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\title{Geometrical Characteristics of the Surfaces on Trapezium-Curved Plans}
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             \author[1]{Dr. V N  Ivanov}

             \author[2]{Imomnazarov  T.S}

             \affil[1]{  Peopleso Friendship University of Russia, Russia.}

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\date{\small \em Received: 11 December 2019 Accepted: 2 January 2020 Published: 15 January 2020}

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\begin{abstract}
        


At the article "Orthogonal curved coordinate system and forming the surfaces on trapezium plans" [1] there is given the method of forming of the orthogonal curved coordinate system at the plane and the methodic of forming of the new forms of the surfaces on the given trapezium curved plans. At the article there are given many pictures of the trapezium curved plans on the base of the different directrix curves and the figures of the surfaces on the given trapezium curved plans and the combinations of the surfaces with conjugated different directrix. The given methodic of the forming of the surfaces may be used in architecture and building for development of thin-walled space constructions in urban and industry building. But for calculation of stress-strain state of thin shell usually there are used the geometrical characteristics of the middle surface of the shell. At this state on the base of the vector equation of the surfaces on the trapezium curved plans there are received the formulas of the coefficients of the fundamental forms and of the curvatures of the surfaces. There are given the examples of the surfaces and there are received the formulas of the coefficients of the fundamental forms and curvatures of the surfaces with concrete directrix and functions of vertical coordinates of the surface.Keywords: plane curve, orthogonal curved coordinate system at the plane, trapezium-curved plan, vector equation of the surface at the trapezium curved plans, geometrical coefficients of the fundamental forms of the surface, curvatures of the surface.

\end{abstract}


\keywords{plane curve, orthogonal curved coordinate system at the plane, trapezium-curved plan, }

