ONE-TOONE NONLINEAR TRANSFORMATION OF THE SPACE WITH IDENTITY PLANE

Dep. «Grafics», Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnipropetrovsk, Ukraine, 49010, tel. +38 (056) 713 56 49, e-mail malyjanatolij@gmail.com, ORCID 0000-0002-2710-7532 Dep. «Grafics», Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnipropetrovsk, Ukraine, 49010, tel.+38 (067) 724 47 22, e-mail ulchenkotv@ya.ru, ORCID 0000-0003-2354-7765 *Dep. «Grafics», Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnipropetrovsk, Ukraine, 49010, tel. +38 (067) 586 45 74, e-mail pro-f@ukr.net, ORCID 0000-0003-1340-0284 Dep. «Grafics», Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnipropetrovsk, Ukraine, 49010, tel. +38 (067) 774 17 47, e-mail 19brit18@ukr.net, ORCID 0000-0002-1383-9863 Dep. «Grafics», Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnipropetrovsk, Ukraine, 49010, tel.+38 (066) 791 35 94,e-mail simatn@rambler.ru, ORCID 0000-0002-3851-9612


Introduction
The idea of relating two objects provides a powerful tool for learning new objects and their properties, as soon as the rules are set -the law of correspondence between these two objects.Regarding the geometry this law is determined by specifying a definite geometric transformation that transforms one object into another.
Geometric transformations are very diverse.We consider the so-called point transformations of space.In this case, each point in space is assigned with another definite point in the same space, and vice versa.This transformation is called one-toone.
Analytically the point transformation is determined by formulas.
( ) where: ( , , ) X Y Z -coordinates of the initial point of the pre-image, and ( ', ', ') X Y Z -coordinates of the transformed point-image.Functions 1 2 3 , , F F F can be linear or nonlinear.In the first case, the transformation will be one-to-one, in the second case, as a rule, multi-value.

