Differential invariants of hypersurfaces and a differentialapproach to detect projective equivalences and symmetries of3−dimensional rational algebraic curves: Algorithms andimplementation

In this thesis, two important equivalence problems are discussed. While the first problem is the Euclidean equivalence of parametric hypersurfaces, the second problem is the detection of the projective equivalences and symmetries of three-dimensional rational algebraic curves. It is aimed to produce solutions for these problems with the effective use of invariant theory. For the solution of the first problem, differential invariants of the hypersurfaces according to the Euclidean group are determined and the complete system of these invariants is obtained. Again, similar results are also investigated for the special Euclidean group. Thanks to the results obtained here, the necessary perspective for the solution of the other problem of the thesis is formed. For the second problem, the rational curve parametrizations are considered in the projective space, thus we avoid the use of rational functions, instead, we study with representations whose inputs are all polynomials. It is known that proper parametrizations of rational curves in reduced form are a uniform representation with respect to bilinear reparametrizations. A method based on projective differential invariants is constructed based on this knowledge. This method is turned into an algorithm and the performance of the algorithm is examined by extensive tests using the Maple computer algebra system.