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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
quation of the surface on the trapezium-curved plan, coefficients of quadratic forms of the surface.\par
The orthogonal curved system of coordinates at the plane there is formed by the system of the straight lines orthogonal to the plane base curve ( ) ( ) ( )j i r u y u x u + = 0 (Fig. \hyperref[fig_0]{1}).\par
So, the curved-orthogonal coordinates there are organized by the system of the equidistant curves parallel to the based curve and the system of the straight lines orthogonal to the system of the equidistant curves.\par
The equation of the curved coordinate system( ) ( ) v u v , u ? ? = 0 r r ,\textbf{(1)}\par
? is a normal to the base curve, v is the coordinate of the generating curves along the normal to the base curve.\par
The positive direction of the coordinate of straight lines there is taken to the side of the convexity of the base curve, because in the direction of the concavity the straight lines may to cross.\par
Assigning some function of vertical coordinate z(u,v), we receive the vector equation of the surface ( ) v , u ? on the base curved-orthogonal coordinate system at the plane( ) ( ) ( )k r v , u z v u v , u + ? = ? 0 ? . (\textbf{2})\par
For deduction the formulas of the coefficients it's necessary to receive the derivatives of the vector equation and to use the formulas of the classic differential geometry \hyperref[b1]{[2]};? s? = ? 0 r ; 0 r? = ? s ; ? ? s k k s = ? = ? ? ; k s k s ? = ; ? s k ? = ? ? . (\textbf{3})\par
Then receive:\par
( )  The coefficients of the first fundamental forms:k u s u z vk s + + ? = ? ? ; k v v z + ? = ? ? . (\textbf{4})( ) ( ) 2 2 u s u u z vk s E + + ? = = ? ? ; ( ) 2 1 v v v z G + = = ? ? ; ( ) v u v v z z F = = ? ? . (\textbf{5})\par
The unit normal vector of the surface( ) ( )( ) ( ) k m + + ? ? ? = × ? = ? ? ? ? v s u v ? z vk s z 1 1 ,\textbf{(6)}( ) ( ) ( ) 2 2 2 2 1 u v s v ? z z vk s F EG + + + ? = × = ? = ? ? ? is a discriminant of the surface.\par
The second derivatives of the equation:( ) ( ) k uu s s s uu z k vk s k v s + + ? + ? + ? ? = ? ? ? ; k uv s uv z k + = ? ? ; k vv vv z = ? . (\textbf{7})\par
The coefficients of the second fundamental form:( ) ( ) ( ) ( ) ? + ? ? + ? + ? + ? ? = = uu s v s s u s uu z vk s z k vk s z k v s L 2 m ? ; ( ) ( ) ? + ? = = vv s vv z vk s N m ? ; ( ) ( ) ? + ? ? ? = = uv s s u uv z vk s k z M m ? . (\textbf{8})\par
The curvatures of the surface:( ) ( ) ( ) ( ) [ ] 2 2 2 u s uu s v s s u s u z vk s z vk s z k vk s z k v s E L k + + ? ? + ? ? + ? + ? + ? ? = = ; ( ) ( ) 2 1 v vv s v z z vk s G N k + ? + ? = = ; ( ) ( ) [ ] ( ) 2 2 2 1 v u s vv s s u uv z z vk s z vk s k z EG M k + + + ? ? + ? ? ? = = . (\textbf{9})\par
The coordinate system of the investigated surfaces isn't orthogonal and isn't conjugated in common, as the coefficients 0 ? M , F and the coordinate system of the surfaces isn't the lines of principle curvatures of the surface.\par
The investigated system of the surfaces is related to the class of normal surfaces \hyperref[b3]{[4]}\hyperref[b4]{[5]}\hyperref[b5]{[6]} -the surfaces with the system of plane coordinate lines (generating curves) at the normal plane of the directrix curve. At the works \hyperref[b3]{[4,}\hyperref[b4]{5]} there was shown, that only for two kinds of normal surfaces the system of generating curves is the system of principle curvatures: 1-surfaces of rotation -directrix is a straight line, generating lines are circles; 2 -normal surfaces with the system of nonchanged generating curve. This type of surfaces is related to the Monge's surfaces \hyperref[b4]{[5,}\hyperref[b6]{[7]}\hyperref[b7]{[8]}\hyperref[b8]{[9]}\hyperref[b9]{[10]}.\par
If z=z(v) -the generating curve doesn't change during moving in normal plane of the directrix (z u =z uu =0), there will be received the Monge's surfaces:( ) 2 s vk s E + ? = ; ( ) 2 1 v v v z G + = = ? ? ; 0 = F ; ( ) 2 1 v s z vk s + + ? = ? ; ( ) 2 1 v v s s z z k vk s L + + ? = ; 2 1 v vv z z N + = ; 0 = M ; ( ) 2 1 1 v s v s z vk s z k k + + ? == ; ( ) 2 3 2 2 1 / v vv z z k + = . (\textbf{10})\par
