# Evaluation of three-dimensional deformation fields of the Earth by methods of the projective differential geometry. The main linear deformations

2016;
: pp. 7 – 17

Authors:
1
Department of Geodesy and Cartography, National University of Water and Environmental Engineering

Aim. The aim is to solve the problem of evaluating the Earth’s topographic surface deformations using
projective differential geometry methods as an expression of the space metric tensor and the group of main linear
deformation parameters in the spatial geocentric coordinate system. Methodology. Solving the problem is based
on using the homeomorphism transformation (mapping) properties of the three-dimensional continuous and
closed domains of the space with the hypothesis that this transformation has a geophysical origin and was caused
by the deformation. If the base functions meet homeomorphism requirements, the functional model
transformation is capable of transmitting the change of metric properties of the domain by different
characteristics that, in the accepted hypotheses, are its deformation parameters. The main carrier of these
characteristics is the metric tensor of three-dimensional Euclidean space. A tensor is formed by the metric form
of the transformed domain of space as the square of the linear element length, which is expressed by differentials
of the transformation domain coordinates and then full differentials of base functions e are taken into account.
Results. Solving the task is carried out on the condition that the transformation domain of space is outlined by
the Earth’s topographic surface and coordinated on a three-dimensional rectangular geocentric system. The
solution results are working formulas for calculating the main spatial linear deformations, which are expressed
by coefficients of elongation, compression, and shear of the topographic surface. Directions of these parameters
are defined in the geocentric polar system. Various coefficients of elongation and their directions are expressed
in metric tensor components. Formulas are obtained for calculating the parameters in any given direction, along
the directions of coordinate axis, on projections to coordinate planes, and for the extreme values triad with the
respective spatial orientation. Scientific novelty and practical significance. It is grounded that studies of the
Earth’s deformation fields by methods of the projective differential geometry has greater potential capabilities
when compared to methods of linear continuum mechanics and also provides generalized solutions. The
homeomorphic functional model as the basis for the formation of the tensor allows the expression of the
deformation of any character. Formulas for expressing the main linear deformations are obtained. Results are
suitable for evaluation of three-dimensional deformation fields of any scale. Deformation parameters are
attributed directly to the topographic surface of the Earth. The sufficient coverage of the Earth by GNSS stations
and representational observational data that defines the completeness of functional model construction, together
with the obtained results are able to provide the evaluation and interpretation of the real deformations, but not

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