This paper looks at the use of a Clifford (or geometric) algebra for handling both rotations and translations in Euclidean space. The algebra is constructed over the real numbers using four basis vectors. Three of these generate a subalgebra which models three-dimensional space; the fourth acts as a projective coordinate. Spatial displacements are represented by bivectors of a certain form. The application to the generation of smooth motions using Bezier and B-spline techniques is illustrated.