*3D rotations made easy in Julia*

This package implements various 3D rotation parameterizations and defines
conversions between them. At their heart, each rotation parameterization *is*
a 3×3 unitary (orthogonal) matrix (based on the StaticArrays.jl package),
and acts to rotate a 3-vector about the origin through matrix-vector multiplication.

While the `RotMatrix`

type is a dense representation of a `3×3`

matrix, we also
have sparse (or computed, rather) representations such as quaternions,
angle-axis parameterizations, and Euler angles.

For composing rotations about the origin with other transformations, see also the CoordinateTransformations.jl or AffineTransforms.jl packages.

```
using Rotations, StaticArrays
# create the null rotation (identity matrix)
id = one(RotMatrix{3, Float64})
# create a random rotation matrix (uniformly distributed over all 3D rotations)
r = rand(RotMatrix{3}) # uses Float64 by default
# create a point
p = SVector(1.0, 2.0, 3.0) # from StaticArrays.jl, but could use any AbstractVector...
# convert to a quaternion (Quat) and rotate the point
q = Quat(r)
p_rotated = q * p
# Compose rotations
q2 = rand(Quat)
q3 = q * q2
# Take the inverse (equivalent to transpose)
q_inv = transpose(q)
q_inv == inv(q)
p ≈ q_inv * (q * p)
q4 = q3 / q2 # q4 = q3 * inv(q2)
q5 = q3 \ q2 # q5 = inv(q3) * q2
# convert to a Stereographic quaternion projection (recommended for applications with differentiation)
spq = SPQuat(r)
# convert to the Angle-axis parameterization, or related Rodrigues vector
aa = AngleAxis(r)
rv = RodriguesVec(r)
ϕ = rotation_angle(r)
v = rotation_axis(r)
# convert to Euler angles, composed of X/Y/Z axis rotations (Z applied first)
# (all combinations of "RotABC" are defined)
r_xyz = RotXYZ(r)
# Rotation about the X axis by 0.1 radians
r_x = RotX(0.1)
# Composing axis rotations together automatically results in Euler parameterization
RotX(0.1) * RotY(0.2) * RotZ(0.3) === RotXYZ(0.1, 0.2, 0.3)
# Can calculate Jacobian - derivatives of rotations with respect to parameters
j1 = Rotations.jacobian(RotMatrix, q) # How does the matrix change w.r.t the 4 Quat parameters?
j2 = Rotations.jacobian(q, p) # How does the rotated point q*p change w.r.t. the 4 Quat parameters?
# ... all Jacobian's involving RotMatrix, SPQuat and Quat are implemented
# (SPQuat is ideal for optimization purposes - no constaints/singularities)
```

**Rotation Matrix**`RotMatrix{N, T}`

An N x N rotation matrix storing the rotation. This is a simple wrapper for a StaticArrays

`SMatrix{N,N,T}`

. A rotation matrix`R`

should have the property`I = R * R'`

, but this isn't enforced by the constructor. On the other hand, all the types below are guaranteed to be "proper" rotations for all input parameters (equivalently: parity conserving, in*SO(3)*,`det(r) = 1`

, or a rotation without reflection).**Arbitrary Axis Rotation**`AngleAxis{T}`

A 3D rotation with fields

`theta`

,`axis_x`

,`axis_y`

, and`axis_z`

to store the rotation angle and axis of the rotation. Like all other types in this package, once it is constructed it acts and behaves as a 3×3`AbstractMatrix`

. The axis will be automatically renormalized by the constructor to be a unit vector, so that`theta`

always represents the rotation angle in radians.**Quaternions**`Quat{T}`

A 3D rotation parameterized by a unit quaternion. Note that the constructor will renormalize the quaternion to be a unit quaternion, and that although they follow the same multiplicative

*algebra*as quaternions, it is better to think of`Quat`

as a 3×3 matrix rather than as a quaternion*number*.**Rodrigues Vector**`RodriguesVec{T}`

A 3D rotation encoded by an angle-axis representation as

`angle * axis`

. This type is used in packages such as OpenCV.Note: If you're differentiating a Rodrigues Vector check the result is what you expect at theta = 0. The first derivative of the rotation

*should*behave, but higher-order derivatives of it (as well as parameterization conversions) should be tested. The Stereographic Quaternion Projection is the recommended three parameter format for differentiation.**Stereographic Projection of a unit Quaternion**`SPQuat{T}`

A 3D rotation encoded by the stereographic projection of a unit quaternion. This projection can be visualized as a pin hole camera, with the pin hole matching the quaternion

`[-1,0,0,0]`

and the image plane containing the origin and having normal direction`[1,0,0,0]`

. The "null rotation"`Quaternion(1.0,0,0,0)`

then maps to the`SPQuat(0,0,0)`

These are similar to the Rodrigues vector in that the axis direction is stored in an unnormalized form, and the rotation angle is encoded in the length of the axis. This type has the nice property that the derivatives of the rotation matrix w.r.t. the

`SPQuat`

parameters are rational functions, making the`SPQuat`

type a good choice for differentiation / optimization.**Cardinal axis rotations**`RotX{T}`

,`RotY{T}`

,`RotZ{T}`

Sparse representations of 3D rotations about the X, Y, or Z axis, respectively.

**Two-axis rotations**`RotXY{T}`

, etcConceptually, these are compositions of two of the cardinal axis rotations above, so that

`RotXY(x, y) == RotX(x) * RotY(y)`

(note that the order of application to a vector is right-to-left, as-in matrix-matrix-vector multiplication:`RotXY(x, y) * v == RotX(x) * (RotY(y) * v)`

).**Euler Angles - Three-axis rotations**`RotXYZ{T}`

,`RotXYX{T}`

, etcA composition of 3 cardinal axis rotations is typically known as a Euler angle parameterization of a 3D rotation. The rotations with 3 unique axes, such as

`RotXYZ`

, are said to follow the**Tait Bryan**angle ordering, while those which repeat (e.g.`EulerXYX`

) are said to use**Proper Euler**angle ordering.Like the two-angle versions, the order of application to a vector is right-to-left, so that

`RotXYZ(x, y, z) * v == RotX(x) * (RotY(y) * (RotZ(z) * v))`

. This may be interpreted as an "extrinsic" rotation about the Z, Y, and X axes or as an "intrinsic" rotation about the X, Y, and Z axes. Similarly,`RotZYX(z, y, x)`

may be interpreted as an "extrinsic" rotation about the X, Y, and Z axes or an "intrinsic" rotation about the Z, Y, and X axes.

All parameterizations can be converted to and from (mutable or immutable) 3×3 matrices, e.g.

```
using StaticArrays, Rotations
# export
q = Quat(1.0,0,0,0)
matrix_mutable = Array(q)
matrix_immutable = SMatrix{3,3}(q)
# import
q2 = Quaternion(matrix_mutable)
q3 = Quaternion(matrix_immutable)
```

This package assumes active (right handed) rotations where applicable.

They're faster (Julia's `Array`

and BLAS aren't great for 3x3 matrices) and
don't need preallocating or garbage collection. For example, see this benchmark
case where we get a 20× speedup:

```
julia> cd(Pkg.dir("Rotations") * "/test")
julia> include("benchmark.jl")
julia > BenchMarkRotations.benchmark_mutable()
Rotating using mutables (Base.Matrix and Base.Vector):
0.124035 seconds (2 allocations: 224 bytes)
Rotating using immutables (Rotations.RotMatrix and StaticArrays.SVector):
0.006006 seconds
```

03/09/2016

30 days ago

193 commits