Rodrigues' rotation formula - AbsoluteAstronomy.com

In the theory of three-dimensional rotation, Rodrigues' rotation formula (named after Olinde Rodrigues

Olinde Rodrigues

Benjamin Olinde Rodrigues , more commonly known as Olinde Rodrigues, was a French banker, mathematician, and social reformer.Rodrigues was born into a well-to-do Sephardi Jewish family in Bordeaux....

) is an efficient algorithm for rotating a vector in space, given an axis

Axis angle

The axis-angle representation of a rotation, also known as the exponential coordinates of a rotation, parameterizes a rotation by two values: a unit vector indicating the direction of a directed axis , and an angle describing the magnitude of the rotation about the axis...

and angle of rotation

Angle of rotation

In mathematics, the angle of rotation is a measurement of the amount, the angle, that a figure is rotated about a fixed point, often the center of a circle....

. By extension, this can be used to transform all three basis vectors to compute a rotation matrix from an axis-angle representation. In other words, the Rodrigues formula provides an algorithm to compute the exponential map

Exponential map

In differential geometry, the exponential map is a generalization of the ordinary exponential function of mathematical analysis to all differentiable manifolds with an affine connection....

from so(3) to SO(3) without computing the full matrix exponent.

If v is a vector in

and k is a unit vector describing an axis of rotation about which we want to rotate v by an angle θ (in a right-handed sense

Right-hand rule

In mathematics and physics, the right-hand rule is a common mnemonic for understanding notation conventions for vectors in 3 dimensions. It was invented for use in electromagnetism by British physicist John Ambrose Fleming in the late 19th century....

), the Rodrigues formula is:

Derivation

Given a rotation axis z represented by a unit vector k = (k_X, k_Y, k_Z), and a vector v = (v_X, v_Y, v_Z) that we wish to rotate about z, the vector

is the component of v parallel to z, also called the vector projection of v on k, and the vector

is the projection of v onto the xy plane orthogonal to z, also called the vector rejection of v from k.

Notice that we chose to define a reference frame

Reference frame

Reference frame may refer to:*Frame of reference, in physics*Reference frame , frames of a compressed video that are used to define future frames...

xyz in which the z axis is aligned with the rotation axis, and the x axis with the rejection of v from k. This simplifies the demonstration, as it implies that v lies on the xz plane, and its component v_y is null (see figure). However, xyz does not coincide with the reference frame XYZ in which vectors v, k, v_x, and v_z are actually represented. For instance, v = (v_X, v_Y, v_Z) ≠ (v_x, v_y, v_z). In other words, Rodrigues' formula is independent of the orientation in space

Orientation (geometry)

In geometry the orientation, angular position, or attitude of an object such as a line, plane or rigid body is part of the description of how it is placed in the space it is in....

of the reference frame XYZ in which v and k are represented.

Next let

.

Notice that v_x and w have the same length. By definition of the cross product

Cross product

In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...

, the length of w is:

where Φ denotes the angle between z and v.
Since k has unit length,

This coincides with the length of v_x, computed trigonometrically as follows:

So w can be viewed as a copy of v_x rotated by 90° about z. Using trigonometry, we can now rotate v_x by θ around z to obtain v_{x rot}. Its two components with respect of x and y are v_xcosθ and wsinθ, respectively. Thus,

v_{x rot} can be also described as the projection on xy (or rejection from z) of the rotated vector v_rot (see figure). Since v_z is obviously not affected by a rotation about z, the other component of v_rot (i.e. its projection on z) coincides with v_z. Thus,

as required.

Matrix notation

By representing v and k as column matrices, and defining a matrix

as the "cross-product
matrix" for

, i.e.,

,

Rodrigues' formula can be written in matrix notation:

Using the triple product expansion in can be written as:

since

for a normalized vector.

Conversion to rotation matrix

The equation can be also written as

where I is the 3×3 identity matrix. Thus we have a formula for the rotation matrix R corresponding to an axis angle

Axis angle

representation [k θ]:

.

Noting that, using the outer product

, we have

or, equivalently,

For the inverse mapping, see Log map from SO(3) to so(3).

External links

Johan E. Mebius, Derivation of the Euler-Rodrigues formula for three-dimensional rotations from the general formula for four-dimensional rotations., arXiv General Mathematics 2007.
For a proof that begins with the xyz frame see http://myyn.org/m/article/proof-of-rodrigues-rotation-formula/
For another descriptive example see http://chrishecker.com/Rigid_Body_Dynamics#Physics_Articles, Chris Hecker, physics section, part 4. "The Third Dimension" -- on page 3, section ``Axis and Angle, http://chrishecker.com/images/b/bb/Gdmphys4.pdf
Matthew T. Mason, Representing rotation.

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.

Derivation

Matrix notation

Conversion to rotation matrix

See also

External links