Generating Four Dimensional Rotation Matrices

Generating Four Dimensional Rotation Matrices Mustafa Özdemiry

Melek Erdo¼ gdu

April 7, 2015

Abstract In this paper, Rodrigues formula and Cayley formula are derived for 4 4 skewsymmetric real matrices in E4 . We …nd two di¤erent methods of generating four dimensional rotation matrices. Also, we give a way to …nd the skew-symmetric matrix A for a given rotation matrix R such that R = eA and R = Cay(A): Finally, we classify of the rotation as; simple, double or isoclinic rotation. Keywords: Euclidean 4-Space, Rotation Matrix, Rodrigues Rotation Formula, Cayley Rotation Formula. MSC 2000: 15B10, 15A16, 53B30.

1

Introduction

In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. To perform the rotation using a rotation matrix R, the position of each point must be represented by a column vector v, containing the coordinates of the point. A rotated vector is obtained by using the matrix multiplication Rv. Since matrix multiplication has no e¤ect on the zero vector (the coordinates of the origin), rotation matrices can only be used to describe rotations about the origin of the coordinate system. Rotation matrices are used extensively for computations in geometry, physics, and computer graphics [1-11]. For this reason, the generation of a rotation matrix becomes one of the most important problems in mathematics. There are di¤erent ways to construct rotation matrices in Euclidean 3-space. Quaternions can be used for generating a rotation matrix by a fairly elegant way in [12-19]. On the other hand, Rodrigues rotation formula is a quite useful formula to generate the rotation matrix for a given rotation angle around a given rotation axis [20-23]. This formula allows to compute eA for a 3 3 skew-symmetric matrix A: If we take the skew-symmetric matrix as: 2 3 0 a12 a13 0 a23 5 A = 4 a12 a13 a23 0

Department of Mathematics-Computer Sciences, Necmettin Erbakan University, Konya, TURKEY, e-mail: [email protected] (corresponding author) y Department of Mathematics, Akdeniz University, Antalya, TURKEY, e-mail: [email protected]

1

where u = (a23 ; a13 ; a12 ) is a unit vector, then we get the R=e

A

= I + sin A + (1

cos ) A2

by using the property A3 = A: The matrix R is the rotation matrix where u is the rotation axis and is the rotation angle. A simple way of computing eA for a given skew-symmetric matrix A is stated in [3]. Furthermore, the derivative of a rotation with Rodrigues rotation formula is investigated in [30] by exponential coordinates. Cayley rotation formula is an another formula to construct the rotation matrix for a given skewsymmetric matrix A R = Cay(A) = (I + A)(I A) 1 represents a rotation matrix [24-27]. Unfortunately, there does not appear to be any simple way of obtaining a rotation matrix in four dimensional Euclidean space. In this paper, a Rodrigues rotation formula is derived for 4 4 skew-symmetric real matrices in E4 . For this purpose, we use the decomposition of skew-symmetric matrix A as A = 1 A1 + 2 A2 by two skew-symmetric matrices A1 and A2 satisfying the properties A1 A2 = 0; A31 = A1 and A32 = A2 : First of all, we prove that A1 and A2 are uniquely obtained for a skew-symmetric matrix A. Then, we …nd the rotation matrices with skewsymmetric matrices by Rodrigues rotation formula. And, we give a way to …nd the skewsymmetric matrix A for a given rotation matrix R such that R = eA . Moreover, we give a Cayley rotation formula for 4 4 skew-symmetric real matrices. We …nd the rotation matrices with skew-symmetric matrices by Cayley rotation formula. Then, we give a way to …nd the semi skew-symmetric matrix A for a given rotation matrix R such that R = Cay(A). As a conclusion, we give the classi…cation of rotations, which are obtained by Rodrigues and Cayley rotation formulas, with respect to the values 1 and 2 :

