We are considering the rotation that goes from a coordinate system A to a coordinate system B, noted BfromA. This notation allows to combine transforms without ambiguity, e.g BfromC = BfromA * AfromC, and applies naturally to a point on the right: pointInB = BfromA * pointInA

#### Euler Angles

Here they are defined in the Z-Y-X order, Tait-Bryan, Intrinsics. Note that this is the same as x-y-z, Tait-Bryan, Extrinsics.

BfromA = R(Z) * R(Y) * R(X)

This corresponds to the rotation of an object A in a coordinate system B. Applying the extrinsics sequence this means that a point gets transformed by first applying the rotation around XB, then around YB, then around ZB.

If you are trying to simulate the transform with your fingers, applying these angles in sequence will make your finger-axes go from B to A, not from A to B.

 X = Y = Z =

#### Angle Axis

The rotation is uniquely defined (up to the sign) by a unit axis [X,Y,Z] and an angle Θ around it.

 X = Y = Z = Θ =

#### Quaternion

A unit quaternion is a unique (up to the sign) representation. Quaternions allow easy interpolation and composition. See the wikipedia page.

 X = Y = Z = W =

#### Rotation matrix 3x3

Rotation matrices are 3x3. The row vectors are the coordinates of each axis of B expressed in A. The columns vectors are the coordinates of each axis of A expressed in B.

\$\$[X,Y,Z]_B = [Y, -Z, X]_A\$\$
Tip: you can paste space or comma separated values in the first field of each formalism and it'll get parsed to fill all the fields.