Interpolating data and the application of data mining techniques in nonlinear manifolds plays a significant role in different areas of knowledge, ranging from computer vision and robotics, to industrial and medical requests, and these growing number of applications have sparked the research interest of the scientific community to these topics. The Generalized Essential manifold, briefly, Essential manifold, consisting of the product of the Grassmann manifold of all k-dimensional subspaces of Rn and the Lie group of rotations in Rn, for instance, plays an important role in the problem of recovering the structure and motion from a sequence of images, also known as stereo matching, which is a crucial problem in image processing and computer vision. A well-known recursive procedure to generate interpolating polynomial curves in Euclidean spaces is the classical De Casteljau algorithm, which is a simple and powerful tool widely used in the field of Computer Aided Geometric Design, particularly because it is essentially geometrically based. This algorithm has been generalized to geodesically complete Riemannian manifolds. Thus, having this in mind, in this work we present all the ingredients for a detailed implementation of the generalized De Casteljau algorithm to generate geometric cubic polynomials in the Essential manifold preparing the ground to solve different real interpolation problems in this manifold.

The authors acknowledge Funda¸c˜ao para a Ciˆencia e a Tecnologia (FCT) and COMPETE 2020 program for financial support to projects UIDB/00048/2020 and UIDB/04466/2020.

Cubic polynomials,Essential manifold,De Casteljau algorithm,Geodesics,Data mining,Interpolating data