Integration on hypercircles in r n

(1)

INTEGRATION ON HYPERCIRCLES IN R

ⁿ

Grzegorz Biernat¹, Magdalena Ligus², Jarosław Siedlecki¹

1Institute of Mathematics, Czestochowa University of Technology, Poland

2Faculty of Mechanical Engineering and Computer Science Czestochowa University of Technology, Poland

Abstract. This paper presents the formulas parametrizing hypercircles (the intersections of hyperplanes with the sphere in R , n ≥ 3). A hypothesis concerning the integral on hyper-ⁿ circle Cn–2 of the

n–2

- canonical form ωn–2 is proposed.

1. Parametrization of hypercircles

A plane circle with the centre at 0 and radius r, has the given parametrization

 



=

= t r y

t r x

sin

cos ( 0 ≤ t ≤ 2 π ) (1)

How about the case of a circle in 3-dimensional space? It is easy to imagine, but how to describe it? Certainly, as an intersection of a plane with a sphere. If we restrict our considerations to circles with the centre at 0, then the system of equation is given as

) 0 , (

) 0 ,

( 0

2 2 2 2

2 2 2

>

= + +

>

+ +

= + +

r sphere r

z y x

C B A plane Cz

By

Ax (2)

The parametrizations, mentioned above, are difficult to apply in integration around circle. Thus, parametric description is used. The construction follows.

1° A = B = 0 . Then, the circle is described by parametrization

 

 



=

0 sin cos

z t r y

t r x

( 0 ≤ t ≤ 2 π ) (3)

2° A

²

+ B

²

> 0 . In this case, we construct on the plane Ax + By + Cz = 0 a set of

orthogonal vectors [ B − , A , 0 ] , [ AB , AC , − A

²

− B

²

] and [ A , B , C ] . We obtain the

following orthonormal basis in the space

R : ³

(2)

 





 





+

− +

= , , 0

2 2 2 1 2

B A

A B

A

v B (4)

 





 





+ +

⋅ +

−

− +

+

⋅ + +

+

⋅ +

=

2 2 2 2 2

2 2 2

2 2 2 2 2 2 2 2

2 2

, ,

C B A B A

B A C

B A B A

BC C

B A B A v AC

(5)

 





 





+ + +

+ +

+

=

2 2 2 2 2 2 2 2 2

3

, ,

C B A

C C

B A

B C

B A

v A (6)

The parametrized circle in the above coordinate system has a typical description given by formula (3).

We obtain the system of equations

 







 







=

 







 







⋅

 





 





+ + +

+ +

+

+ + +

−

− +

+ +

+ + +

+

− +

0 sin cos

0

2 2 2 2

2 2 2

2 2

2 2 2 2 2

2 2 2

2 2 2 2 2 2 2 2 2

2 2 2

2

t r

z y x

C B A

C C

B A

B C

B A

A

C B A B A

B A C

B A B A

BC C

B A B A

AC

B A

A B

(7) where the determinant of the coefficient matrix is equal to 1.

The solution of the system looks as follows

 





 





+ + +

+ +

= t

C B A B A t AC B

A r B

x cos sin

2 2 2 2 2 2

2

(8)

 





 





+ + +

+ +

−

= t

C B A B A t BC B

A r A

y cos sin

2 2 2 2 2 2

2

(9)

C t B A

B r A

z

₂ ₂ ₂

sin

2 2

+ +

− +

= (10)

where 0 ≤ t ≤ 2 π .

(3)

Thereafter, we consider the case, where hyperplane H

_n

is defined by 0

2

...

2 1

1

x + A x + + A

_n

x

_n

=

A , A

₁²

+ A

₂²

+ ... + A

_n²

> 0 , n ≥ 3 (11) and sphere S

_n

intersects

R ⁿ

2 2 2

2 2

1

x ... x r

x + + +

_n

= , r > 0 (12)

Let us refer to these as intersections hypercircles C

_n₋₂

(for n = 3 we obtain a circle).

Obviously, hypercircles C

_n₋₂

have the dimension n – 2. Next, the following orthogonal versors can be constructed (the first n – 1 on hyperplane H

_n

).

