aloe.math.euler module

Rotation matrices from Euler Angles

see:

J. B. Kuipers “Quaternions and Rotation Sequences”, Princeton University Press, 1999

aloe.math.euler.Rx(rotation_rad)

Provide the rotation matrix around the X (\(\vec{e_1}\)) axis in a cartesian system, with the input rotation angle in radians.

Meaning of Rx acting from left on a COLUMN of VECTORS:

Transformation matrix U for VECTORS. This matrix rotates a set of “old” basis vectors \((\vec{e_1},\vec{e_2},\vec{e_3})^T\) (column) by +RotAngle (right hand rule) to a new set of basis vectors \((\vec{e_1}',\vec{e_2}',\vec{e_3}')^T\) (column)

Meaning of Rx acting from left a COLUMN of COORDINATE VALUES:

1. (N_P_O): coordinates of a fixed vector in a “New” basis that is rotated by +RotAngle (passive rotation)

2. (O_P_O): active rotation of vector coordinates in the same “Old” basis by -RotAngle

aloe.math.euler.Ry(RotAngle)

Provide the rotation matrix around the X (\(\vec{e_1}\)) axis in a cartesian system, with the input rotation angle in radians.

Meaning of Rx acting from left on a COLUMN of VECTORS:

Transformation matrix U for VECTORS. This matrix rotates a set of “old” basis vectors \((\vec{e_1},\vec{e_2},\vec{e_3})^T\) (column) by +RotAngle (right hand rule) to a new set of basis vectors \((\vec{e_1}',\vec{e_2}',\vec{e_3}')^T\) (column)

Meaning of Rx acting from left a COLUMN of COORDINATE VALUES:

1. (N_P_O): coordinates of a fixed vector in a “New” basis that is rotated by +RotAngle (passive rotation)

2. (O_P_O): active rotation of vector coordinates in the same “Old” basis by -RotAngle

aloe.math.euler.Rz(rotation_rad)

Provide the rotation matrix around the X (\(\vec{e_1}\)) axis in a cartesian system, with the input rotation angle in radians.

Meaning of Rx acting from left on a COLUMN of VECTORS:

Transformation matrix U for VECTORS. This matrix rotates a set of “old” basis vectors \((\vec{e_1},\vec{e_2},\vec{e_3})^T\) (column) by +RotAngle (right hand rule) to a new set of basis vectors \((\vec{e_1}',\vec{e_2}',\vec{e_3}')^T\) (column)

Meaning of Rx acting from left a COLUMN of COORDINATE VALUES:

1. (N_P_O): coordinates of a fixed vector in a “New” basis that is rotated by +RotAngle (passive rotation)

2. (O_P_O): active rotation of vector coordinates in the same “Old” basis by -RotAngle

aloe.math.euler.Rz2D(RotAngle)

provides the Z axis (e3) rotation matrix in cartesian systems, input “RotAngle” in radians

Transformation matrix U for VECTORS. This matrix rotates a set of “old” basis vectors \((\vec{e_1},\vec{e_2},\vec{e_3})^T\) (column) by +RotAngle (right hand rule) to a new set of basis vectors \((\vec{e_1}',\vec{e_2}',\vec{e_3}')^T\) (column)

Meaning of Rx acting from left a COLUMN of COORDINATE VALUES:

1. (N_P_O): coordinates of a fixed vector in a “New” basis that is rotated by +RotAngle (passive rotation)

2. (O_P_O): active rotation of vector coordinates in the same “Old” basis by -RotAngle

aloe.math.euler.euler_tsl2global(phi1_tsl, Phi_tsl, phi2_tsl)

transform euler angles form edax-tsl software to global reference system

see: M. Jackson etal. Integrating Materials and Manufacturing Innovation 2014, 3:4 Page 8 of 12

http://www.immijournal.com/content/3/1/4

aloe.math.euler.rand_rotation_matrix(deflection=1.0, randnums=None)

Creates a random rotation matrix.

deflection: the magnitude of the rotation. For 0, no rotation; for 1, competely random rotation. Small deflection => small perturbation. randnums: 3 random numbers in the range [0, 1]. If None, they will be auto-generated.