扭面方程与箱变坐标系简介（英文）.doc

扭面方程与箱变坐标系简介(英文)_曹玉聘（全面版）资料

A Brief Introduction to Screwy Surfaces Equation and Box Transformation Coordinate System
--Understanding a new type mathematic train of thoughts and method
Cao Yupin1 , Chen Zhanjie2

(No.1 Geological Exploration Institute of Henan Bureau of Geology and Minerals Exploration, Nayang, Henan; 2. Ministry of Land and Resources, Peking 100812)
Abstract: Box transformation coordinate system was a new type coordinate system changing homonymy coordinate unit of conventional rectangular coordinate system from completely equal into completely variable, not only realized accurate expression of various screwy surfaces equation, but also basically solved the problems of various screwy surfaces, non-linear uneven dynamic limited space, function and image dynamic corresponding. Its promotion would cause major breakthrough or reform of modern mathematic and its related subjects.
Keywords: screwy surfaces; positive box body; box transformation coordinate system; variable coefficient; variable function

Introduction: In real life, various screwy surfaces exist everywhere, but it is difficult for current coordinate systems to have accurate expression and relevant accurate operations. Therefore, this author tried a non-traditional box transformation coordinate system, box transformation function and relevant variable operation, and furthermore extended it into a complete set of variable method. Although quite hoarse, this set of method fundamentally revealed certain universal mathematic rules. In order to facilitate exchange and save space, this author briefly described a part of its train of thoughts and method below with the purpose to offer a few commonplace remarks by way of introduction so that others may come up with valuable opinions. This paper focused on a brief introduction to screwy surfaces equation and box transformation coordinate system, the rest could be seen from the book of “Elementary Study on CS Variate Method ”, out of which, due to the descriptions frequently exceeded traditional category, this author had to use some new terms and expected the readers could understand and offer instructions.
1 4 1 4
5 2 5 2
2/ 3(3/) 2/ 3(3/)
6 7 8 6 7 8
J 6/ 7/ (a) J 6/ 7/ (b)
图1 正箱体概念示意图
(a)线性直近箱体 (b)线性直次箱体
1. Screwy surfaces and screwy surfaces equation
The initial train of thoughts for screwy surfaces and screwy surfaces equation came from various positive box bodies.

Chart 1 Sketch Map of Positive Box Body Concept
(a) Linear straight ahead box body (b) Linear straight secondary box body
1.1 Mathematic agreement of positive box body and screwy surfaces
Positive box body referred to convex hexahedron with 4 sides vertical (or direct cross) on a certain plane (could be called base level) (Chart 1). Convex body here referred to space body with none of the internal angle not more than π value.

Chart 2 Sketch Map of Positive Box Body Basic Type
(a)Linear straight ahead box body (b)Linear curve ahead box body (c) Non-linear straight ahead box body (d) Non-linear curve ahead box body
(e) Linear straight secondary box body (f) Linear curve secondary box body (g) Non-linear straight secondary box body (h) Non-linear curve secondary box body

The 4 planes of the positive box body could be plane (Chart 1 1265 surface) or straight line type surface (Chart 2), the distance between the 2 vertical sides and horizontal side was called wide distance and long distance. Top and bottom surfaces could be plane, surface or screwy surfaces (Chart 1 1485 surface). The distance between top and bottom surfaces was called high distance. The so-called screwy surface could be made through proper move with a generatrix along 2 different surface leads or accompanied with certain change (Chart 1 1485 surface). Its 2 different surface leads could be made through proper twisting of conventional plane or same surface lead, hence the name, its screwy angle could be called screwy surface angle or screwy angle. In case that the lead and generatrix were both straight line, it was linear screwy surfaces (Chart 1 Top, bottom surfaces), otherwise, it could be non-linear or linear screwy surfaces (Chart 2 Top surface). In case that screwy surface angle was 0, it was conventional plane and surface. The above positive box body’s top and bottom surfaces were all determined by 4 sidelines (namely its lead and generatrix), and this was screwy surface equation’s seeking method basis.
In case that a certain positive box body had 3 concurrent sides as mutually vertical, it was called standard positive box body (Chart 2), otherwise called ordinary positive box body (could become standard type through dimensions reducing transformation, and dimensions reducing transformation here referred to dimensions reduction corresponding equivalence transformation). The standard type’s 3 mutually vertical sides were called basic sides, the rest 3 sides were called variable sides.
Positive box body could have 8 basic types(as shown by Chart 2, which showed standard type, and also could be ordinary type), out of which, in case positive box body’s 4 sides had rectangle projection on base level, it could be called square box body. Conventional box body was contour square box body.
1.2 Screwy surfaces equation basic seeking method
Now take standard positive box body’s top surface as example, to make a brief introduction to basic seeking method of the 2 screwy surface equations.
Linear screwy surface equation: real line in Chart 3 was a linear straight box body (could be other linear positive box body), out of which, the 2 surfaces of M1, M2 were mutually parallel (LY1=LY2), but the widths were not equal (LX1≠LX2), top surface (H11H12H22H21) was a simple linears crewy surface. Try to seek the screwy surface equation. In the flowing 2 cases:
Case 1, when the 2 surfaces of M1, M2 were parallel and had equal width, (broken line in Chart 3), it was square box body, and the high distance arithmetic formula of various sides shown by Table 1

