Canonization of the hypersurfaces of the first and the second degree and application for the maximal absolute and relative inaccuracies

(1)

RESEARCH ARTICLE

CANONIZATION OF THE HYPERSURFACES OF THE FIRST AND THE SECOND DEGREE

AND APPLICATION FOR THE MAXIMAL ABSOLUTE AND RELATIVE INACCURACIES

1

Kiril Kolikov,

2,*

Radka Koleva and

1

Yordan Epitropov

1

_{Plovdiv University “Paisii Hilendarski”, Tzar Assen str. 24, Plovdiv, Bulgaria}

2

_{University of Food Technologies, 26 Maritsa Blvd., Plovdiv, Bulgaria}

ARTICLE INFO ABSTRACT

In this paper we reduce a linear homogeneous form of n unknowns with an orthogonal transformation of the unknowns in a linear homogeneous form of one unknown with an exactly definite positive coefficient. As an application of this result we find the canonical form of an arbitrary hyperplane in the real n-dimensional affine Euclidean space En. This method for the obtaining of the canonical

forms of the hyperplanes is new, since until nowa canonical form of hyperplanes is not defined and is not considered.Besides we find an effective canonical form of the surfaces of the second degree in

n

Е

. Ourmethodfor the canonization of the surfaces of the second degree in

E

_nis effective since it gives the exact coefficients of the canonical form of the surface in a dependent ofthe coefficients of the givensurface equation. As an application we give a canonization of the hypersurfaces of the maximal absolute and relative inaccuracies (errors). Besides this method is different from the known approach in the case of the obtaining of a cylinder.

Copyright©2017, Kiril Kolikov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

INTRODUCTION

The canonization of the surfaces of the second degree in the real n-dimensional affine Euclidean space is well known ((Efimov, 2005; Konstantinov, 2000), and (Shafarevich, 2013)).In the known canonizations geometric interpretations are made while in our paper we offer an algebraic approach. In this paper we reduce with orthogonal transformation of the unknowns a real linear non-zero form of n unknowns in a linear homogenous form of a unknown

with a positive coefficient Giving a definition of a canonical form of a planes in , we apply the above result

and we find the canonical form of a surface of these hyperplanes. Until now a canonical form of hyperplanes is not defined and

it is not considered. We obtain a effective canonical forms of the surfaces of the second degree in . Our approach for an

obtaining of the canonical form in of a surface of the second degree is different from the known methods in a case of a cylinder. Besides this method is effective since it gives the exact coefficients of the canonical form in a dependent of the

coefficients of . As an application of the obtained results we give the canonical forms of the maximal absolute and relative inaccuracies (errors).

1.Reduction of a linear homogenous form in a linear homogenous form of one unknown

In the following result with an orthogonal transformation of the unknowns we reduce a linear homogenous form of unknownsin a linear homogenous formof one unknown.

*Corresponding author: Radka Koleva,

University of Food Technologies, 26 Maritsa Blvd., Plovdiv, Bulgaria. 1 1

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ISSN: 0975-833X

International Journal of Current Research

Vol. 9, Issue, 10, pp.58521-58529, October, 2017

OF CURRENT RESEARCH

Article History:

Received 06thJuly, 2017 Received in revised form 23rd_{August, 2017}

Accepted 28th_{September, 2017}

Published online 17thOctober, 2017

Citation: Kiril Kolikov, Radka Koleva and Yordan Epitropov. 2017.“Canonization of the hypersurfaces of the first and the second degree and application for the maximal absolute and relative inaccuracies”, International Journal of Current Research, 9, (10), 58521-58529.

Key words:

Linear Homogeneous form of n

Unknowns, Orthogonal Transformation of the Unknowns, Canonical forms of the Surfaces of the first and of the second degree, Affine Euclidean space, Maximal Absolute and Relative Inaccuracies.

RESEARCH ARTICLE

CANONIZATION OF THE HYPERSURFACES OF THE FIRST AND THE SECOND DEGREE

AND APPLICATION FOR THE MAXIMAL ABSOLUTE AND RELATIVE INACCURACIES

1

Kiril Kolikov,

2,*

Radka Koleva and

1

Yordan Epitropov

1

_{Plovdiv University “Paisii Hilendarski”, Tzar Assen str. 24, Plovdiv, Bulgaria}

2

_{University of Food Technologies, 26 Maritsa Blvd., Plovdiv, Bulgaria}

n

Е

E

INTRODUCTION

University of Food Technologies, 26 Maritsa Blvd., Plovdiv, Bulgaria. 1

,...,

n

x

f

ISSN: 0975-833X

Key words:

Unknowns, Orthogonal Transformation of the Unknowns, Canonical forms of the Surfaces of the first and of the second degree, Affine Euclidean space, Maximal Absolute and Relative Inaccuracies.

