User:Prof McCarthy/Screw axis

Geometric argument
Let D: R3 →R3 define an orientation preserving rigid motion of R3, and let x denote an arbitrary vector of R3. Now let the image of 0 be the vector D(0)= d, and define the new rigid motion A: R3 →R3 such that A(x) := D(x) - d for all x in R3. This guarantees that A(0) =  0.

Because A is an orientation preserving rigid motion of R3 and A(0) =  0, then A must be a rotation. If A is the identity I, then for this case, the screw motion is merely a translation by the vector d and does not involve a rotation. From here on we assume A is not the identity I.

Each rotation of R3 must have an axis L (a bi-infinite straight line) that is pointwise fixed by the rotation, therefore this is true for the rotation A(x) = D(x) - d, hence
 * $$ D(\mathbf{x})=A(\mathbf{x}) + \mathbf{d}. $$

for all x in R3. This shows that the orientation preserving rigid motion D is the result of applying a rotation A followed by a translation by the vector d.

The translation vector d can be resolved into a sum of two vectors, one parallel to the axis L of the rotation A, and the other in the plane perpendicular to L, as follows:
 * $$\mathbf{d}=\mathbf{d}_L + \mathbf{d}_{\perp}.$$

Therefore the rigid motion takes the form
 * $$ D(\mathbf{x})=(A(\mathbf{x}) + \mathbf{d}_{\perp}) + \mathbf{d}_L. $$

Now, the orientation preserving rigid motion D* = A(x) + d⊥ transforms all the points of R3 so that they remain in planes perpendicular to L'. For a rigid motion of this type there is a unique point c in the plane P perpendicular to L through 0, such that
 * $$ D^*(\mathbf{c})=A(\mathbf{c})+\mathbf{d}_{\perp}=\mathbf{c}.$$

The point c can be calculated as
 * $$ \mathbf{c}=[I-A]^{-1}\mathbf{d}_{\perp},$$

because d⊥ does not have a component in the direction of the axis of A.

A rigid motion D*  with a fixed point must be a rotation of around the axis Lc through the point c'. Therefore, the rigid motion
 * $$ D(\mathbf{x})=D^*(\mathbf{x}) + \mathbf{d}_L,$$

consists of a rotation about the line Lc followed by a translation by the vector dL in the direction of the line   Lc.

Conclusion: every rigid motion of R3 is the result of a rotation of R3 about a line Lc followed by a translation in the direction of the line. The combination of a rotation about a line and translation along the line is called a screw motion.