Rotation formalisms in three dimensions

In geometry, various formalisms exist to express a rotation in three dimensions as a mathematical transformation.

An example where rotation representation is used is in computer vision, where an automated observer needs to track a target.

The basic problem is to specify the orientation of these three unit vectors, and hence the rigid body, with respect to the observer's coordinate system, regarded as a reference placement in space.

Any proper motion of the Euclidean space decomposes to a rotation around the origin and a translation.

Whichever the order of their composition will be, the "pure" rotation component wouldn't change, uniquely determined by the complete motion.

The three unit vectors, û, v̂ and ŵ, that form the rotated basis each consist of 3 coordinates, yielding a total of 9 parameters.

Typically, the coordinates of each of these vectors are arranged along a column of the matrix (however, beware that an alternative definition of rotation matrix exists and is widely used, where the vectors' coordinates defined above are arranged by rows[2])

These statements comprise a total of 6 conditions (the cross product contains 3), leaving the rotation matrix with just 3 degrees of freedom, as required.

Two successive rotations represented by matrices A1 and A2 are easily combined as elements of a group,

The axis is the unit vector (unique except for sign) which remains unchanged by the rotation.

It is best to employ the rotation matrix or quaternion notation, calculate the product, and then convert back to Euler axis and angle.

The middle matrix represents a rotation around an intermediate axis called line of nodes.

The convention being used is usually indicated by specifying the axes about which the consecutive rotations (before being composed) take place, referring to them by index (1, 2, 3) or letter (X, Y, Z).

These 12 combinations avoid consecutive rotations around the same axis (such as XXY) which would reduce the degrees of freedom that can be represented.

In aviation orientation of the aircraft is usually expressed as intrinsic Tait-Bryan angles following the z-y′-x″ convention, which are called heading, elevation, and bank (or synonymously, yaw, pitch, and roll).

Quaternions, which form a four-dimensional vector space, have proven very useful in representing rotations due to several advantages over the other representations mentioned in this article.

Applying the same procedure n times will take a 2n-tangled object back to the untangled or 0 turn state.

This representation is a higher-dimensional analog of the gnomonic projection, mapping unit quaternions from a 3-sphere onto the 3-dimensional pure-vector hyperplane.

Today, the most straightforward way to prove this formula is in the (faithful) doublet representation, where g = n̂ tan a, etc.

The modified Rodrigues vector is a stereographic projection mapping unit quaternions from a 3-sphere onto the 3-dimensional pure-vector hyperplane.

Active rotations of a 3D vector p in Euclidean space around an axis n over an angle η can be easily written in terms of dot and cross products as follows:

Using the x-convention, the 3-1-3 extrinsic Euler angles φ, θ and ψ (around the z-axis, x-axis and again the

For the x-convention the rotations are about the x-, y- and z-axes with angles ϕ, θ and ψ, the individual matrices are as follows:

Note: This is valid for a right-hand system, which is the convention used in almost all engineering and physics disciplines.

The Euler axis can be also found using singular value decomposition since it is the normalized vector spanning the null-space of the matrix I − A.

A quaternion equivalent to yaw (ψ), pitch (θ) and roll (ϕ) angles.

the x-convention 3-1-3 extrinsic Euler Angles (φ, θ, ψ) can be computed by

The formalism of geometric algebra (GA) provides an extension and interpretation of the quaternion method.

This bivector describes the plane perpendicular to what the cross product of the vectors would return.

In three-dimensional space, however, it is often simpler to leave the expression for B̂ = iv̂, using the fact that i commutes with all objects in 3D and also squares to −1.