The coordinate system of the Monge's surfaces is lines of principle curvatures of the surface.\par
If the generating curve will be a straight line ? = vtg z (? is an angle of slope of generating strait line to the plane of base curve), then there will be received the torus surface of constant slope \hyperref[b9]{[10]}\hyperref[b10]{[11]}\hyperref[b11]{[12]}. Then we'll receive:? = tg z v ; 0 = vv z ; ? = + 2 2 1 1 cos z v ; ? + ? = ? cos vk s s ; ( ) 2 s vk s E + ? = ; ? = 2 1 cos G ; ( ) ? + ? = sin k vk s L s s ; N=0; s v s vk s sin z k k + ? ? == 1 ; k 2 =0. (\textbf{11})\par
If the angle of slope of the generating strait line ?=0, z=0, then will be received the trapezium-curved plate:( )\textbf{2}s vk s E + ? = ; 1 = G ; L=N=0; k 1 =k 2 =0. (\textbf{12})\par
The geometric characteristics of surfaces with concrete directrix and generating curves will be received on the base of the common formulas of coefficient of the surfaces on trapezium-curved plans \hyperref[b2]{(3)}\hyperref[b3]{(4)}\hyperref[b4]{(5)}\hyperref[b5]{(6)}\hyperref[b6]{(7)}\hyperref[b7]{(8)}\hyperref[b8]{(9)}\hyperref[b9]{(10)}\hyperref[b10]{(11)}. On the fig. \hyperref[fig_1]{2} there is shown the surface with ellipse as directrix and generating sine with linier change of its amplitude:( ) ( ) ( )j i r u Y u X u + = 0 ; ( ) u cos a u X = ; ( ) u sin b u Y = ; ( ) d v sin u c v , u z ? ? = 2 ; ? ÷ = 2 0 u ; d v ÷ = 0\par
, c is maximum amplitude of sine curve; d is the width of trapezium curved plan.\par
Determine parameters of the directrix ellipse and derivatives of generative curve:  ? = ? + ? = ? a Y X s 2 2 ; u cos u sin 2 2 2 ? + = ? ; a b = ? ; ? ?? = ? ? 2 a s ; ( ) u sin 2 1 2 ? + = ?? ; 2 3 3 / a s Y X Y X k ? ? = ? ? ? ? ? ? ? ? = ; ? ? = ? = k s k s ; ( ) 2 2 2 2 1 ? ? + ? ? = ?? ? ? ? = ? u sin k s ; d v sin c z u ? ? = 2 ; 0 = uu z ; d v cos d c z uv ? = 2 ; d v cos u d c z v ? = 2 ; d v sin u d c z vv ? ? ? = 2 2 . 
\section[{Coefficients of the fundamental forms:}]{Coefficients of the fundamental forms:}v sin c v a E ? ? + ? ? ? ? ? ? ? ? ? ? + ? = 2 2 2 2 4 ; d v cos u d c G ? + = 2 2 2 2 4 1 ; d v sin u d c F ? ? = 2 8 2 ; d v sin c d v cos u d c v a ? ? + ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? + ? = ? 2 2 2 2 2 2 2 2 4 4 1 - ( ) ? ? µ ? ? ? ? ? ? ? ? ? ? ? + ? + ? ? ? ? ? ? ? ? ? ? ? + ? ? + = = 2 2 2 2 1 2 2 2 d v cos u d v a d v sin u sin v u sin a c L s uu m ? ; ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? = d v sin u v a d c N 2 2 ; ? ? ? ? ? ? ? ? ? ? ? ? + ? + ? ? ? ? ? = 2 1 1 d v cos v a d d v sin c M . (\textbf{12})\par
On the fig. \hyperref[fig_1]{2} there is shown the Monge's surface with evolvent of the circle as directrix:\par
( ) ( )   u sin u u cos a u X + = ; ( ) ( ) u cos u u sin a u Y ? =d v cos d b G v v 2 2 2 1 + = = ? ? ; ( ) d v cos d b v au 2 2 2 1 + + = ? ; ( ) d v cos d b d d v cos v au b L 2 2 2 1 + + = ; d v cos d b d d v sin b N 2 2 2 2 1 + ? = ; ( ) d v cos d b v au d d v cos b k 2 2 2 1 1 + + == ; 2 3 2 2 2 2 2 1 / d v cos d b d d v sin b k ? ? ? ? ? ? ? ? + ? = . (\textbf{13})\par
On fig. \hyperref[fig_6]{4} there is shown the torus surface of constant slope with Bernoulli's lemniscate as directrix:( ) ( ) u cos u aR u X = ; ( ) ( ) u sin u aR u Y = ; ( ) u cos u R 2 2 = , ( )\textbf{4}1 1 / u ? ÷ ? = . ( ) u R a s 2 = ? ; ( ) a u R k 2 3 = ; 3 = s k .\par
The coefficients of fundamental forms and the curvatures of the surface:( ) 2 3 2 ? ? ? ? ? ? ? ? + = v u R a E ; ? = 2 1 cos G ; ( ) ? ? ? ? ? ? ? ? ? + = sin v u R a L 3 2 3 ; ( ) ( )v u R a sin u R k 3 2 3 1 + ? = ; 0 2 = k . (14)\par
Let us consider the linier surfaces which aren't surfaces of constant slope. On fig.  {\ref 5} there are shone wavy linear surfaces with different directrix curves.   ? + + ? ? ? ? ? ? ? ? + ? ? ? ? ? ? + = ? ( ) ( ) ( ) ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? + ? ? ? ? ? ? + + = 1 2 2 2 2 2 2 