Let us assume that A

₁

≠ 0 . The first versor is defined as

 





 





+

− +

= , , 0 ,..., 0

2 2 2 1

1 2

2 2 1 2

1

A A

A

v A (13)

Certainly,

₁

v is orthogonal to the vector [ A

₁

, A

₂

,..., A

_n

] and v ⊂

₁

H

_n

. We determine versor v

₂

by adopting the following formula

 



 +

+ +

−

 





+ + +

=

0 ,..., 0 , ) )(

(

, ) )(

( , ) )(

(

2 3 2 2 2 1 2 2 2 1

2 2 2 1

2 3 2 2 2 1 2 2 2 1

3 2 2

3 2 2 2 1 2 2 2 1

3 1 2

A A A A A

A A

A A A A A

A A A

A A A A

A v A

(14)

versor v

₂

is orthogonal to vectors [ A

₁

, A

₂

,..., A

_n

] , v

₁

and v ⊂

₂

H

_n

. The third versor is defined as follows

( )( ) ( )( )

( )( ) ( )( )   



 +

+ + +

+

−

− +

+ + +

+

 







+ + + +

+ +

+ + +

+

=

0 ,..., 0 , ,

, ,

2 4 2 3 2 2 2 1 2 3 2 2 2 1

2 3 2 2 2 1 2

4 2 3 2 2 2 1 2 3 2 2 2 1

4 3

2 4 2 3 2 2 2 1 2 3 2 2 2 1

4 2 2

4 2 3 2 2 2 1 2 3 2 2 2 1

4 1 3

A A A A A A A

A A A A

A A A A A A

A A

A A A A A A A

A A A

A A A A A A

A v A

(15)

Versor v

₃

is orthogonal to vectors [ A

₁

, A

₂

,..., A

_n

] , v

₁

, v

₂

and v ⊂

₃

H

_n

.

(4)

Generally, we obtain (applying the vector product [1, 5])

( )( ) ( )( )

( )( )   



 +

+ +

+

−

 







+ + +

+ +

+

=

+

+ +

0 ,..., 0 , ,

, ,

2 1 2

1 2 2

1

2 2

1

2 1 2

1 2 2

1

1 2

1 2 2

1

1 1

k k

k

k k

A A

A A A

A v A

K K

K

K K

K

(16)

for 1 ≤ k ≤ n − 1 (if k = n − 1 , then the last component is not 0).

Evidently versor v

_k

is orthogonal to the vectors [ A

₁

, A

₂

,..., A

_n

] , v

₁

,…, v

_k₋₁

and

n

k

H

v ⊂ . The last versor

 





 





+ + + +

+ + +

+ +

=

2 2

2 2 1 2

2 2 2 1

2 2

2 2 2 1

1

...

,..., ...

,

...

_n

n n

n

A A A

A A

v A (17)

is orthogonal to hyperplane H

_n

. Let us define matrix U as follows

[ v v v v

_n

]

^T

U : =

₁

,

₂

,

₃

,..., (18)

Next, we recall sphere parameterization [1, 2, 4]

1 1 2 1

1 3

2 1

1 2

1 1

sin cos sin

cos ...

cos sin

cos ...

cos sin

cos ...

cos cos

−

=

n n

n n n

t r x

t t r x

t t t r x

...

... (19)

where 0 ≤ t

₁

≤ 2 π ,

2 2

π π ≤ ≤

− t

_j

, 2 ≤ j ≤ n − 1 .

We calculate the hypercircle parametrization by solving the following system of

equations

(5)

 





 





=

 





 





⋅

−

0 sin

..

...

cos ...

cos sin

cos ...

cos sin

cos ...

cos cos

...

2 2 2

1

2 2

1

2 2

1

1 3 2 1

n n n n

n

r t

t t t r

x x

x x x

U (20)

Because matrix U is orthogonal, the solution is given by vector

 





 





⋅

=

 





 





−

0 sin

..

...

cos ...

cos sin

cos ...

cos sin

cos ...

cos cos

...

2 2 2

1

2 2

1

2 2

1

1 3 2 1

n n n n

T

n

r t

t t t r

U

x x

x x x

(21)

2. Integration around hypercircles

We calculate some integrals around circle C

1

in R

³

and hypercircle C

2

in R

⁴

[6].

Using the results from the previous section (three dimensional case of intersection of a sphere with the equation x

²

+ y

²

+ z

²

= r

²

of the plane Ax + By + Cz = 0 where A

²

+ B

²

+ C

²

> 0) we can calculate the following integral:

∫ ₊ ⁻ ₊ ⁺ ₊ ⁻ ₊ ⁺ ₊ ⁻ ₊

1

2 2 2 2

2 2 2

2 2 C

z dz y x

Bx dy Ay

z y x

Az dx Cx

z y x

Cy

Bz (22)

The above integral is given by the formula

∫ ∫

∫

+ +

=

− =

⋅

=

− + ⋅

= + +

+

+ = + + − +

+ + − +

+

−

π π

π

2

0

2

0

2 2 2 2

2

0

2 2 2 2

0

2 2 2

2 2 2 2

2 2 2

2 2

2 cos '

sin sin ' cos

1 0 0

'

cos sin

'

sin cos

1 ' 1

1

C B A dt

C t dt

t t r t

r C

dt C

t r t r B

t r t r A z U

y dt x

dt z dz C

dt y dy B

dt x dx A

z y x

z dz y x

Bx dy Ay

z y x

Az dx Cx

z y x

Cy Bz

T C

(23)

(6)

where

 







 







⋅

=

 







 







C B A U C

B A

' ' '

(24)

Thus, integral (22) is equal to the length of a circle with the radius A

²

+ B

²

+ C

²

[6].

