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Also included in Octave’s geometry functions are primitive functions to enable vector rotations in 3-dimensional space. Separate functions are provided for rotation about each of the principle axes, x, y, and z. According to Euler’s rotation theorem, any arbitrary rotation, R, of any vector, p, can be expressed as a product of the three principle rotations:
p' = Rp = Rz*Ry*Rx*p
rotx
returns the 3x3 transformation matrix corresponding to an active
rotation of a vector about the x-axis by the specified angle, given in
degrees, where a positive angle corresponds to a counterclockwise
rotation when viewing the y-z plane from the positive x side.
The form of the transformation matrix is:
| 1 0 0 | T = | 0 cos(angle) -sin(angle) | | 0 sin(angle) cos(angle) |
This rotation matrix is intended to be used as a left-multiplying matrix
when acting on a column vector, using the notation
v = T*u
.
For example, a vector, u, pointing along the positive y-axis, rotated
90-degrees about the x-axis, will result in a vector pointing along the
positive z-axis:
>> u = [0 1 0]' u = 0 1 0 >> T = rotx (90) T = 1.00000 0.00000 0.00000 0.00000 0.00000 -1.00000 0.00000 1.00000 0.00000 >> v = T*u v = 0.00000 0.00000 1.00000
roty
returns the 3x3 transformation matrix corresponding to an active
rotation of a vector about the y-axis by the specified angle, given in
degrees, where a positive angle corresponds to a counterclockwise
rotation when viewing the z-x plane from the positive y side.
The form of the transformation matrix is:
| cos(angle) 0 sin(angle) | T = | 0 1 0 | | -sin(angle) 0 cos(angle) |
This rotation matrix is intended to be used as a left-multiplying matrix
when acting on a column vector, using the notation
v = T*u
.
For example, a vector, u, pointing along the positive z-axis, rotated
90-degrees about the y-axis, will result in a vector pointing along the
positive x-axis:
>> u = [0 0 1]' u = 0 0 1 >> T = roty (90) T = 0.00000 0.00000 1.00000 0.00000 1.00000 0.00000 -1.00000 0.00000 0.00000 >> v = T*u v = 1.00000 0.00000 0.00000
rotz
returns the 3x3 transformation matrix corresponding to an active
rotation of a vector about the z-axis by the specified angle, given in
degrees, where a positive angle corresponds to a counterclockwise
rotation when viewing the x-y plane from the positive z side.
The form of the transformation matrix is:
| cos(angle) -sin(angle) 0 | T = | sin(angle) cos(angle) 0 | | 0 0 1 |
This rotation matrix is intended to be used as a left-multiplying matrix
when acting on a column vector, using the notation
v = T*u
.
For example, a vector, u, pointing along the positive x-axis, rotated
90-degrees about the z-axis, will result in a vector pointing along the
positive y-axis:
>> u = [1 0 0]' u = 1 0 0 >> T = rotz (90) T = 0.00000 -1.00000 0.00000 1.00000 0.00000 0.00000 0.00000 0.00000 1.00000 >> v = T*u v = 0.00000 1.00000 0.00000
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