2

Preliminaries

Four dimensional Euclidean space, which is denoted by E4 ; is provided with the inner product hu; vi = u1 v1 + u2 v2 + u3 v3 + u4 v4 for vectors u = (u1 ; u1 ; u3 ; u4 ), v = (v1 ; v2 ; v3 ; v4 ) 2 E4 : For any R 2 M4 4 (R); if hRu; Rui = hu; ui for all vectors u 2 E4 , then R is called a orthogonal matrix. That is, orthogonal matrices preserve the length of vectors in E4 and columns (or rows) of the a orthogonal matrix form an orthonormal base of E4 : Moreover, R is a orthogonal matrix if and only if RRT = RT R = I where I is the identity matrix of order 4. Taking the determinant of both sides of this equation gives det (R) = 1: If det R = 1; then R is called a rotation matrix in E4 . The set of the rotation matrices of E4 can be expressed as follows: SO (4) = fR 2 M4 (R) : RRT = RT R = I and det R = 1g: On the other hand, for any A 2 M4 4 (R); if A satis…es the equality AT = A; then A is called a skew-symmetric matrix in E4 : Skew-symmetric matrices are very important for the generating a rotation matrix using Rodrigues or Cayley rotation formulas [20-27]. Instead of axis of rotations, there is a plane of rotation which is an abstract object and used to 2

describe rotations in E4 . A plane of rotation for a rotation is a plane that is mapped to itself by the rotation. The plane is not …xed, but all vectors in the plane are mapped to other vectors in the same plane by the rotation. They are related to the eigenvalues and eigenvectors of the rotation matrix. Since the rotations in E4 have at most two planes of rotations, then we can classify the rotations depending on plane of the rotation. i. Simple Rotation: A rotation with only one plane of rotation is a simple rotation. In a simple rotation there is a …xed plane, and rotation can be said to take place about this plane, so points as they rotate do not change their distance from this plane. The plane of rotation is orthogonal to this plane, and the rotation can be said to take place in this plane. For example; the matrix 3 2 1 0 0 0 7 6 0 1 0 0 7 6 4 0 0 cos sin 5 0 0 sin cos

…xes the xy-plane. The plane of rotation is the zw-plane, points in this plane are rotated through an angle : ii. Double Rotation: A rotation with two plane of rotations is a double rotation. The rotation can be said to take place in both planes of rotation. These planes are orthogonal. A double rotation has two angles of rotation, one for each plane of rotation. The rotation is speci…ed by giving the two planes and two non-zero angles and (if either angle is zero the rotation is simple). Points in the …rst plane rotate through , while points in the second plane rotate through . All other points rotate through an angle between and . For a general double rotation the planes of rotation and angles are unique. For example; 3 2 cos sin 0 0 7 6 sin cos 0 0 7 6 4 0 0 cos sin 5 0 0 sin cos

is a double rotations where the plane of rotations are xy and zw planes with the angles and ; respectively. iii. Isoclinic Rotation: Isoclinic rotations can be considered as a special case of the double rotation, when the angles are equal, that is = 6= 0. It di¤ers from a general double rotation, since all non-zero points are rotated through the same angle in an isoclinic rotation. Furthermore, the planes of rotation are not uniquely identi…ed. There are in…nitely many number of pairs of orthogonal planes that can be treated as planes of rotation [29]. Firstly, we will prove that any 4 eigenvalues of the form 1 i; 1 i;

4 nonzero skew-symmetric matrix A has the set of 2 2 i; 2 2 i : 1 + 2 > and a unique decomposition as A=

1 A1

+

2 A2

where A1 and A2 are also skew-symmetric matrices and satisfy the properties A31 =

A1 ; A32 =

A2 ; A1 A2 = A2 A1 = 0: 3

This is a well known property and given in the study [4] for n dimensional case. But, we will give the proof of this property by a di¤erent approach. Let A = (aij ) 2 M4 4 (R) be any skew-symmetric matrix that is aij = aji for i; j = 1; 2; 3; 4: Some tedious and direct computations from det ( I A) = 0; we can …nd the eigenvalues of A as 1 i; 1 i; 2 i and 2 i where p q 2 p (K1 K2 )(K1 + K2 ) K1 ; 1 = p2 q p 2 K1 (K1 K2 )(K1 + K2 ): 2 = 2 Here

K1 =

X

a2ij and K2 = 2(a12 a34

a13 a24 + a14 a23 ):

i 0 : Then A can be decomposed as A=

1 A1

+

2 A2

where A1 and A2 are skew-symmetric matrices satisfying the properties A1 A2 = A2 A1 = 0; A31 =

A1 and A32 =

A2 :

(1)