RESEARCH ARTICLE

CANONIZATION OF THE HYPERSURFACES OF THE FIRST AND THE SECOND DEGREE

AND APPLICATION FOR THE MAXIMAL ABSOLUTE AND RELATIVE INACCURACIES

1

Kiril Kolikov,

2,*

Radka Koleva and

1

Yordan Epitropov

1

_{Plovdiv University “Paisii Hilendarski”, Tzar Assen str. 24, Plovdiv, Bulgaria}

2

_{University of Food Technologies, 26 Maritsa Blvd., Plovdiv, Bulgaria}

n

Е

E

INTRODUCTION

University of Food Technologies, 26 Maritsa Blvd., Plovdiv, Bulgaria.

n

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d u

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ISSN: 0975-833X

Key words:

(2)

Theorem1.If is a non-zeroreal linear homogenous formof unknowns , then thereexists an

orthogonaltransformation of the unknownswhich reduces inа linear homogenous form ,where

is a positive real number.

Proof.Let , i.e. . If , thenobviously , and the theorem is proved. If

, then we apply the orthogonal transformation and we obtain , .

Therefore,the theorem holds for .

Suppose that . Let, forаdetermination, it holds .

At first we reduce the form withan orthogonal transformation of , in the linear form

, where are unknowns, i.e. we exceptthe unknown . Namely, we use the normed vector

,

and we form thelinear transformation

, (1)

It is easily to verify, thatthis transformation is orthogonal and that ,i.e. the indicatedorthogonal

transformation excepts theunknown in .

Suppose, that bythe orthogonal transformation from the form (1), where ,it is obtained the linear form

,

where are unknowns and .

In the linear form we exceptthe unknown with theorthogonal transformation

, ,

, where , т.е. .In this way we obtain the linear form

,

where are new unknowns.

The induction is ended. Finally we obtainthe linear formof the kind ,where is anunknownand

is a positive real number. The linear form is obtained with a sequential realizationof the transforms

, i.e. with the product . However it is well known, that the product of orthogonal transformations of

the unknowns is an orthogonal transformation of the original unknowns. Therefore, is an orthogonal transformation of

, whichreduces in , where .

The theorem is proved.

2.Canonical forms of the hypersurfaces of the first degree

The surfaces of the first degree in the real affine Euclidean space are given bythe equation

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(3)

(2)

where and at least .

Definition1.We say, that a plain has a canonical form (canonical equation) if this plain has the form , where is an unknown. Let in an orthogonal coordinate system is given. We notethat the equation in is theplain

defined by the coordinate axes , since every point of satisfies this equation.

In the following result we obtain the canonical form of the plain given by (2).

Theorem 2.In the many-dimensional real affine Euclidean space the following cases hold for the plain .

1)If at least one coefficient is non-zero, then the canonical form of is given by the equation , i.e. this canonical form

the plain, defined by the coordinate axes of some coordinate system of .

2)If , then is the space .

3)If и , then is the empty set.

Proof.Statements 2) and 3) of the theorem are trivial. We shall prove the statement 1). We represent the surface in the form

, where Theorem1 impliesthat there exists an orthogonal transformationwhich reduces

in , .In the last form of we replace with the unknowns

. Consequently obtains the form . We make the transformation which wesupplementwith

, where are unknowns.In this way we obtain a translation in and obtains theform

, i.e. .

The theorem is proved.

4. Canonical forms of the hypersurfaces of the second degree

In the many-dimensional real affine Euclidean space the surfaces of the second degree are given bythe equation

, , (3)

where and at least and are unknowns.

The forms

, , ,

are called a linear and a quadratic part of , respectively. Denote by the rang of the quadratic part of . If , i.e.

in a space , the canonical form of is well known and absolutely trivial.

If and the canonical form of is also well known. We shall consider , in order to do an analogy.We put

and ,

where and are systems of unknowns.

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At first we shall obtain the canonical form of reducing the quadratic part in a canonical form by an orthogonal transformation

, , , . (4)

We replace every by . Thenthe linear part of obtain the form

, , . (5)

In view of Theorem2,we shall suppose, that at least one coefficient , i.e. .Let the characteristic roots

of the quadratic part of aredifferent from zero and . Then we set

. (6)

In the following result we shall obtain the canonical form of the surface of the second degree in by orthogonal

transformations of the unknowns and by translations. We shall suppose that is given by equation (3).

