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 L_c through the point c. Therefore, the rigid motion

$D(\mathbf {x} )=D^{*}(\mathbf {x} )+\mathbf {d} _{L},$

consists of a rotation about the line L_c followed by a translation by the vector d_L in the direction of the line L_c.

Conclusion: every rigid motion of R³ is the result of a rotation of R³ about a line L_c 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.