Moreover, the skew-symmetric matrices A1 and A2 are uniquely obtained as 1

A1 =

1(

2 2

2 1)

2 2A

+ A3

1

and A2 =

2 1

2(

2 2)

2 1A

+ A3 :

Proof. Suppose that A is a given 4 4 nonzero skew-symmetric matrix with the set of 2 2 eigenvalues 1 i; 1 i; 2 i; 2 i : 1 + 2 > 0 : Then A has four linearly independent eigenvectors. That is A is diagonalizable. Therefore, we have 1

A = SDS where D = diagf 1 i;

1 i;

2 i;

2 ig:

If we de…ne D1 = i diagf1; 1; 0; 0g and D2 = i diagf0; 0; 1; 1g; then we may write D=

1 D1

+

2 D2 :

Let us de…ne the skew-symmetric matrices A1 = SD1 S A=

1 A1

+

A3 = ( 1 A1 +

2 A2 )( 1 A1 2 2 2 A2 )( 1 A1

+ +

4

and A2 = SD2 S

2 A2 :

2 A2 ) 2 2 2 A2 )

1:

Thus we get (2)

Since D13 = D1 and D23 = D2 ; it follows that A31 = the properties in (1), we have A2 = ( 1 A1 +

1

= =

A1 and A32 =

A2 : By using the

2 2 2 2 1 A1 + 2 A2 3 3 1 A1 2 A2

(3)

Solving the equations (2) and (3), we …nd A1 =

3

1 1(

2 2A

2 1)

2 2

+ A3

and A2 =

1 2(

2 1A

2 2)

2 1

+ A3 :

Rotations by Rodrigues Formula

Rodrigues rotation formula, which is named after Olinde Rodrigues, gives an e¢ cient method for computing rotation matrices in three dimensional Euclidean space by exponential of skew symmetric matrices. In this part, we will give Rodrigues rotation formula for four dimensional Euclidean space by using the decomposition of skew-symmetric matrices in Lemma 1. Theorem 1 Let A = the set of eigenvalues

1 A1

+

2 A2

1 i;

1 i;

R = eA = I + sin

1 A1

be a given 4 2 i; 2i :

+ (1

cos

2 1

2 1 )A1

4 nonzero skew-symmetric-matrix with + 22 > 0 : Then + sin

2 A2

+ (1

cos

2 2 )A2

is a rotation matrix in E4 ; which is generated by Rodrigues rotation formula, with the set of eigenvalues fe 1 i ; e 1 i ; e 2 i ; e 2 i g: Proof. Suppose that A = 1 A1 + 2 A2 is a given 4 4 nonzero skew-symmetric-matrix 2 2 A with the set of eigenvalues 1 i; 1 i; 2 i; 2 i : 1 + 2 > 0 : Let us compute e by using the decomposition of the skew-symmetric matrix A: Since A1 A2 = A2 A1 ; we may write eA = e 1 A1 + 2 A2 = e 1 A1 e 2 A2 : The property A31 = e

1 A1

A1 implies that

=I+

X k 1

=I+

k k 1 A1

k!

1

3 1

1!

3!

= I + sin

1 A1

Similarly, the property A32 = e

2 A2

=I+

X k 1

=I+

5 1

A1

5!

+ (1

cos

2 1

2!

+

4 1

6 1

4!

6!

4 2

6 2

4!

6!

A21

2 1 ) A1 :

A2 implies that

k k 2 A2

k!

2

3 2

1!

3!

= I + sin

+

2 A2

+

+ (1

5 2

A2

5! cos

2 2

2!

+

A22

2 2 ) A2 :

Using the property A1 A2 = A2 A1 = 04 ; we obtain R = [I + sin = I + sin

1 A1 1 A1

+ (1

cos

+ (1

cos

2 1 ) A1 ][I + sin 2 A2 + 2 1 )A1 + sin 2 A2 + (1

5

(1 cos

2 2 ) A2 ] 2 2 )A2 :

cos

Now, we need to show that RT R = I and det R = 1: By using the properties A1 A2 = A2 A1 = 0 and AT1 = A1 and AT2 = A2 ; we get RT R = I + sin

cos 1 )A21 + sin 2 A2 + (1 cos 2 )A22 sin 1 A1 1 A1 + (1 sin2 1 A21 sin 1 (1 cos 1 )A31 + (1 cos 1 )A21 + (1 cos 1 ) sin 1 A31 + (1 cos 1 )2 A41 sin 2 A2 sin2 2 A22 sin 2 (1 cos 2 )A32 + (1 cos 2 )A22 + sin 2 (1 cos 2 )A32 + (1 cos 2 )2 A42 = I + (2-2 cos 1 - sin2 1 -(1- cos 1 )2 )A21 +(2-2 cos 2 - sin2 2 -(1- cos 2 )2 )A22

