GEOMETRIC TOOLS FOR COMPUTER GRAPHICS

(MIRI)

Computing Euler angles: an example

Vera Sacristán

The old and the new framese1 = 81, 0, 0<;

e2 = 80, 1, 0<;

e3 = 80, 0, 1<;

v3 = 81, 1, 1<

81, 1, 1<

u3 = v3�Sqrt@v3.v3D

:1

3,

1

3,

1

3>

v2 = Cross@v3, 81, 2, 4<D

82, -3, 1<

u2 = v2�Sqrt@v2.v2D

:2

7, -

3

14,

1

14>

u1 = Cross@u2, u3D

:-22

21, -

1

42,

5

42>

The line of nodes

The line of nodesn = Cross@e3, u3D

:-

1

3,

1

3, 0>

nn = n�Sqrt@n.nD

:-

1

2,

1

2, 0>

The anglescosGamma = nn.u1

�3

7

2

signGamma = Sign@Det@8nn, u1, 80, 0, 1<<DD

1

sinGamma = signGamma Sqrt@1 - cosGamma^2D

5

2 7

cosBeta = e3.u3

1

3

signBeta = 1

1

sinBeta = signBeta Sqrt@1 - cosBeta^2D

2

3

cosAlpha = e1.nn

-

1

2

signAlpha = Sign@Det@8e1, nn, 80, 0, 1<<DD

1

sinAlpha = signAlpha Sqrt@1 - cosAlpha^2D

1

2

The rotations

The rotationsMatrixForm@RotOzGamma = 88cosGamma, -sinGamma, 0<, 8sinGamma, cosGamma, 0<, 80, 0, 1<<D

�3

7

2-

5

2 70

5

2 7

�3

7

20

0 0 1

MatrixForm@RotOxBeta = 881, 0, 0<, 80, cosBeta, -sinBeta<, 80, sinBeta, cosBeta<<D

1 0 0

0 1

3- �

2

3

0 �2

3

1

3

MatrixForm@RotOzAlpha = 88cosAlpha, -sinAlpha, 0<, 8sinAlpha, cosAlpha, 0<, 80, 0, 1<<D

-1

2-

1

20

1

2-

1

20

0 0 1

The resultMatrixForm@Result = Simplify@RotOzAlpha.RotOxBeta.RotOzGammaDD

-2 2

21�2

7

1

3

-1

42-

3

14

1

3

5

42

1

14

1

3

MatrixForm@matrixA = Transpose@8u1, u2, u3<DD

-2 2

21�2

7

1

3

-1

42-

3

14

1

3

5

42

1

14

1

3

MatrixForm@Simplify@Result - matrixADD

0 0 0

0 0 0

0 0 0