\section[{Conclusion}]{Conclusion}\par
The surfaces on the trapezium curved plans are formed by the moving of some generating curve at the normal plane of directrix curve. The generative curve may change its form when it is moving along the directrix, but has the constant wide of the plan. At the article there is received the vector equation of the surfaces on trapezium curved plans. On the base of the vector equation there are received the coefficients of the fundamental forms and the curvatures of the surfaces. If the function of vertical coordinates depends on coordinate parameter of the directrix (the form of generating curve changes at moving along the directrix), then the coordinate system of the surface isn't orthogonal and isn't conjugated. If along directrix there moving unchangeable curve then the coordinate lines of the surface are lines of principle curvatures and this type of surfaces is applied to the class of Monge's surfaces \hyperref[b4]{[5,}\hyperref[b6]{[7]}\hyperref[b7]{[8]}\hyperref[b8]{[9]}. On the base of common formulas there are received the formulas of geometric characteristics of the Monge's surfaces. If at the normal plane of the directrix there is moving a straight line with the constant slope to the directrix plane, then there will be received the torus surface of constant slope. Those type of surfaces belong to the class of Monge's surfaces as well.\par
On the base of common formulas of investigated class there are received the formulas of the surfaces and their geometric characteristics of the surfaces with concrete directrix and generating curves, as for surfaces of common type and so for Monge's and surfaces of constant slope. The using of common formulas made more simple the proses for receiving formulas for concrete surfaces. For every investigated surface there are given their figures.\par
Also there was investigated the type of wavy surfaces formed by the generating straight line which make oscillations at the normal plane of directrix. There are received the formulas of the geometric characteristics of this type of surfaces and given the figures of wavy line surfaces with some directrix lines.\par
The figures of the surfaces were made with using of vector equations of the surfaces in the "MathCad" system \hyperref[b4]{[5,}\hyperref[b12]{13]} \begin{figure}[htbp]
\noindent\textbf{1}\includegraphics[]{image-2.png}
\caption{\label{fig_0}Fig. 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2}\includegraphics[]{image-3.png}
\caption{\label{fig_1}Fig. 2 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-4.png}
\caption{\label{fig_2}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-5.png}
\caption{\label{fig_3}b}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-6.png}
\caption{\label{fig_4}.}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{3}\includegraphics[]{image-7.png}
\caption{\label{fig_5}Fig. 3 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{4}\includegraphics[]{image-8.png}
\caption{\label{fig_6}Fig. 4 :}\end{figure}
 			\footnote{© 2020 Global Journals} 			\footnote{Journal of Researches in Engineering ( ) Volume Xx X Issue II V ersion I E © 2020 Global Journals Geometrical Characteristics of the Surfaces on Trapezium-Curved Plans} 			\footnote{© 2020 Global Journals Geometrical Characteristics of the Surfaces on Trapezium-Curved Plans} 		 		\backmatter  			 \par
The generating straight line at its moving along directrix made a wavy motion at the normal plane of the directrix: a) ( ) ( ) The coefficients of the fundamental forms and curvatures of these surfaces are determined on common formulas (4-9) with using formulas (19) and the fact that N=0, k v =0. If to take cycloid as the directrix (fig.  {\ref 5,d} 			 			  				\begin{bibitemlist}{1}
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