The identities cos2 1 + sin2 1 = 1 and cos2 2 + sin2 2 = 1 give us RT R = I: Thus R is a orthogonal matrix. Since A is a skew-symmetric matrix, then trace(A) = 0: By using the well known property of exponential matrix det R = det eA = etrace(A) = e0 = 1: Since A has distinct eigenvalues, the eigenvalues of R = eA are e

1i

;e

1i

;e

2i

and e

2i

:

For further information about exponential of a matrix, the readers are referred to [6]. It is essential to note that, construction of rotation matrices with Rodrigues rotation formula was discussed for four dimensional Euclidean space under the condition K1 = 1 in the study [28]. Now, we will express the way of …nding the skew-symmetric matrix A such that eA is a rotation matrix. Firstly, we give the cases that 1 ; 2 6= in the following theorem. Then, we discuss the special case 1 = or 2 = in Remark 1. Theorem 2 Let R be a rotation matrix E4 with the set of eigenvalues fe 1 i ; e 1 i ; ei 2 ; e i 2 g where 1 ; 2 6= : Then we can …nd the skew-symmetric matrix A such that R = eA by the following way. i. If 1 = 0 and 2 6= 0; then A=

ii. If

1

6= 0 and

2

1

6= 0 and

A=

R2 1

(R

RT ):

(R

RT ):

2

= 0; then A=

iii. If

2

2 sin

2

1

2 sin

1

6= 0; then T

R2 2 cos 2 (R RT ) + 4 sin 1 (cos 1 cos 2 )

R2 2

T

R2 2 cos 1 (R RT ) : 4 sin 2 (cos 2 cos 1 )

Proof. Suppose that R is a given rotation matrix in E4 with the set of eigenvalues e 1 i ; e 1 i ; e 2 i ; e 2 i where 1 ; 2 6= : i. If 1 = 0 and 2 6= 0; then we have R = I + sin

2 A2

+ (1

6

cos

2 2 )A2 :

Using the property AT2 =

A2 ; we get RT = I

sin

We obtain

2 A2

1 2 sin

A2 =

+ (1

cos

(R

RT ):

2 2 )A2 :

2

That is A= ii. If

1

6= 0 and

2

2 A2

2

=

RT ):

(R

2 sin

2

= 0; then R = I + sin

By the property AT1 =

1 A1

2 2 )A1

+ (1

cos

(R

RT ):

(R

RT ):

A1 ; we obtain 1 2 sin

A1 =

1

Thus, we have 1

A= iii. If

1

6= 0 and

2

2 sin

1

6= 0; then we have

R = I + sin

1 A1

+ (1

2 1 )A1

cos

+ sin

Since A1 and A2 are skew-symmetric, we have AT1 = identities and properties in (1), we …nd RT = I

sin

1 A1

+ (1

cos

2 1 )A1

2 A2

+ (1

cos

A1 and AT2 =

sin

2 A2

+ (1

2 2 )A2 :

A2 : If we use these

cos

2 2 )A2

So, we obtain RT = 2 sin

R

1

A1 + 2 sin

2

A2 :

(4)

On the other hand, we get R2 = I + 2 sin

1 cos 1 A1

+ 2 sin2

2 1 A1

+ 2 sin

2 cos 2 A2

+ 2 sin2

2 2 A2

and R2

T

=I

2 sin

1 cos 1 A1

+ 2 sin2

2 1 A1

2 sin

2 cos 2 A2

+ 2 sin2

2 2 A2 :

Therefore, we …nd R2

R2

T

= 4 sin

1 cos 1

A1 + 4 sin

2 cos 2

If we solve the equations (4) and (5), then we obtain A1 =

R2

T

2 cos 2 (R RT ) R2 : 4 sin 1 (cos 1 cos 2 ) 7

A2 :

