Built-in Mathematica Symbol

RotationTransform

RotationTransform[

]
gives a TransformationFunction that represents a rotation in 2D by

radians about the origin.

RotationTransform[

, p]
gives a 2D rotation about the 2D point p.

RotationTransform[

, w]
gives a 3D rotation around the direction of the 3D vector w.

RotationTransform[

, 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[

, {u, v}, ...]
gives a rotation by

radians in the hyperplane spanned by u and v.

MORE INFORMATION

RotationTransform gives a TransformationFunction which can be applied to vectors.

Degree or ° specifies an angle in degrees.

RotationTransform[, {u, v}, p] can be used to specify any rotation about any point p, in any number of dimensions.

Positive in RotationTransform[, {u, v}, p] corresponds to going from the direction of u toward the direction of v.

RotationTransform[] is equivalent to RotationTransform[, {{1, 0}, {0, 1}}].

RotationTransform[, w] is equivalent to RotationTransform[, {u, v}], where uw, vw and u, v, w form a right-handed coordinate system.

RotationTransform[, {u, v}] can effectively specify any element of the -dimensional rotation group . RotationTransform[, {u, v}, p] can effectively specify any element of the n-dimensional special Euclidean group.

EXAMPLES

CLOSE ALL

Basic Examples (3)

A 2D rotation transform by

radians:

Rotate a vector:

Rotate around the z axis:

Rotate a 2D graphic by 30° about the origin:

A 2D rotation transform by

radians:

Rotate a vector:

Rotate around the z axis:

Rotate a 2D graphic by 30° about the origin:

In[1]:=

Out[1]=

Scope (9)

Rotation by

radians about the point {px, py}:

Rotation by

radians around the x axis about the point {px, py, pz}:

A 2D rotation by

in the x, y plane:

A 3D rotation by

in the x, y plane:

A 4D rotation by

in the x, y 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 (1)

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:

Using GeometricTransformation:

Properties & Relations (7)

The inverse of RotationTransform[

, {u, v}] is RotationTransform[-

, {u, v}]:

The inverse of RotationTransform[

, {u, v}] is RotationTransform[

, {v, u}]:

If u or v is not real, the relationship is more complex:

The inverse of RotationTransform[

] is given by RotationTransform[-

]:

The inverse of RotationTransform[

, w] is given by RotationTransform[-

, w]:

The inverse of RotationTransform[

, w] is also given by RotationTransform[

, -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 result of the two possible orders; the result is not zero:

Neat Examples (1)

Rotating a character: