Rodrigues' rotation formula

In geometry, Rodrigues' rotation formula (named after Olinde Rodrigues) is a vector formula for a rotation in space, given its axis and angle of rotation.

Say u,v $in$ R³ and we want to obtain a representation for the rotation v_rot of the vector v around the vector u (which is assumed to have unit length) by an angle θ in the counterclockwise (i.e. positive) direction. Rodrigues' formula reads as follows:

$mathbf{v}_{rot} = mathbf{v} cdot cos heta + mathbf{u} imes mathbf{v} cdot sin heta + langle mathbf{u}, mathbf{v}
angle mathbf{u} cdot (1 - cos heta).$

Proof of the formula

Take the vector w = v − <u,v>u, which is the projection of v on the plane orthogonal to u, and the cross product of the vectors u and v: z = u×v. Turn the vector w by the angle θ around the base of the vector u to obtain the projection of the rotated vector v_rot:

$egin{align} mathbf{w}_{rot} &= mathbf{w} cdot cos heta + mathbf{z} cdot sin heta \ &= (mathbf{v} - langle mathbf{u}, mathbf{v}
angle mathbf{u}) cdot cos heta + mathbf{u} imes mathbf{v} cdot sin heta.end{align}$

Notice that both the vectors w and z have the same length: |w|,|z| = |v - <u,v>u|, because the vector u is of unit length. To get the rotated vector v, we have to add back the adjustment <u,v>u. Hence

$egin{align} mathbf{v}_{rot} &= (mathbf{v} - langle mathbf{u}, mathbf{v}
angle mathbf{u}) cdot cos heta + mathbf{u} imes mathbf{v} cdot sin heta + langle mathbf{u}, mathbf{v}
angle mathbf{u} \ &= mathbf{v} cdot cos heta + mathbf{u} imes mathbf{v} cdot sin heta + langle mathbf{u}, mathbf{v}
angle mathbf{u} cdot (1 - cos heta),end{align}$

which is exactly what we were looking for.

Using the annotation $mathbf{u}^T mathbf{v}$ for the scalar product, we get:

$egin{align} mathbf{v}_{rot} &= cos heta cdot mathbf{v} + sin heta cdot mathbf{u} imes mathbf{v} + (1 - cos heta) cdot mathbf{u} cdot mathbf{u}^T mathbf{v}end{align}$

Substituting the cross product with matrix multiplication:

$egin{align} mathbf{u} imes mathbf{v} &=left(egin{array}{ccc}0 & -u_3 & u_2 \u_3 & 0 & -u_1 \-u_2 & u_1 & 0end{array}
ight) mathbf{v} = left [ mathbf{u}
ight] _ imes mathbf{v}end{align}$

and multiplying with the identity matrix "I", we get

$egin{align} mathbf{v}_{rot} &= cos heta cdot I mathbf{v} + sin heta cdot left [ mathbf{u}
ight] _ imes mathbf{v} + (1 - cos heta) cdot mathbf{u} cdot mathbf{u}^T mathbf{v} \ &= left( cos heta I + sin heta left [ mathbf{u}
ight] _ imes + (1 - cos heta) mathbf{u} mathbf{u}^T
ight) mathbf{v} end{align}$

where the expression in the paranthesis can be identified as the rotation matrix "R":

$egin{align} R = cos heta I + sin heta left [ mathbf{u}
ight] _ imes + (1 - cos heta) mathbf{u} mathbf{u}^Tend{align}$

External links

For another descriptive example see www.d6.com, Chris Hecker, physics section, part 4. "The Third Dimension" -- on page 3, section ``Axis and Angle", http://www.d6.com/users/checker/pdfs/gdmphys4.pdf

Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

Rodrigues — may refer to:*Rodrigues (island), one of the Mascarene Islands, in the Indian Ocean *Rodrigues Triple Point, a tectonic triple junction near Rodrigues Island *Rodrigues College, a secondary school on Rodrigues Island *Fr. Conceicao Rodrigues… … Wikipedia
Rotation (mathematics) — Rotation of an object in two dimensions around a point O. In geometry and linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a rigid body around a fixed point. A rotation is different from a… … Wikipedia
Rodrigues' formula — has two possible meanings: *In geometry, short for Rodrigues rotation formula. *A formula for producing a series of expressions by repeated differentiation of some other functions. A typical application is in producing a series of orthogonal… … Wikipedia
Rotation matrix — In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix rotates points in the xy Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian… … Wikipedia
Rotation group — This article is about rotations in three dimensional Euclidean space. For rotations in four dimensional Euclidean space, see SO(4). For rotations in higher dimensions, see orthogonal group. In mechanics and geometry, the rotation group is the… … Wikipedia
Olinde Rodrigues — Benjamin Olinde Rodrigues (1795–1851), 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[1] in Bordeaux. Rodrigues was awarded a doctorate … Wikipedia
Rotation representation (mathematics) — In geometry a rotation representation expresses the orientation of an object (or coordinate frame) relative to a coordinate reference frame. This concept extends to classical mechanics where rotational (or angular) kinematics is the science of… … Wikipedia
List of mathematics articles (R) — NOTOC R R. A. Fisher Lectureship Rabdology Rabin automaton Rabin signature algorithm Rabinovich Fabrikant equations Rabinowitsch trick Racah polynomials Racah W coefficient Racetrack (game) Racks and quandles Radar chart Rademacher complexity… … Wikipedia
Logarithm of a matrix — In mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix. It is thus a generalization of the scalar logarithm and in some sense an inverse function of the matrix… … Wikipedia
Формула поворота Родрига — формула, связывающая два вектора с общим началом, один из которых получен поворотом другого на известный угол вокруг оси, проходящей через их общее начало: где исходный вектор, результирующий вектор, единичный вектор оси поворо … Википедия

Academic Dictionaries and Encyclopedias

Rodrigues' rotation formula

Look at other dictionaries:

Share the article and excerpts

Academic Dictionaries and Encyclopedias

Wikipedia

Rodrigues' rotation formula

Look at other dictionaries:

Share the article and excerpts

Direct link