Final notes about the construction Euler also points out that can be found by intersecting the perpendicular bisector of with the angle bisector of , a construction that might be easier in practice. He also proposed the intersection of two planes: the symmetry plane of the angle (which passes through the center of the sphere), and the symmetry plane of the arc (which also passes through ). Proposition. These two planes intersect in a diameter. This diameter is the one we are looking for. Proof. Let us call either of the endpoints (there are two) of this diameter over the sphere surface. Since is mapped on and the triangles have the same angles, it follows that the triangle is transported onto the triangle . Therefore the point has to remain fixed under the movement. Corollaries. This also shows that the rotation of the sphere can be seen as two consecutive reflections about the two planes described above. Points in a mirror plane are invariant under reflection, and hence the points on their intersection (a line: the axis of rotation) are invariant under both the reflections, and hence under the rotation. Another simple way to find the rotation axis is by considering the plane on which the points , , lie. The rotation axis is obviously orthogonal to this plane, and passes through the center of the sphere. Given that for a rigid body any movement that leaves an axis invariant is a rotation, this also proves that any arbitrary composition of rotations is equivalent to a single rotation around a new axis. Matrix proof A spatial rotation is a linear map in one-to-one correspondence with a rotation matrix that transforms a coordinate vector into , that is . Therefore, another version of Euler's theorem is that for every rotation , there is a nonzero vector for which ; this is exactly the claim that is an eigenvector of associated with the eigenvalue 1. Hence it suffices to prove that 1 is an eigenvalue of ; the rotation axis of will be the line , where is the eigenvector with eigenvalue 1. A rotation matrix has the fundamental property that its inverse is its transpose, that is where is the identity matrix and superscript T indicates the transposed matrix. Compute the determinant of this relation to find that a rotation matrix has determinant ±1. In particular, A rotation matrix with determinant