We can now consider and calculate the following integrate on hypercircle C

₂

in R

⁴

:

( ) ( )

( )

, 4

' 4 2 ' 2

sin '

cos '

cos 1 '

0 0

0 '

cos 0

sin '

sin sin cos

cos cos

sin '

sin cos cos

sin cos

cos ' 1

1

2 4 2 3 2 2 2 1 4

4 2

0 2

2

0

2

2 2

0

2 1 4 2

2

1 2 2 4

1 2 2 3 3 4

2

0 2

2

1 2

4

2 2

3

2 1 2

1 2

2 1 2

1 2

1 1

2 3 2

2

0 2

2

1 2

2 4 1

4 4 4

2 3 1

3 3 3

2 2 1

2 2 2

2 1 1 1 1 1

2 3 2 4 2 3 2 2 2 1

4 3 2 3 2 4 2 3 2 2 2 1

1 2

4 2 2 3 2 4 2 3 2 2 2 1

1 3

3 2 2 3 2 4 2 3 2 2 2 1

1 4

4 1 2 3 2 4 2 3 2 2 2 1

2 3

3 1 2 3 2 4 2 3 2 2 2 1

2 4

2 1 2 3 2 4 2 3 2 2 2 1

3 4

2

A A A A A

A

t dt A dt dt t A

dt dt t r r A

dt dt A

t r t

r A

t t r t t r t t r A

t t r t t r t t r A U r

dt dt

t x t x x A

x x x x

dx dx x x x x

x x

A A dx

dx x x x x

x x

A A

dx dx x x x x

x x

A A dx

dx x x x x

x x

A A

dx dx x x x x

x x

A A dx

dx x x x x

x x

A A

T C

+ + +

=

⋅

=

⋅

=

⋅

− =

−

⋅

=

∂

⋅ + + +

=

∧ +

+ +

−

∧ +

+ +

−

∧ +

+ + +

∧ +

+ +

+

∧ +

+ +

−

∧ +

+ +

∫ ∫ ∫ ∫ ∫

∫ ∫

∫

− − −

−

π π

π

π π

π

π π

π

π π

π

π π

π

(25)

(7)

where

 





 





⋅

=

 





 





4 3 2 1

' ' ' '

A A A A U A A A A

(26)

We observe that integral (25) is equal to the surface of a sphere with the radius

2 4 2 3 2 2 2

1

A A A

A + + + [6]. The above results suggest the following hypothesis.

3. Hypothesis

Integral around hypercircle C

_n₋₂

of the n − 2 - canonical form [3]

( )

( ) ⁽ ⁾

2 2

1 2 1

2 1

2 2 1 2 1

2 2 1

2 1

...

det

, , ,..., ...

1

2 1 2 1

,..., , , ... 1

2 1 1 2 2

1

2

sgn

−

∧

 ∧



 





× +

+

= −

₋

≠

≤

<

≤ ≤

<

≤

−

∑

n n n

i i

i j j

j j

n

i i i j j

n j n j

i i i n n n n

dx dx

x dx x

A A

j j i i x

x ω

(27)

is equal to

∫

−

+ +

 +





 



−

= −

− 2

2 2

2 2 1

2

...

] 1 [ )

2

n

(

C

n

A A A

sphere unit imensional d

n

of surface

ω (28)

Parametrizations and some integrals around hyperspheres will be presented in the next paper.

References

[1] Sikorski R., Rachunek róŜniczkowy i całkowy funkcji wielu zmiennych, PWN, Warszawa 1980.

[2] Birkholc A., Analiza matematyczna. Funkcje wielu zmiennych, WN PWN, Warszawa 2002.

[3] Spivak M., Analiza na rozmaitościach, WN PWN, Warszawa 1997.

[4] Leja F., Rachunek róŜniczkowy i całkowy, PWN, Warszawa 1973.

[5] Musielak J., Wstęp do analizy funkcjonalnej, PWN, Warszawa 1989.

[6] Ligus M., Całkowanie na hiperokręgach w Rⁿ (praca dyplomowa), Instytut Matematyki Politech- niki Częstochowskiej, Częstochowa 2009.