Theorem3.Let be the rang of the quadratic part of the surface of the second degree in the real affine Euclidean space

, .Then the following cases hold, in which the numbers and , , are defined by (5)and (6), respectively.

1)Let .Then the surface is with a canonical equation

, (7)

where is defined by (6) for . Besides the following subcases hold.

1.1)If have the same sign and , then is an ellipsoid(an ellipse,if ).

1.2)Letat least two of the signs of are different and . Then is ahyperboloid (a hyperbola, if ).

1.3)If ,then is acone. Besides, if have the same signs, then is an imaginary cone with only one real point

(0,0,…,0). (If and the signs of and are different, then is a pair intersecting straight lines.)

2)Let .We have two subcases.

2.1)Let . Then thesurface is aparaboloid(a parabola, if ) with a canonical equation

. (8)

Besides if are with the same signs, then is an elliptical paraboloidandif at least two of signs of are

different, then is ahyperbolic paraboloid.

2.2) Let .Thenthesurface is acylinderwith a canonical equation

. (9)

3)Let .Then the surface is acylinder.Besides the following subcases hold.

3.1)If at least one of the coefficients is non-zero, , then is a parabolic cylinder and has a canonical equation

. (10)

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n

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0

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b

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n

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1 1

z

...

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z

n 1

2 b z

n n

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 

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1

,...,

n 1



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1

,...,



n1

f

0

n

b



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2 2

1 1

z

...

n 1

z

n 1

c



 



 

 

1   

r

n

2 f

i

b

i

 

r

1,...,

n

f

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r

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(5)

3.2)If ,then the canonical form of is

. (11)

Proof.Afterthereductionof inacanonicalformby orthogonal transformation (4), equation (2), in view of (5), obtains the form

.

In this equation we make the following transformation

, , .

(12) We add the last equalities with and we obtain a translation of the surface (12). In this way (12) obtains the form

, (13)

where .

We shall consider the different cases of the theorem.

1)Let . Then (13) obtains the form (7) andcase 1 of the theorem is fulfilled together with the consider subcases. 2)Let . Equation (13) obtains the form

. (14)

We consider the following subcases.

2.1) Let . Since

,

then by making the translation , , (14) obtains form (8). Besides the subcaseshold for

elliptical and hyperbolic paraboloids, i.e. statement 2.1 of the theorem is fulfilled.

2.2) Let . Then (14) obtains form (9), i.e. is a cylinder and condition 2.2of the theorem is fulfilled.

3) Let . We consider the following subcases.

3.1) Let at leastone coefficient in (13) is non-zero, .Then, byTheorem1,

wetransformthelinearhomogenouspartof(13) inalinearhomogenousform of one unknown , applying an orthogonal

transformation of the unknowns , expressed by and supplemented with . In this way we

obtain the equation

.

For this equation we make the translation , .Then (13) Obtain form (10),i.e. is a cylinder

and subcase 3.1 of the theorem holds.

3.2) Let . Then is also a cylinder and (13) obtain form (11), i.e. subcase 3.2 of the theorem is fulfilled.

The theorem is proved.

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r

2 2 2 2

1 1

t

...

r r

t

st

n

c

0 ,

s

b

r 1

...

b

n

0 

 





 





 



1 1

,...,

n 1 n 1

u



t

u

_



t

_

u

_n

t

_n

c

s

 

f

1

...

0

r n

(6)

We note that , then the surface belong to aparabolicclass.

5. Hypersurfaces of the maximal absolute and relative inaccuracies

In this section instead of an error of a physical experiment we shall use the concept an inaccuracy ((Kolikov et al., 2010; Kolikov et al., 2010; Kolikov et al., 2015)).

Let be an indirectly measurable variable depending on the directly measurable variables . Denote by the

real function of arguments such that . Let observations of of

are made in an experimental investigation.

The maximal absolute inaccuracy by the method of (Kolikov et al., 2010) is

, (15)

where

(16)

and

, (17)

where are the maximal absolute inaccuracies of the directly measurable variables.

The maximal relative inaccuracy of , according to (Kolikov et al., 2012), is

, (18)

where

(19)

and

, (20)

where are the maximal relative inaccuracies of the directly measurable variables.

We note, that and in (16) and (19) are the values of and ,

respectively, calculated in the observation and and are the arithmetic mean of these values for .

