gives a TransformationFunction that represents a rotation in 2D by θ radians about the origin.
RotationTransform [θ,p]
gives a 2D rotation about the 2D point p.
RotationTransform [θ,w]
gives a 3D rotation around the direction of the 3D vector w.
RotationTransform [θ,w,p]
gives a 3D rotation around the axis w anchored at the point p.
RotationTransform [{u,v}]
gives a rotation about the origin that transforms the vector u to the direction of the vector v.
RotationTransform [{u,v},p]
gives a rotation about the point p that transforms u to the direction of v.
RotationTransform [θ,{u,v},…]
gives a rotation by θ radians in the plane spanned by u and v.
RotationTransform
gives a TransformationFunction that represents a rotation in 2D by θ radians about the origin.
RotationTransform [θ,p]
gives a 2D rotation about the 2D point p.
RotationTransform [θ,w]
gives a 3D rotation around the direction of the 3D vector w.
RotationTransform [θ,w,p]
gives a 3D rotation around the axis w anchored at the point p.
RotationTransform [{u,v}]
gives a rotation about the origin that transforms the vector u to the direction of the vector v.
RotationTransform [{u,v},p]
gives a rotation about the point p that transforms u to the direction of v.
RotationTransform [θ,{u,v},…]
gives a rotation by θ radians in the plane spanned by u and v.
Details
- RotationTransform gives a TransformationFunction that can be applied to vectors.
- θDegree or θ° specifies an angle in degrees.
- RotationTransform [θ,{u,v},p] can be used to specify any rotation about any point p, in any number of dimensions.
- Positive θ in RotationTransform [θ,{u,v},p] corresponds to going from the direction of u toward the direction of v.
- RotationTransform [θ] is equivalent to RotationTransform [θ,{{1,0},{0,1}}].
- RotationTransform [θ,w] is equivalent to RotationTransform [θ,{u,v}], where u⊥w, v⊥w, and u,v,w form a right-handed coordinate system.
- RotationTransform [θ,{u,v}] can effectively specify any element of the -dimensional rotation group . RotationTransform [θ,{u,v},p] can effectively specify any element of the -dimensional special Euclidean group.
Examples
open all close allBasic Examples (4)
A 2D rotation transform by θ radians:
Rotate a vector:
Rotate around the axis:
Rotate a 2D graphic by 30° about the origin:
Rotate around the axis:
Scope (9)
Rotation by θ radians about the point {px,py}:
Rotation by θ radians around the axis about the point {px,py,pz}:
A 2D rotation by θ in the plane:
A 3D rotation by θ in the plane:
A 4D rotation by θ in the plane:
A 3D rotation by θ in the plane parametrized by s{1,-1,1}+t{1,1,1}:
This rotates the vector {1,1,1}:
Generate the transform for a symbolic vector, assuming that all quantities are real:
Simplify the result further by assuming that {x,y,z} is a unit vector:
Transformation applied to a 2D shape:
Transformation applied to a 3D shape:
Applications (5)
Basic (2)
Parametrize a great circle passing through the points u and v on a sphere:
The great circle passing through {1,-1,1}/√3 and {1,1,1}/√3 :
This plots the great circle:
This shows the great circle and points on the unit sphere:
Using GeometricTransformation :
Rotating a character:
Image Transformations (3)
Rotate an image about its {0,0} origin using RotationTransform :
Rotate about the image center:
Specify a different center of rotation in the standard image coordinate system:
Rotate a 3D image around the axis:
Rotate a 3D image around the axis:
Properties & Relations (9)
The rotation transformation is an isometric transform, i.e. preserves distances:
The linear part of a rotation transform is given by RotationMatrix :
The matrix for the linear part is OrthogonalMatrixQ for real rotations:
The inverse of RotationTransform [θ,{u,v}] is RotationTransform [-θ,{u,v}]:
The inverse of RotationTransform [θ,{u,v}] is RotationTransform [θ,{v,u}]:
If u or v is not real, the relationship is more complex:
The inverse of RotationTransform [θ] is given by RotationTransform [-θ]:
The inverse of RotationTransform [θ,w] is given by RotationTransform [-θ,w]:
The inverse of RotationTransform [θ,w] is also given by RotationTransform [θ,-w]:
If w is not real, the relationship is more complex:
The composition of rotations is a rotation:
For graphics transformation, use Rotate :
Possible Issues (1)
The order in which rotations are applied is important:
Compare the results of the two possible orders; the result is not zero:
Neat Examples (1)
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:
Related Workflows
- Rotate, Pan and Zoom 3D Graphics
Related Demonstrations
History
Text
Wolfram Research (2007), RotationTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/RotationTransform.html.
CMS
Wolfram Language. 2007. "RotationTransform." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RotationTransform.html.
APA
Wolfram Language. (2007). RotationTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RotationTransform.html
BibTeX
@misc{reference.wolfram_2025_rotationtransform, author="Wolfram Research", title="{RotationTransform}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/RotationTransform.html}", note=[Accessed: 05-December-2025]}
BibLaTeX
@online{reference.wolfram_2025_rotationtransform, organization={Wolfram Research}, title={RotationTransform}, year={2007}, url={https://reference.wolfram.com/language/ref/RotationTransform.html}, note=[Accessed: 05-December-2025]}