(5)

and A2 =

R2

T

R2 2 cos 1 (R RT ) : 4 sin 2 (cos 2 cos 1 )

Thus, we get A=

T

R2 2 cos 2 (R RT ) + 4 sin 1 (cos 1 cos 2 )

R2 1

R2 2

T

R2 2 cos 1 (R RT ) : 4 sin 2 (cos 2 cos 1 )

Remark 1 Let R be a given rotation matrix in E4 with the set of eigenvalues fe ei ; e i g: This is the case 2 = : Case:1 If 1 = 0; then we have

1i

;e

1i

;

R = I + 2A22 : Thus, we have 1 A22 = (R 2

I):

1 (R I) by B: Now, we need to solve A22 = B where B is known and skew2 symmetric matrix A2 satis…es A32 = A2 : Since A2 has the set of eigenvalues f0; 0; i; ig; then B has the set of eigenvalues f0; 0; 1; 1g: There exists an orthogonal matrix P such that B = P KP T

Let us denote

where K = diagf0; 0; 1; 1g: If we choose 2 0 0 6 0 0 E=6 4 0 0 0 0

3 0 0 7 7: 1 5 0

0 0 0 1

then, we have

K = E2

and

E3 =

E:

So, we get A22 = P E 2 P T = P EP T P EP T : That is A2 = P EP T : Case:2

If

1

6= 0; ; then we have R = I + sin

By using the properties AT1 =

1 A1

+ (1

A1 and AT2 = R

cos

A2 we get

RT = 2 sin

8

2 1 )A1

1 A1 :

+ 2A22 :

(6)

So, we have A1 =

1 2 sin

RT ):

(R 1

If we substitute A1 in equation (6), we get A22 = Let us denote

1 1 T (R + R 2 2

1 1 T (R + R 2 2

2I) +

2I) +

cos 1 1 2 (R 4 sin2 1

cos 1 1 2 (R 4 sin2 1

(RT )2 ) :

(RT )2 )

by C: Thus, we need to solve A22 = C where C is known and skew-symmetric matrix A2 satis…es A32 = A2 : We can …nd A2 by similar way given in previous case. It is essential to note that we can …nd the skew-symmetric matrix A in the case by following the similar way given above remark.

4

1

=

Rotations by Cayley Formula

In 1846, Arthur Cayley discovered a formula to express special orthogonal matrices with skew-symmetric matrices, which is called Cayley rotation formula. In this part, we will describe Cayley rotation formula for four dimensional Euclidean space. In this manner, we will use the decomposition of skew-symmetric matrices, which is given in Lemma 1. Theorem 3 Let A = the set of eigenvalues

1 A1

R = (I + A)(I

A)

+

1 i; 1

2 A2

be a given 4 1 i; 2 i; 2i :

=I+

2 1

4 nonzero skew-symmetric-matrix with + 22 > 0 : Then

2 12 2 2 2 22 2 1 2 A + A + A + A2 1 2 1 + 12 1 + 12 1 1 + 22 1 + 22 2

is a rotation matrix in E4 ; which is generated by Cayley rotation formula. Proof. Suppose that A = 1 A1 + 2 A2 is a given 4 4 nonzero skew-symmetric-matrix with 2 2 the set of eigenvalues 1 i; A) 1 1 i; 2 i; 2 i : 1 + 2 > 0 : Let us compute (I by using the decomposition of the skew-symmetric matrix A: Using the properties in (1), we have A2 = A3 = A4 = A5 = .. .

