In this article we discuss the 2D–3D pose estimation problem of 3D free-form contours. We observe objects of any 3D shape in an image of a calibrated camera. Pose estimation means estimating the relative position and orientation of the 3D object to the reference camera system. While cycloidal curves are derived as orbits of coupled twist transformations, we apply a spectral domain representation of 3D contours as an extension of cycloidal curves. Their Fourier descriptors are also related to twist representations. A twist is an element of se(3) and is a pair containing two 3D vectors. In a matrix representation, its exponential leads to an element of SE(3) and therefore to a rigid motion. We show that twist representations of objects can numerically efficiently and easily be applied to the free-form pose estimation problem. The pose problem itself is formalized as an implicit problem and we gain constraint equations, which have to be fulfilled with respect to the unknown rigid body motion.