15, 20, 21, 22, 23;
Examples obtained at 24 and 30 are shown.

【図７】請求項３の方法で、弦に正接する多角辺の総和
と、円弧長率の１／２を、Ｎ＝２５で、求めた例を示
す．
FIG. 7 shows an example in which the sum of polygonal sides tangent to a chord and の of the arc length ratio are obtained at N = 25 by the method of claim 3.

【図８】請求項４の方法で、応接面の１／４折込みで
の、球面率Ｗを、Ｎ＝２、５、６、１０、２０として、

FIG. 8 is a diagram illustrating a method according to claim 4, wherein the spherical surface ratio W at the quarter folding of the contact surface is N = 2, 5, 6, 10, and 20;
Here is an example.

【図９】請求項４の方法で、応接面の１／４折込みで
の、球面率Ｗを、Ｎ＝２００として、求めた例を示す．
FIG. 9 shows an example in which the sphericity W at a quarter folding of the contact surface is set to N = 200 by the method of claim 4.

【図１０】請求項４の方法で、応接面の１／４折込みで
の、球面率Ｗを、Ｎ＝１０，０００として、求めた例を

FIG. 10 shows an example in which the sphericity W at a quarter-folding of the contact surface is set as N = 10,000 by the method of claim 4.

【図１１】請求項４の方法で、応接面の１／４折込みで
の、球面率Ｗを、Ｎ＝８０，０００として、求めた例を

FIG. 11 shows an example in which the spherical surface ratio W at a quarter-folding of the contact surface is set to N = 80,000 by the method of claim 4.

【図１２】請求項４の方法で、応接面の１／４折込みで
の、球面率Ｗを、Ｎ＝４００，０００として、求めた例
を示す．
FIG. 12 shows an example in which the spherical surface ratio W at the quarter-folding of the contact surface is set to N = 400,000 by the method of claim 4.

【図１３】請求項４の方法で、応接面の１／４折込みで
の、球面率Ｗを、Ｎ＝８００，０００として、求めた例
を示す．
FIG. 13 shows an example in which the sphericity W at a quarter-folding of the contact surface is set to N = 800,000 by the method of claim 4.

【図１４】請求項４の方法で、応接面の１／４折込みで
の、球面率Ｗを、Ｎ＝１０，０００，０００として、求
めた例を示す．
FIG. 14 shows an example in which the spherical surface ratio W at the quarter-folding of the contact surface is set to N = 10,000,000 by the method of claim 4.

【符号の説明】[Explanation of symbols]

Ａ 線分間の角度である． Ｃ、Ｃ、Ｃｎ＋１ 円弧とｎ回目、ｎ＋１回目の近似

A is the angle between line segments. C, C n , and C n + 1 represent the partial side lengths of the arc and the n-th and n + 1-th approximate arcs. E represents n, E n + 1 n th, the area of the n + 1 th element triangle. L, L n, L n + 1 lines and n-th represents a partial side line length of (n + 1) th approximate arc. Represents n times. R The radius of the circle in each figure.

## Claims (8)

【特許請求の範囲】[Claims] 【請求項１】図１の要素関係で、延長２＊Ｒの線分の中

０、底辺長Ｌ＝２＊Ｒ、正多角形の全辺周長＝Ｌ
２＝４＊Ｒとし、ｎ＋１回目で、個数＝２＾（ｎ＋
１）、面積Ｅｎ＋１＝Ｒ＊Ｌ／４、底辺長Ｌｎ＋１
（（Ｒ−ｈ）＾２＋（Ｌｎ／２）＾２）＾（１／
２）、底辺に対する高さｈｎ＋１＝２＊Ｅｎ＋１／Ｌ
ｎ＋１、正多角形の全辺周長＝Ｌｎ＋１＊２＾（ｎ＋
１）、近似円周率＝全辺周長／２／Ｒ、として、必要な