Chart 3 Linear straight ahead box body

Simplified table of mathematic characteristics of various sides of linear square box body Table 1
Side
High distance
Wide distance
Side high distance arithmetic formula
Remarks
Horizontal 1
H11，H12
LX1
HX1=HX1(X)=AHX1+BHX1X
AHX1=H11,BHX1=(H12-H11)/LX1
Horizontal 2
H21，H22
LX2=LX1
HX2=HX2(X)=AHX2+BHX2X
AHX2=H21,BHX2=(H22-H21)/LX1
Vertical 1
H11，H21
LY1
HY1=HY1(Y)=AHY1+BHY1Y
AHY1=H11,BHY1=(H21-H11)/LY1
Vertical 2
H12，H22
LY2=LY1
HY2=HY2(Y)=AHY2+BHY2Y
AHY2=H12,BHY2=(H22-H12)/LY1
Through the high distance arithmetic formula of the 4 sides in the table, could gain linear screwy surface equation as:
H(X,Y)=HX1+(HX2-HX1)Y/LY1 or H(X,Y)=HY1+(HY2-HY1)X/LX1， upon completion, as:
H(X,Y)=AH+BHxX+BHyY+CHxyXY, (1)
formula ,AH =H11, BHx=BHX1=(H12-H11)/LX1, BHy=BHY1=(H21-H11)/LY1, CHxy=(H22-H21-H12+H11)/LX1/LY1。
Formula (1) was called one-off function, which meant complete power multinomial single and independent variable maximum value. For multi-element function, when it was proper to consider an independent variable, it could be called deflection function (unary function was an exception). Therefore, linear screwy surfaces equation should be deflection function maximum complete power multinomial.
Case 2, when the 2 surfaces of M1,M2 were parallel but with different width or with equal width but not parallel or neither parallel nor equal width, it would be difficult for the screwy surface equation to adopt the above conventional method for direct seeking. However, if density between surfaces was properly changed so as to make them have equal length and width, or coordinate unit was properly changed so as to make vertical and horizontal coordinate unit between surfaces completely corresponding and equal, it would be possible to follow Case 1 to use conventional method (adopting corresponding coordinate unit) for direct seeking for screwy surface equation [shown by (1) formula]. The basic method was: first set up a certain norm coordinate unit (such as coordinate unit on coordinate axis), then multiply a properly changed certain function of coordinate unit (could be called coordinate coefficient, its seeking method could be seen below). Then, mathematic characteristics of various sides could be seen from Table 2.
The above were initial train of thoughts for solution to the problems of screwy surfaces.
Simplified table of mathematic characteristics of various sides of linear positive box body Table 2
Side
High distance
Wide distance
Side high distance arithmetic formula
Remarks
Horizontal 1
H11,H12
LX1
HX1=HX1(X)=AHX1+BHX1X
AHX1=H11, BHX1=(H12-H11)/LX1
Horizontal 2
H21,H22
LX2≠LX1
HX2=HX2(X)=AHX2+BHX2X
AHX2=H21, BHX2=(H22-H21)/LX1
Vertical 1
H11,H21
LY1
HY1=HY1(Y)=AHY1+BHY1Y
AHY1=H11, BHY1=(H21-H11)/LY1
Vertical 2
H12,H22
LY2≠LY1
HY2=HY2(Y)=AHY2+BHY2Y
AHY2=H12, BHY2=(H22-H12)/LY1

Non-linear screwy surface equation: Chart 4 showed a non-linear straight box body (could also be other non-linear positive box body), H11H12H22H21 were a non-linear screwy surfaces, changes on various sides took secondary parabola as example (such as Table 3, could also be other non-linear change). Try to seek for that screwy surfaces equation