2 2 1 A1 + 3 1 A1 4 2 1 A1 5 1 A1 +

9

2 2 2 A2 ; 3 2 A2 ; 4 2 2 A2 ; 5 2 A2 ;

Thus, we may write (I

A)

1

=I+

X

Ak

k 1

=I +( +(

2

=I+

1 3 2

+

1

1+

3 5 1 + 1 5 2

)A2 + 2 1

2 A1 +

1+

1

2 2

)A2 +

2 1 4 2 2

2 2 A1 +

1+

1

+

4 6 1 + 1 6 2

2 A2 +

2

A21 A22 2 2

1+

2 2 A2 2

Using the properties in (1), we obtain R = (I + =I+

1 A1

+

2 A2 )

I+

1

1+

2 A1 +

1

2 1

1+

2 2 A1 +

1

2

1+

2 A2 +

2

2 2

1+

2 2 A2 2

2 12 2 2 2 22 2 1 A1 + A21 + A2 + A2 : 2 2 2 1+ 1 1+ 1 1+ 2 1 + 22 2

Now, we need to show that RT R = I and det R = 1: By using the properties A1 A2 = A2 A1 = 0 and AT1 = A1 and AT2 = A2 ; we get 2 1 2 12 A1 + A2 2 1+ 1 1 + 12 1

RT = I

2 2 2 22 A2 + A2 : 2 1+ 2 1 + 22 2

If we use the properties in (1), then we can easily see that RT R = I: Thus R is a orthogonal matrix. Some tedious and direct computations, we can see that det(I + A) = det(I A): So, we get R is a rotation matrix. Lemma 2 Let A be a given 4 ues 1 i; 1 i; 2 i; 2i : f

2 1

4 nonzero skew-symmetric matrix with the set of eigenval+ 22 > 0 : Then R = Cay(A) has the set of eigenvalues

2 (1 + 2 (1 2 (1 + 1 i)2 (1 1 i) 2 i) 2 i) ; ; ; g: 2 2 2 2 1+ 1 1+ 1 1+ 2 1+ 2

Proof. Suppose that A is a given 4 4 nonzero skew-symmetric matrix with the set of 2 2 eigenvalues 1 i; 1 i; 2 i; 2 i : 1 + 2 > 0 : Then A has four linearly independent eigenvectors. That is A is diagonalizable. Therefore, we have A = P DP

1

where D = diagf 1 i;

1 i;

2 i;

2 ig:

So, we can write R = (I + P DP

1

)(I

P DP

1

)

1

= P (I + D)(I

D)

1

P

1

:

By long and tedious calculations, we …nd (I + D)(I

D)

1

= diag(

2 (1 + 2 (1 2 (1 + 1 i)2 (1 1 i) 2 i) 2 i) ; ; ; ): 1 + 12 1 + 12 1 + 22 1 + 22

10

Thus, R is also diagonalizable with the set of eigenvalues 2 (1 + 2 (1 2 (1 + 1 i)2 (1 1 i) 2 i) 2 i) ; ; ; 1 + 12 1 + 12 1 + 22 1 + 22

:

Theorem 4 Let R be a rotation matrix in E4 with the set of eigenvalues f

2 (1 + 2 (1 2 (1 + 1 i)2 (1 1 i) 2 i) 2 i) ; ; ; g: 1 + 12 1 + 12 1 + 22 1 + 22

Then we can …nd the skew-symmetric matrix A such that R = Cay(A) by the following way. i. If 1 6= 0 and 2 6= 0; then (1 + 12 )2 (1 + 22 ) R2 2) 16( 22 1 (1 + 22 )2 (1 + 12 ) R2 + 2 2 16( 1 2)

A=

ii. If

iii. If

1

1

= 0 and

6= 0 and

2

2

2(1 1+ 2(1 1+

(R2 )T (R2 )T

2 2) 2 (R 2 2 1) 2 (R 1

RT ) RT ) :

6= 0; then A=

1+ 4

2 2

A=

1+ 4

2 1

(R

RT ):

(R

RT ):

= 0; then

Proof. Suppose that R is a rotation matrix E4 with the set of eigenvalues f i. If

1

6= 0 and

2

2 (1 + 2 (1 2 (1 + 1 i)2 (1 1 i) 2 i) 2 i) ; ; ; g: 2 2 2 2 1+ 1 1+ 1 1+ 2 1+ 2

6= 0; then we have

R=I+

2 1 2 12 2 2 2 22 A1 + A21 + A2 + A2 : 2 2 2 1+ 1 1+ 1 1+ 2 1 + 22 2

Since A1 and A2 are skew-symmetric, we have AT1 = identities and properties in (1), we …nd R R2

A1 and AT2 =

4 2 4 1 A1 + A2 ; 2 1+ 1 1 + 22 2 8 2 (1 8 1 (1 1) (R2 )T = A1 + 2 2 (1 + 1 ) (1 + RT =

11

A2 : If we use these

(7) 2 2) 2 2 A2 : 2)

