This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# RotationTransform

 RotationTransform[] gives a TransformationFunction that represents a rotation in 2D by radians about the origin. RotationTransformgives a 2D rotation about the 2D point p. RotationTransformgives a 3D rotation around the direction of the 3D vector w. RotationTransformgives a 3D rotation around the axis w anchored at the point p. RotationTransformgives a rotation about the origin that transforms the vector u to the direction of the vector v. RotationTransformgives a rotation about the point p that transforms u to the direction of v. RotationTransformgives a rotation by radians in the hyperplane spanned by u and v.
• Degree or specifies an angle in degrees.
• RotationTransform can be used to specify any rotation about any point p, in any number of dimensions.
• Positive in RotationTransform corresponds to going from the direction of u toward the direction of v.
• RotationTransform can effectively specify any element of the -dimensional rotation group . RotationTransform can effectively specify any element of the -dimensional special Euclidean group.
A 2D rotation transform by radians:
Rotate a vector:
Rotate around the axis:
Rotate a 2D graphic by 30° about the origin:
Rotate around the axis:
A 2D rotation transform by radians:
 Out[1]=
Rotate a vector:
 Out[2]=

Rotate around the axis:
 Out[1]=
 Out[2]=

Rotate a 2D graphic by 30° about the origin:
 Out[1]=

Rotate around the axis:
 Out[2]= Play Animation ▪
 Scope   (9)
A 2D rotation by in the plane:
A 3D rotation by in the plane:
A 4D rotation by in the plane:
A 3D rotation by in the plane parametrized by :
This rotates the vector :
Generate the transform for a symbolic vector assuming that all quantities are real:
Simplify the result further by assuming that is a unit vector:
Transformation applied to a 2D shape:
Transformation applied to a 3D shape:
 Applications   (2)
Parametrize a great circle passing through the points u and v on a sphere:
The great circle passing through and :
This plots the great circle:
This shows the great circle and points on the unit sphere:
Rotating a character:
If u or v is not real, the relationship is more complex:
The inverse of RotationTransform[] is given by RotationTransform:
The inverse of RotationTransform is given by RotationTransform:
The inverse of RotationTransform is also given by RotationTransform:
If w is not real, the relationship is more complex:
The composition of rotations is a rotation:
For graphics transformation use Rotate:
The order in which rotations are applied is important:
Compare the result of the two possible orders; the result is not zero:
Rotate a 3D object about a point p:
Rotate around the axis, in the plane:
Rotate around the axis, in the plane:
Rotate around the axis, in the plane:
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