Extended Coordinate Representation: Encoding the Orientation of the Plane of Rotation at the Coordinate Level

Conventionally, the orientation of the plane of a spatial rotation is provided as an extrinsic parameter and is not part of the coordinates on which the rotation acts. In this paper, we ask whether this orientation can be embedded directly into an extended coordinate system, in the same coordinate‐extension approach that we developed previously for Lorentz boosts. Starting with a rotation between the X Y ‐plane and the Z ‐axis, we decompose the z ‐coordinate into the components that contribute to the transformations of the x ‐ and y ‐coordinates, namely z ₁ = z cos ϕ and z ₂ = z sin ϕ , where ϕ denotes the orientation of the rotation plane with respect to the X ‐axis, thereby extending the coordinate vector ( x , y , z ) to ( x , y , z ₁ , z ₂ ). This construction gives an explicit 4 × 4 orthogonal matrix with determinant +1, originally defined for points in the rotation plane, and extended to general points in three‐dimensional space. We also apply the extended formulation to the composition of two successive rotations about orthogonal axes and demonstrate, by direct matrix multiplication, that it correctly reproduces the residual axial rotation arising from the noncommutativity of the rotation operators, as expected from the Lie‐algebraic structure of S O (3). The suggested formulation incorporates directly the orientation of the rotation plane in the coordinate representation, providing a transparent and elementary framework for describing the orthogonality and composition properties of spatial rotations, and is particularly well‐suited to problems involving continuously varying rotation axes.