(8)

If we solve the equations (7) and (8), then we obtain (1 + 12 )2 (1 + 22 ) 2 16 1 ( 22 1) (1 + 22 )2 (1 + 12 ) A2 = 2 16 2 ( 12 2)

A1 =

R2

(R2 )T

R2

(R2 )T

2(1 1+ 2(1 1+

2 2) 2 (R 2 2 1) 2 (R 1

RT )

and

RT ) :

That is A=

1 A1

2 A2 2 2 (1 + 1 ) (1 + 2 16( 22 1) (1 + 22 )2 (1 + 2 16( 12 2)

= +

ii. If

1

= 0 and

2

+

2 2) 2 1)

R2

(R2 )T

R2

(R2 )T

2(1 1+ 2(1 1+

2 2) 2 (R 2 2 1) 2 (R 1

6= 0; then we have R=I+

By the property AT2 =

2 2 2 22 A + A2 2 1 + 22 1 + 22 2

A2 ; we obtain RT =

R

4 2 A2 1 + 22

Thus, we have A= iii. If

1

6= 0 and

2

2 A2

=

1+ 4

2 2

(R

RT ):

= 0; then we have R=I+

Using the property AT1 =

2 1 2 12 A + A2 : 1 1 + 12 1 + 12 1

A1 ; we obtain R

RT =

4 1 A1 : 1 + 12

So, we get A=

1 A1

=

1+ 4

12

2 1

(R

RT ):

RT ) RT ) :

5

Applications

In this part, we will give an algorithm to generate rotations by Rodrigues and Cayley formulas with mathematica. The algorithm starts to work by giving entries of the upper triangular part of the 4 4 skew-symmetric matrix. Firstly, it …nds the values of 1 and 2 and gives the type of rotations. Then, it obtains the skew-symmetric matrices A1 and A2 and generates two di¤erent rotation matrices, namely Rrod and Rcay : This simple algorithm is given as follows: >f[i_, j_] = Which[i == j, 0, i < j, a[i, j], i > j, -a[j, i]]; >A = Table[f[i, j], {i, 4}, {j, 4}]; >A // MatrixForm >

1

= Min[SingularValueList[A, 4]]

>

2

= Max[SingularValueList[A, 4]]

>If[

1

If[

== 0, Print[This is a simple rotation], 1

==

2,

Print[This is an isoclinic rotation], Print[This is a double rotation]]]

>A1 = If[

1

== 0, DiagonalMatrix[{0; 0; 0; 0}],

>A2 = If[

1

== 0,

1 2

A,

1 2 2( 1

2 2)

1 2 1( 2

2 1)

( 22 A+MatrixPower[A,3])];

( 12 A+MatrixPower[A,3])];

>A1 // MatrixForm >A2 // MatrixForm >Rrod =[IdentityMatrix[4]+Sin[ 1 ]A1 +(1-Cos[ 1 ])MatrixPower[A1 ,2] +Sin[ 2 ]A2 +(1-Cos[ 2 ])MatrixPower[A2 ,2])]; >Rrod // MatrixForm 2

2

2 1 1 >Rcay =[IdentityMatrix[4]+( 1+ 2 )A1 +( 1+ 2 )MatrixPower[A1 ,2] 1

1

2 2 2 2 +( 1+ 2 )A2 +( 1+ 2 )MatrixPower[A2 ,2])]; 2

2

>Rcay // MatrixForm Now, let us consider a numerical example. For given values a12 = 1; a13 = 1; a14 = 1; a23 = 1; a24 = 0 and a34 = 1; we obtain the following skew-symmetric matrix 2 3 0 1 1 1 6 7 6 1 0 1 0 7 7 6 A=6 7 6 1 1 0 1 7 5 4 1

0

13

1 0

where the set of eigenvalues of 2 0 1 6 16 6 1 0 A1 = 6 36 1 1 4 1

2

A is fi; 1

1

i; 2i;

