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Clarify the interpretation of composed Euler angle rotations (#100)
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danielmatz authored and c42f committed Aug 13, 2019
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Expand Up @@ -131,14 +131,10 @@ j2 = Rotations.jacobian(q, p) # How does the rotated point q*p change w.r.t. the

A 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 Byran**](https://en.wikipedia.org/wiki/Euler_angles#Tait.E2.80.93Bryan_angles) angle ordering,
such as `RotXYZ`, are said to follow the [**Tait Bryan**](https://en.wikipedia.org/wiki/Euler_angles#Tait.E2.80.93Bryan_angles) angle ordering,
while those which repeat (e.g. `EulerXYX`) are said to use [**Proper Euler**](https://en.wikipedia.org/wiki/Euler_angles#Conventions) angle ordering.

Like the two-angle versions, read the application of the rotations along the
static cardinal axes to a vector from right-to-left, so that `RotXYZ(x, y, z) * v == RotX(x) * (RotY(y) * (RotZ(z) * v))`.
This is the "extrinsic" representation of an Euler-angle rotation, though
if you prefer the "intrinsic" form it is easy to use the corresponding
`RotZYX(z, y, x)`.
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.

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