るＮ回、計算を繰返し、正多角形の全辺周長と近似円周

1. With respect to the element relation of FIG. 1, the center of a line segment of an extension 2 * R is set as a center, and the isosceles of 2 ＾ N sets of isosceles triangles having a length R of two sides sharing this point. With a polygon, the initial value is
The number n = 2, the area E 1 = 0, and the height h 1 =
0, base length L 1 = 2 * R, circumference of all sides of regular polygon = L 1 *
2 = 4 * R, and at the (n + 1) -th time, the number = 2 ＾ (n +
1), area E n + 1 = R * L n / 4, base length L n + 1 =
((R-h n) ^ 2 + (Ln / 2) ^ 2) ^ (1 /
2), height hn + 1 = 2 * En + 1 / L with respect to the base
n + 1 , circumference of all sides of regular polygon = L n + 1 * 2 ＊ (n +
1) The calculation is repeated N times corresponding to the required accuracy ratio [approximate expression] 1 ± 1/2 ＾ (2 * N + 1), assuming that the approximate circumference ratio = perimeter of all sides / 2 / R, A method to find the circumference and approximate pi of all sides of a polygon.
【請求項２】図２の要素関係で、半径１の単位円の角度
に相当する円弧の長さＣから、その両端を半径１の線分
で、囲んだ３辺形で、必要な精度率［近似式１±１／２
＾（２＊Ｎ＋５）］に相当するＮで、初期値をＣ＝円

＝２＊（Ｃ＾２−（Ｃ＾４／４））＾（１／２）と
して、Ｎ回、計算を繰返し、単位円の円弧端点間の近似

2. In the element relation shown in FIG. 2, a required accuracy ratio is defined as a triangle surrounded by a line segment having a radius of 1 at both ends from an arc length C corresponding to an angle of a unit circle having a radius of 1. [Approximation formula 1 ± 1/2
{(2 * N + 5)], the initial value is C 0 = arc length C / 2 ＾ (N−1), and at the (n + 1) th time, C n + 1
= 2 * (C n ＾ 2− (C n ＾ 4/4)) ＾ (）), the calculation is repeated N times, and the approximate distance between the arc end points of the unit circle is obtained. How to find the length.
【請求項３】図３の要素関係で、半径１の単位円に内接
する線分の長さＬを初期値Ｌとして、ｎ＋１回目で、
ｎ＋１＝（Ｌ＾２＋（１−（１−Ｌ＾２）＾（１
／２））＾２）＾（１／２）／２、総辺長を２＾（ｎ＋
１）倍して、必要な精度率［近似式１±１／（２＾（２
＊Ｎ＋３）］に相当するＮ回、計算を繰返し、単位円の

An element relationship wherein Figure 3, the line segments inscribed in a unit circle of radius 1 and length L as the initial value L 0, (n + 1) th,
L n + 1 = (L n ^ 2 + (1- (1-L n ^ 2) ^ (1
/ 2)) ＾ 2) ＾ (1/2) / 2, and the total side length is 2 ＾ (n +
1) times the required accuracy rate [approximate expression 1 ± 1 / (2 ＾ (2
* N + 3)] A method in which the calculation is repeated N times corresponding to the above, and an approximate arc length corresponding to the arc length of the unit circle is obtained.
【請求項４】図４の要素関係で、微小片分割法によっ
て、全内径をＮ個に分割し、微小角Ａ＝円周率／２／Ｎ
とし、［微小台形の上辺長率］と、［その上部微小台形
の下辺長率］の和を、ｎ個目で、２＊ＣＯＳ（ｎ＊Ａ）
として、ｎ＝０からｎ＝Ｎまで、順次計算して、Ｎ個の

