Let  be an orientation-preserving rigid motion of R. The set of these transformations is a subgroup of Euclidean motions known as the special Euclidean group SE(3). These rigid motions are defined by transformations of x in R given by
consisting of a three-dimensional rotation A followed by a translation by the vector d.
A three-dimensional rotation A has a unique axis that defines a line L. Let the unit vector along this line be S so that the translation vector d can be resolved into a sum of two vectors, one parallel and one perpendicular to the axis L, that is,
In this case, the rigid motion takes the form
Now, the orientation preserving rigid motion D* = A(x) + d transforms all the points of R 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
The point C can be calculated as
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 through the point c. Therefore, the rigid motion
consists of a rotation about the line L followed by a translation by the vector d in the direction of the line L.
Conclusion: every rigid motion of R is the result of a rotation of R about a line L 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.