Methodology
The most important, in our view, are one-toone transformations.They allow exploring and studying the properties of the transformed object using the properties of the original object (line, surface, figure) and the properties of the transformation.
Cremona transformations occupy a special place in the set of one-to-one nonlinear transformations; they are named after L.Cremona, who presented a coherent theory of plane non-linear transformations.The fundamental theorem of Cremona plane transformations about the ability to factorize any transformation into quadratic product was proven in the late XIX century.An attempt to prove a similar theorem for Cremona space transformations have been to date unsuccessful.In this regard, we study only some groups of transformations and their particular types.Without going deeply into the theory of Cremona transformations we refer the interested reader to the sources [6,11,12].
At present, much attention is paid to the study and construction of the so-called stratifiable transformations.[2,4,5,13].Construction of oneparameter (stratifiable) transformations is carried out as one-parameter set of plane transformations, both linear and non-linear ones.The plane, in which the specific transformation is prescribed, moves in space by a certain law forming a oneparameter set of planes.The set of such plane transformations makes up the space transformation.
The problem of studying such transformations is relevant both in theoretical terms and in terms of application for constructing the technical forms of parts and aggregates, construction machines running in the flow of liquid or gas (bridge supports, the surface of the turbulent blades of water and gas turbines, surfaces of shells) etc.
The purpose of this work is to design and study the space transformations on the basis of plane transformations that transform straight lines into algebraic curves of any order with ( 1) n − multiple singular point and vice versa.
Before proceeding to the design of space transformations we give some information from the theory of algebraic curves [1,10].
1. Plane algebraic line is a line defined by an algebraic function of the coordinates of its points in the form of: Another way to define the curve is a parametric representation for which its current coordinates are set individually as a function of some parameter: Excluding the parameter t from the equations (2), we obtain the equation of the same curve in the form (1) and vice versa.From equations (1) we can obtain the parametric representation of curve.
2. The highest degree of the polynomial ( , ) F X Z is called the curve order (1).The curve order is determined by the number of curve intersection points with an arbitrary line.
3. The algebraic curve of m -th order, is gener- ally determined by ( ) 4. Two algebraic curves d and h of the order m and n meet at the points , M N respectively.5.The multiple point (irregularity) of curve is called the point, where several curve branches meet: forming double, triple, etc. points according to the order n of the curve.Indecomposable curve of n order cannot have points of multiplicity higher 1 n − and more than ( )( ) An algebraic curve may not have multiple points at all, or have less than the specified limits.
Curve genus or genre is the number p that is the difference between the largest number of double points, which may belong to the curve of this order, and their actual number on a given curve.This definition is equally valid if the curve has the points of higher multiplicity, providing that k is for ( ) If the curve is of zero genre (i.e., it has the maximum possible number of double points), it has an important property: the coordinates of its points can be expressed as rational functions of some parameter.
These curves are called unicursal.Every curve having a point of the highest possible multiplicity ( 1) n − is a unicursal curve.Any line passing through this point intersects the curve only in one more point.Consequently, between the points of this curve and any line we can establish one-to-one correspondence with central projection, if the point ( 1) n − of multiplicity is taken as the projection centre.
6.If the multiple point ( 1) n − is taken as the origin of coordinates, then the curve equation can be written in the following form where n F and ( 1) relation to X and Y to n and 1 n − power respectively.
Let us now construct a stratifiable space transformation generated by the curves of the type (3).In the space rectangular coordinate system 0 xyz (Fig. 1) we plot the curve s in the frontal plane φ .In this system, the curve s will have the equation Let us plot ( 1) n − -the multiple point on the Yaxis at the point 0' , and through the point of its intersection with the axis 1 X A − draw the horizontal projecting line t .
Any line d , passing through the origin of coordinates will intersect the curve s at a single point ' A ( s -unicursal curve), and the straight t at the point A .Thus, all the points of the curve s can be projected at the point of the line t and vice versa, i.e. one-to-one transformation is recognized.In this transformation the straight line t will correspond to the curve s, and vice versa.The curve s will correspond to the straight line t.
The equation of the line d : where K -slope of the straight line.
Solving the combination of equations ( 4) and ( 5) we obtain the coordinates of the point ' A .
( ) ( ) . These coordinates will correspond to the coordinates of the point ( , ) A X Y .Since the coordinate X of the point A on the line t equals the coordinate ' X of the point ' A on the curve s , then the first of them can be determined as the coordinate of the intersection point of the curve s with the axis X .It has to be done in each case of the transformation, having the defined curve s .
For example, the representative of the set of curves s (see Fig. 1) is Maclaurin trisector, thirdorder curve with a double point (( 1) n − -fold )0 ' : To determine the point of intersection of this curve with the axis X we suppose 0 Z = , and then we have: A with its axis X .
Thus, in each particular case we can determine the transformation formulas in the plane.φ .

Findings
In each of the planes φ there is the same plane transformation, that is why the space transformation is stratifiable, and the coordinates Y of the corresponding points remain unchanged.
Any curve s when moving circumscribes the cylinder with a cross-section s .This cylinder within the space transformation corresponds to the profile plane β .Let us construct, first geometrically, a particular form of space transformation.The circumference will act as a unicursal curve.In the space rectangular coordinate system 0 xyz (Fig. 2) we define an arbitrary point  In this plane, on the segment 1 2 A T , as on di- ameter, we draw a circumference f .It is tangent to the lines t and 3 ϕ .We plot the half ray T to the circumference f .Thus, one-to-one transformation between the points t and the circumference s was found.Each point A of the straight line t corresponds on the circumference f a single point ' i A and vice versa.The point 1 T corresponds to the infinite point i A ∞ .The whole circumference f corresponds to the straight line t .Each frontal plane has a similar correspondence, and their set makes a AA are transformed into the cylinder elements.
The algorithm for building the corresponding points on the complex drawing: 1. Produce the frontal plane φ through the set point 1 2