The maximal absolute inaccuracy and the maximal relative inaccuracy of the second order of

according to (Kolikov et al., 2015) are

r



n

f

Y

X X

₁

,

₂

,...,

X

_n

f

i

X



i



1, 2,...,

n



Y



f X X



₁

,

₂

,...,

X

_n



k

x x

_i₁

,

_i₂

,...,

x

_ik

i

X



i



1, 2,...,

n



1

1 n

i i

i

Y

A

X



 







1



1

1 ,...,

,...,

,

1,...,

k

i m im nm

m i

f

A

x

i

n

k



X









1

1 ,

1, 2,...,

k

i ij

j

X

x

i

n

k













ij

x



1

Y



Y

1

1 n

i i

X

Y

B

Y



X



_









1



1 1

1 ,...,

,

1,...,

,...,

k

im

i m nm

m m nm i

x

f

B

x

i

n

k



f x

x

X









1

1 ,

1, 2,...,

k ij i

j

i ij

x

X

i

n

X

k



x









ij

x





1m

,...,

nm



i

f

x

X



_

_



1



1

,..., ,...,

im

m nm m nm i

x f

x x f x x X



 i

f X

 

i i X f

f X

 

th

m

A

_i

B

_i

m



1, 2,...,

k

2

Y



2

Y





1, 2,..., n



(7)

и , (21)

respectively, where of and of are given by the equalities:

(22)

and

(23)

respectively.

In (22) and (23) and arethe values of and , respectively,

calculatedin them-thobservationand is the arithmetic mean of these values for .

The maximal absolute inaccuracy of of the second approximation we call the function

. (24)

The maximal relatively inaccuracy of of the second approximation we call the function

. (25)

In (Kolikov et al., 2010; Kolikov, 2012; Kolikov, 2015) we assume that and in (15) and (18) are unknown

values with constant coefficients. Then (15) and (21) imply, that (24) has the form

. (26)

From (18) and (21) weobtainanalogously, that(25) has the form

. (27)

Wechange with and with in (26) and, analogously, with and with in (27) .

Thenwe obtain, that the maximal absolute and relative inaccuracies of of the second approximation in -dimensional

affine Euclidean space have the general kind

, (28)

where and are non-negative constants, and at least one of the coefficients is distinct from zero ( ).

We shall make an algebraic classification of (28), i.e. a classification of the surfaces of the maximal inaccuracies of of the second approximationasa corollary from the canonization of the surfaces in the many-dimensional affine Euclidean space , i.e. as acorollary of Theorems 2 and 3.

2

, 1 n

ij i j

i j

Y

A

X



 





2

, 1 n

j i ij

i j i j

X

Y

A

Y



X



_

_

ij

A



2

Y

2

Y







2 1 1

1 ,...,

, ,

1, 2,...,

k

ij m nm

m i j

f

A

x

i j

n

k



X X





 











2 1 1 1

1 ,...,

, ,

1, 2,...,

,...,

k

im jm

ij m nm

m m nm i j

x x

f

A

x

i j

n

k



f x

x

X X





 







2

1m,..., nm i j f x x X X   









2 1 1 ,..., ,..., im jm m nm m nm i j

x x f

x x f x x X X

   2 i j f X X    2 i j i j

X X f f X X

  

ij

A

m



1, 2,...,

k

Y



Y

1

2

2 Y

Y

    

Y



Y

1 2

1

2 Y

Y







i X  i i X X



_

_i__{1, 2,...,}_n

_

1 , 1

1 .

.

2

n n

i i ij i j

i i j

Y

A

X

A

X

 

 











1 , 1

1 .

.

2

n n j i i i ij

i i i j i j

X

Y

B

Y



X



X



_

_

_

_

Y



y

n1



X

i

x

i

Y



1 n

y

_ i

i

X



i

x



i



1, 2,...,

n



Y



n



1 

1 n

E

_

1

, 1 1

2

n n

n ij i j i i

i j i

y

_

a x x

a x

 









ij

a

_i

a

_ij



a

_ji

i j

,



1, 2,...,

n

Y

(8)

Theorem 4.Let be the rang of the quadratic part of of the surface ofthe maximal absolute (relative) inaccuracy of the

second approximation in the real affine Euclidean space , . Then is from parabolic typeand the following cases hold.

(i)Let . Then the canonical equation of is

, (29)

i.e. is a paraboloid. Besides, ifthecharacteristicroots of arewithsamesigns, then isanellipticparaboloidand if at

least twocharacteristicrootsof are with opposite signs, then is a hyperbolicparaboloid.

(ii)Let . Then the canonical equation of is

, (30)

where aredefinedby (5), i.e. is acylinder.

(iii)If , then the canonical form of is , i.e. is a hyperplane, defined from thecoordinate axes of

same coordinate system of .

