Affine Differential Invariants of Curves

The theory of curves had been developed extensively for plane and space. In this spaces, the equivalence problem of curves had been solved by locally. But this hadn’t been solved by globally. Consequently, investigation of equivalence problem in global aspect is very important. When curves have singular points, differential invariants obtained Frenet formulas are not enough in investigation of equivalence problem. Because of this, the investigation problem of all polinom and rational function of differential invariants of curves comes out. In this study, the G-equivalence problem of parametric curves for groups G=SL(n;R), GL(n;R), SAff(n;R), Aff(n;R) is solved globally. In here, first for groups giving up, the generators of G-invariant differential polinomials ring and G-invariant rational functions field are researched. Then by using these generators the solution of equivalence problem are done. Finally, it is shown that these generators are differential independent.