2ig i.e. 2

3

1

= 1 and

0 6 7 1 2 7 16 6 1 7 and A = 6 7 2 36 2 0 1 7 4 5 1 0 1

1 0 1 1

= 2: We …nd 3 2 1 7 1 1 7 7 7: 0 1 7 5 2

1

0

Here, we can easily see that skew-symmetric matrices A1 and A2 satisfying the properties given in (1). The rotation matrix, which is generated by Rodrigues rotation formula, is obtained as follows; 2 3 0:097 33 0:902 41 0:325 71 0:264 77 6 7 6 0:264 77 0:221 49 0:902 41 0:257 88 7 6 7 Rrod = 6 7: 6 0:325 71 0:264 77 0:097 33 0:902 41 7 4 5 0:902 41

0:257 88

0:264 77 0:221 49

The eigenvectors of the rotation matrix corresponding to the eigenvalues 2i 2 are found as 3 =e , 4 =e u1 = (1; 1 + i; 2:0653 u2 = (1; 1 + i; 2:036

10

u4 = (2; 1 + i;

10

1:6965

= ei ;

2

= e i;

+ i; 1 + i);

28

10

u3 = (2; 1 + i; 3:9566

28

1

i; 1

30

i);

2i; 1 + i); 29

10

+ 2i; 1

i);

respectively. Thus, the plane of rotations are obtained as P1 = span(u1 ; u2 )? and P2 = span(u3 ; u4 )? with the rotation angles hand, the rotation matrix

1

= 1 + 2k

Rcay

and

2

2

2

= 2 + 2k ; respectively. On the other

4

6 16 6 2 = 6 56 1 4 4

1

1

2

7 2 7 7 7 4 7 5 1

4

2

2

2

2

3

is generated by Cayley rotation formula with the given skew-symmetric matrix A: The eigenvectors of the rotation matrix corresponding to the eigenvalues 1 = i; 2 = i; 3 4 3 4 3 = 5 + 5 i, 4 = 5 + 5 i are found as v1 = ( 1

i; 2i; 1

i; 2);

v2 = ( 1 + i; 2i; 1 + i; 2); v3 = (1

i; i;

v4 = (1 + i;

1 i;

14

i; 1); 1

i; 1);

respectively. Thus, the plane of rotations are P1 = span(v1 ; v2 )? and P2 = span(v3 ; v4 )? with the rotation angles 1 = 90 + 2k and 2 = 127 + 2k : Notice that, since 1 , 2 are nonzero and 1 6= 2 , then there are two plane of rotations, which means that both of the rotation matrices represent double rotations.

6

Conclusion

In this paper, it is proved that a given nonzero skew symmetric matrix A 2 M4 (R) has the set of eigenvalues 2 2 1 i; 1 i; 2 i; 2i : 1 + 2 > 0 and there are two nonzero skew-symmetric matrices A1 and A2 such that A=

1 A1

+

A1 A2 = A2 A1 = 0 and A3i =

2 A2 ;

Ai

for i = 1; 2 by a new approach. By using this decomposition of skew-symmetric matrices, we give two di¤erent methods to generating rotation matrices with skew-symmetric matrices in E4 : One of them is called Rodrigues rotation formula and the other one is called Cayley rotation formula. The explicit form of the rotation matrices, which are generated by these formulas, are obtained as follows; R = eA = I +

2 X

sin

k

Ak + (1

cos

2 k )Ak ;

k=1

R = (I + A)(I

1

A)

2 X 2 kj =I+ Aj ; 1 + k2 k k;j=1

with the set of eigenvalues

f

n e

1i

;e

1i

;e

2i

;e

2i

o

;

2 (1 + 2 (1 2 (1 + 1 i)2 (1 1 i) 2 i) 2 i) ; ; ; g; 1 + 12 1 + 12 1 + 22 1 + 22

respectively. As a conclusion of this result, we can classify generating rotations with respect to the values 1 and 2 as follows: i. If ii. If iii. If

i

= 0 and 1, 1,

2 2

j

6= 0 (i 6= j); then formulas generates simple rotations;

are nonzero and are nonzero and

1 1

6= =

2; 2;

then formulas generates double rotations; then formulas generates isoclinic rotations.

On the other hand, we give way of …nding the skew-symmetric matrix, which generates a given rotation matrix by Rodrigues and Cayley rotation formulas, separately. Furthermore, we give an example to show how these methods work. 15

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