Proof.For the obtaining of the result we shall applyTheorem 3 and 2. Under the use of these theoremswe have to change by

, by and by . The equations (28) and (29) we shall obtain directlyfrom the cases 2.1 и 3.1of the defined

Theorem 3.

Really, if , then the canonical equation (8) (of case 2.1) of Theorem 3 obtains the form (29),sinse . The case(i) is

completed.

Let . It holds case 3 only withsubcase 3.1 with equation (10) which obtains the form (30), i.e.case (ii) is completed. If , then case(iii) is obtained by case 1 of Theorem 2.

The proof is competed.

A significant part of the proved Theorem 4 canbeobtainedfromtheknowncanonicalformsofthehyperspacesoftheseconddegree ((Efimov, 2010), (Konstantinov, 2000) and (Shafarevich, 2013)), but indirectly, by same non-trivial additional reasonings. Wenoteexplicitly, thatcase (iii) ofTheorem 4 cannotbeobtainedfromtheknowncanonizationsofthehypersurfaces ((Efimov, 2005), (Konstantinov, 2000) и(Shafarevich, 2013)), sincecase (iii)is obtained from our original Theorem 2 which gives the canonical form of an arbitraryhyperplane in . Furthermore, we indicate in Theorem 4 the exact parameters of the canonical forms.

Corollary 5.If the rang of the quadraticpart of the hypersur face of the maximalabsolute (relative)

inaccuracyofthesecondapproximationin is 1, then the canonical equation of this surface is the parabola , where is

non-zero characteristic root of .

Proof.It holds only case(i)of formulated Theorem4and the canonical equation is obtained from (29) by the changes

and .

Conclusion

Our approach for a canonization of the hypersurfaces of the second degree in the many-dimensional space stress more to the algebraic part of the question since the geometric interpretations are well known. This approach is effective since it gives the exact coefficients of the given equation of the surface. Namely this effectiveness together with Theorem 2, gives us a possibility to obtain the canonical equation of the hypersurfaces of the maximal inaccuracies (errors).

REFERENCES

Efimov, N. And Rosendorn, E. 2005. Linear algebra and multidimensional geometry, Fizmatlit. (in Russian)

Kolikov, K., Krastev, G., Epitropov, Y., Hristozov, D. 2010. Analytically determining of the absolute inaccuracy (error) of indirectly measurable variable and dimensionless scale characterising the quality of the experiment, Chemometr Intell Lab, 102, pp. 15-19 http://dx.doi.org/10.1016/j.chemolab.2010.03.001

r

f

₂

f

1 n

E

_

n



1 f

r



n

f

2 2

1 1

z

...

n

z

n

z

n 1



 







f



₁

,...,



_n

f

₂

f

2

f

1   

r

n

1 f

2 2 2 2

1 1 1

1 ...

,

...

4

r r n r n

z

Sz

S

b



 



 





 



i

b

f

0 r



f

z

_n_₁



0 f

O z

_{1 1}

,...,

O z

₁ _n

1 1 2

...

n 1

O z z

z

_

E

_n_₁

n

E

1

n

E



n



1 b

n

1

1 2

n

b 

r



n

1

1 0 2

n

b  

1   

r

n

1

0 r



n

E

r

f

₂

2

F

y





1

x

2



2

f

2 1

y





x

1

(9)

Kolikov, K., Krastev, G., Epitropov, Y., Hristozov, D. 2010. Method for analytical representation of the maximum inaccuracies of indirectly measurable variable (survey), Proceedings of the Anniversary International Conference “Research and Education

in Mathematics, Informatics andtheir Applications” –REMIA 2010, Bulgaria, Plovdiv, 10-12 December, pp. 159-166, 2010. http://hdl.handle.net/10525/1449

Kolikov, K., Krastev, G., Epitropov, Y., Corlat A. 2012. Analytically determining of the relative inaccuracy (error) of indirectly measurable variable and dimensionless scale characterising the quality of the experiment, CSJM, vol. 20, no. 1, pp. 314-331.http://www.math.md/en/publications/csjm/issues/v20-n1/11035/

Kolikov, K., Epitropov, Y., Corlat, A., Krastev, G. 2015. Maximum Inaccuracies of Second Order, CSJM v.23, no.1 (67). http://www.math.md/en/publications/csjm/issues/v23-n1/11891/

Konstantinov, M., Elements of analytical geometry: Curves and surfaces of the second degree, Sofia, 2000. (in Bulgarian) Shafarevich, I. & Remizov, A., Linear algebra and geometry, Springer, 2013