In the master thesis we consider the construction of rigid body motion in dual quaternion space. We show some classical examples of interpolation procedures on the smooth manifold $SE(3)$, where we use several known methods from differential geometry and the theory of Lie groups. In the procedure we often split the construction of the motion in the rotational and translational part, where we put more effort into the construction of the rotational part since translational movement of the rigid body is almost trivial using standard interpolation procedures in ${\mathbb R}^3$. From the theory of Clifford algebra we construct the space of dual quaternions. We search for a submanifold of $\mathbb {DH}$ which is isomorphic to the Euclidean group $SE(3)$, where rigid body movement transformations are represented. Using special projections from the Euclidean space ${\mathbb R}^8$ onto the Study quadric, which is a special submanifold of Dual quaternions representing body transformations, we develop several interpolations schemes which enables us to interpolate rotations, translations and rigid body twists. Twists are objects representing the angular velocity and the velocity of the moving frame.
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