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Additional resources for An Invitation to 3-D Vision: From Images to Geometric Models
Since the triple product corresponds to the volume of the parallelepiped spanned by the three vectors, rigid-body motion also preserves volumes. • How do these properties help us describe a rigid-body motion concisely? The fact that distances and orientations are preserved by a rigid-body motion means that individual points cannot move relative to each other. As a consequence, a rigid-body motion can be described by the motion of a chosen point on the body and the rotation of a coordinate frame attached to that point.
Given a trajectory R(t) : R -+ 80(3) that describes a continuous rotational motion, the rotation must satisfy the following constraint Computing the derivative of the above equation with respect to time t and noticing that the right-hand side is a constant matrix, we obtain R(t)RT(t) + R(t)RT(t) =0 R(t)RT(t) =} = -(R(t)RT(t)f . The resulting equation reflects the fact that the matrix R(t)RT(t) E R 3x 3 is a skew-symmetric matrix. 4, there must exist a vector, say w(t) E R 3 , such that R(t)RT(t) = w(t).
V)11 = Ilvll for all free vectors v E ]R3. A map that preserves the distance is called a Euclidean transformation. In the 3-D space, the set all Euclidean transformations is denoted by E(3) . Note that preserving distances between points is not sufficient to characterize a rigid object moving in space. In fact, there are transformations that preserve distances, and yet they are not physically realizable. For instance, the map f: [Xl, X 2 , X3]T f-+ [Xl, X 2 , -X3]T preserves distances but not orientations.