This paper presents a new method for the recognition and reconstruction of surfaces from 3D data. Line element geometry, which generalizes both line geometry and the Laguerre geometry of oriented planes, enables us to recognize a wide class of surfaces (spiral surfaces, cones, helical surfaces, rotational surfaces, cylinders, etc.) by fitting linear subspaces in an appropriate seven-dimensional image space. In combination with standard techniques such as PCA and RANSAC, line element geometry is employed to effectively perform the segmentation of complex objects according to surface type. Examples show applications in reverse engineering of CAD models and testing mathematical hypotheses concerning the exponential growth of sea shells.
We understand and reconstruct special surfaces from 3D data with line geometry methods. Based on estimated surface normals we use approximation techniques in line space to recognize and reconstruct rotational, helical, developable and other surfaces, which are characterized by the configuration of locally intersecting surface normals. For the computational solution we use a modified version of the Klein model of line space. Obvious applications of these methods lie in Reverse Engineering. We have tested our algorithms on real world data obtained from objects as antique pottery, gear wheels, and a surface of the ankle joint.
Invariant notions of a class of Segre varieties S (m) (2) of PG(2 m − 1, 2) that are direct products of m copies of PG(1, 2), m being any positive integer, are established and studied. We first demonstrate that there exists a hyperbolic quadric that contains S (m) (2) and is invariant under its projective stabiliser group G S (m) (2) . By embedding PG(2 m − 1, 2) into PG(2 m − 1, 4), a basis of the latter space is constructed that is invariant under G S (m) (2) as well. Such a basis can be split into two subsets of an odd and even parity whose spans are either real or complex-conjugate subspaces according as m is even or odd. In the latter case, these spans can, in addition, be viewed as indicator sets of a G S (m) (2) -invariant geometric spread of lines of PG(2 m − 1, 2). This spread is also related with a G S (m) (2) -invariant non-singular Hermitian variety.The case m = 3 is examined in detail to illustrate the theory. Here, the lines of the invariant spread are found to fall into four distinct orbits under G S (3) (2) , while the points of PG(7, 2) form five orbits. Classification (2010): 51E20, 05B25, 15A69
Mathematics Subject
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.