４折込みでの、近似球面率Ｗを求める方法．
4. According to the element relationship shown in FIG. 4, the entire inner diameter is divided into N pieces by a minute piece dividing method, and a small angle A = perimeter / 2 / N
The sum of [the upper side length ratio of the minute trapezoid] and [the lower side length ratio of the upper minute trapezoid] is 2 * COS (n * A) for the n-th piece.
Is calculated sequentially from n = 0 to n = N, 1 is added to the sum of the N calculations, divided by 2 * N, and
Method for obtaining approximate spherical ratio W in four folding.
【請求項５】従来の計算法で、使用する数値の確定域が

5. In the conventional calculation method, when the definite range of a numerical value to be used is unknown, 5 is added to the digit next to the last digit, and the calculation is performed.
The calculation result is a calculation method in which digits corresponding to the 5 digits are truncated.
【請求項６】従来の１０進数値表示法で、小数点．に相

←を表記数字の上部に表示する数字列表示法．
6. A conventional decimal value display method, wherein a decimal point. If the numerical value has a fixed area at the position corresponding to, use z. If there is a cyclic part, use g as well. If the cyclic part is a single digit, enter. → to the end point
Numeric string display method in which ← is displayed above the notation number.
【請求項７】従来の１０進数値表示法を１０＾Ｎ進法に

０進法で、＋１００を＠０、−１００をＪ０、１０＾Ｎ

−１０＾Ｎを［Ｊと、Ｎ−１個の０］で表示し、−１、
−２、−３、−４、−５、−６、−７、−８、−９に相

Ｇ、Ｈ、Ｉの英字で表現する数値表記法．
7. The conventional decimal value notation is replaced by 10 ＾ N notation, and in decimal notation, +10 represents ＠, -10 represents J,
In 0-base, +100 is {0, -100 is J0, 10} N
In decimal notation, +10 表示 N is represented by [＠ and N−1 zeros],
−10 ＾ N is represented by [J and N−1 0], and −1,
Negative integers corresponding to -2, -3, -4, -5, -6, -7, -8, and -9 are represented by A, B, C, D, E, F,
Numerical notation expressed by alphabets of G, H and I.
【請求項８】請求項７で、数値表示列の前に、１０進法
では＠＠を、１０＾Ｎ進法では、＠Ｎ＠を付記し、その

8. A method according to claim 7, wherein before the numerical display sequence, ＠＠ is added in decimal notation, and {N} is added in 10 ＾ N-decimal notation. Thereafter, an integer part R and a decimal part m are added. In addition, [＠＠ R. m ±]
Or [{N} R. [m ±].
JP29597698A 1998-09-11 1998-09-11 Method for obtaining function values of circle and sphere with arbitrary accuracy and method for calculating numerical stream for arbitrary number of digits Pending JP2000090072A (en)

## Priority Applications (1)

Application Number Priority Date Filing Date Title
JP29597698A JP2000090072A (en) 1998-09-11 1998-09-11 Method for obtaining function values of circle and sphere with arbitrary accuracy and method for calculating numerical stream for arbitrary number of digits

## Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
JP29597698A JP2000090072A (en) 1998-09-11 1998-09-11 Method for obtaining function values of circle and sphere with arbitrary accuracy and method for calculating numerical stream for arbitrary number of digits

## Publications (1)

Publication Number Publication Date
JP2000090072A true JP2000090072A (en) 2000-03-31

# Family

## Family Applications (1)

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## Country Status (1)

Country Link
JP (1) JP2000090072A (en)

## Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
KR100357439B1 (en) * 2000-08-24 2002-10-19 대한민국(창원대학교) Control method of cam system for manufacturing nc-code of spur gear
AU774975B2 (en) * 2000-08-14 2004-07-15 Canon Kabushiki Kaisha Method and apparatus for determining the area enclosed by a planar curve

## Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
AU774975B2 (en) * 2000-08-14 2004-07-15 Canon Kabushiki Kaisha Method and apparatus for determining the area enclosed by a planar curve
KR100357439B1 (en) * 2000-08-24 2002-10-19 대한민국(창원대학교) Control method of cam system for manufacturing nc-code of spur gear

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