( , )
A A A (Fig. 3); 2. Draw a circumference at x coordinates of this point, as on diameter; 3. Through the point 1 2 ( , ) T T T belonging to φ , draw the straight line A correspond to each other in this transformation [8].Now we form the equation of this transformation.The circle We transfer the origin of coordinates to the point T .
( ) Let us transform this expression Equation ( 6) is the equation of the circle We solve together the equations ( 6), (7) and get , but since 2r X = (Fig. 2), we have: A by the coordinates , X Z of the initial point A .
Substituting into the equation ( 6) instead of the variable X its value from (7) and producing a transformation similar to the above, we obtain: Let us write the formulas of direct space transformation: The same procedure is for the formulas of inversion transformation: .

Originality and practical value
The transformation formulas ( 8) and (9) show that the third order (cubic) transformation transforms the profile plane 2 X r = into the frontal projecting cylinder.This is easily seen by substituting x in its equation with its expression from the first transformation formula (9): The set of front-projecting lines of this plane is transformed into the set of cylinder elements, and the set of horizontally-projecting straight linesinto the set of cylinder circumferences.
The horizontal plane is transformed into the surface of the third order.The complex figure (Fig. 4) shows a horizontal plane γ .Let us con- sider the transformation in the plane ( ) We take an arbitrary point ( ) After completing elementary algebraic transformations of this equation, we obtain the following cissoid equation: This equation shows that the cissoid is an algebraic curve of the 3rd order.It is symmetrically relative to the axis Z , and the line 2 Z a = is its asymptote, and the origin of coordinates is a cusp of the 1st kind [3].
If the plane φ is continuously moved parallel to itself, it results in occurrence of the surface, whose carcass is the set of cissoids and the set of front-projecting straight lines (Figure 5) [9,7].2. These graphic algorithms allow graphically depicting the transformed lines and surfaces.
3. The considered procedure of drawing up analytical formulas of specific transformations allows us to study the transformed surfaces and lines using the methods of analytic geometry.

point 1 A 2 ∞
of the curve 1 s along the axis X we obtain a set of horizontally projecting straight lines Наука та прогрес транспорту.Вісник Дніпропетровського національного університету залізничного транспорту, 2016, № 3 (63) ТРАНСПОРТНЕ БУДІВНИЦТВО doi 10.15802/stp2016/74768 © A. D. Malyi, T. V. Ulchenko, A. S. Shcherbak, Yu.Ya.Popudniak, 2016∩ and the corresponding bundle of curves s .Moving the plane φ with the transformation set on it, parallel to itself so that the multiplicity point( 1)   n − of the curve s would move along the axis Y , we obtain a space transformation, in which 2 ∞ of the projecting lines will match the 2 ∞ of the curves s .The set of projecting straight lines and curves φ are perspective regarding the horizontal projection plane, so this plane in the transformation remains fixed and standard.

Fig. 1
Fig. 1 . The coordinate planes 0 xy and 0 xz are taken as the horizontal and frontal planes of the projections, respectively.Let us plot the frontal plane 1 3( , ) ϕ ϕ ϕ through the point A .

Fig. 2 Fig. 3
Fig. 2 crosses the circumference f at the point ' A .In other words, we have built a central projection of the point A from the centre 1 d d d in the plane φ ; 4. The line d passes through the point A and crosses the circumference s at the point ' A ; 5. Points A and ' the line d relative to the same origin , '

2 ' 2 '
on the plane γ in the planeφ .And according to the known algorithm we graphically build its image 1 ( ' , ' ) A A A .To do this, we draw through the origin of coordinates the line 1 in the transformation.The set of points ' A will make up the curve of the third order -cissoid of Diocles.Using the transformation formulas (9) we write its equation as an image of the straight line 2 Z a = .In the equation of the line we substitute the coordinate Z with its value from the third formula (9):