Chart 4 Non-linear box body

Simplified table of mathematic characteristics of various sides of non-linear positive box body Table 3
Side
High distance
Wide distance
Positive thickness equation
Remarks
Horizontal 1
H11,H12
LX1
HX1=HX1(X)=AHX1+BHX1X+CHX1X2
Various equations could adopt parabola method to seek, out of which, X, Y were coordinate value or coordinate unit amount on corresponding coordinate axis.
Horizontal 2
H21,H22
LX2≠LX1
HX2=HX2(X)=AHX2+BHX2X+CHX2X2
Vertical 1
H11,H21
LY1
HY1=HY1(Y)=AHY1+BHY1Y+CHY1Y2
Vertical 2
H12,H22
LY2≠LY1
HY2=HY2(Y)=AHY2+BHY2Y+CHY2Y2

It was difficult for such equation to directly seek, however, if vertical and horizontal one-way non-linear change (HX2Y,HY2X) were sought according to the above train of thoughts and method, then deduct a two-way linear change(HXY), the problem would be solved without any difficulty. Its ordinary seeking method was shown as follows: H(X,Y)=HX2Y+HY2X-HXY， (2)
Among HX2Y(X,Y)=HX1+(HX2-HX1)Y/LY1，HY2X(X,Y)=HY1+(HY2-HY1)X/LX1，HXY(X,Y) was shown as formula(1)
Insert the above 3 formulas into formula (2), upon completion, could gain screwy surfaces equation as:
H(X,Y)=AH+BHxX+BHyY+CHxxX2+CHxyXY+CHyyY2+DHxxyX2Y+DHxyyXY2 (2.1)
Formula，AH=H11，BHx=BHX1，BHy=BHY1，CHxx=CHX1，CHxy=(BHX2-BHX1)/LY1+(BHY2-BHY1)/LX1-(H22-H21 -H12+H11)/LX1/LY1，CHyy=CHY1，DHxxy=(CHX2-CHX1)/LY1，DHxyy=(CHY2-CHY1)/LX1。
The above seeking method took lead in realizing various screwy surface equation accurate expression and the relevant operations.
2. A brief introduction to box transformation coordinate system
Apply the above seeking train of thoughts and method to conventional rectangular coordinate system, and gain universal box transformation coordinate system, the following is a brief introduction:
2.1 Box transformation coordinate system’s mathematic agreement

Chart 5 Linear box transformation coordinate system
KX(Y)=1+KXBY,KY(X)=1+KYBX,KH(X,Y)=1+KHxX+KHyY+KHxyXY

Box transformation coordinate system could take a certain standard positive box body (real line in Chart 5) as norm or primary type(called base box body), and was made through corresponding net form cutting according to geometric proportion(broken line in Chart 5), out of which, coordinate net corresponding to base coordinate net was called norm net. The box transformation coordinate system also had plane (box surfaces) and space (box body) coordinate systems. In box transformation coordinate system, base box body’s length, width and height on the 3 coordinate axes were respectively called base length, base width and base height, which could uniformly called base distance.

In box transformation coordinate system, coordinate unit on the coordinate axis was called basic value unit, basic value coordinate or coordinate unit amount on the same coordinate net line or net surface were completely equal. Homonymy coordinate units outside coordinate axis were completely variable(called variable unit), corresponding coordinate value was called variable coordinate; and accordingly, homonymy coordinate units within conventional rectangular coordinate system were completely equal (could be called equivalence unit), corresponding coordinate value could be called equivalence coordinate. Equivalence coordinate and variable coordinate in the same box transformation coordinate system should have the same position and equivalence. The proportion between variable unit and homonymy basic value unit was called coordinate coefficient or variable coefficient(usually expressed by K ),namely, variable coefficient = variable unit / basic value unit = variable coordinate / basic value coordinate, or :variable coordinate = basic value coordinate ´ variable coefficient, this formula could be called variable formula. In order to facilitate distinction, basic value coordinate was usually expressed with capital letters (such as X, Y, Z).
Various coordinate coefficients of box transformation coordinate system could have first item as a non-0 constant unary or binary elementary function, out of which, first item (KA) expressed equivalence part (first item usually as 1), the rest various items were called surplus item (KR or R) expressed variable part. In case of K≡KA≡1, box transformation coordinate system would change into Cartesianism equivalence coordinate system.
2.2 Box transformation coordinate system establishing method
To establish box transformation coordinate system, it is necessary to establish Cartesianism coordinate system, then seek for its coordinate coefficient and make proper marks. The key here lied in how to seek for coordinate coefficient, usually with 2 methods, namely, setting method and base box method. Setting method (in case there was no base box body) could directly set coordinate coefficient as a certain continuous elementary function with its first item as 1 according to practical situation, then, select a group of base distances and gain a certain base box body. Base box method (in case there was base box body) usually combine a certain base box body’s base sides and coordinate plane, and seek for function formula of the long distance, wide distance and high distance (change side equation into variable coordinate),then, divided by homonymy base distance (function