This paper is devoted to the complete classification of space curves under affine transformations in the view of Cartans theorem. Spivak has introduced the method but has not found the invariants. Furthermore, for the first time, we propound a necess
ary and sufficient condition for the invariants. Then, we study the shapes of space curves with constant curvatures in detail and suggest their applications in physics, computer vision and image processing.
In this paper, the symmetry group of a differential system of n quadratic homogeneous first order ODEs of n variables is studied. For this purpose, we consider the action of both point and contact transformations to signify the corresponding Lie alge
bras. We also find the independent differential invariants of these actions.
Classification of curves up to affine transformation in a finite dimensional space was studied by some different methods. In this paper, we achieve the exact formulas of affine invariants via the equivalence problem and in the view of Cartans lemma a
nd then, state a necessary and sufficient condition for classification of n--Curves.
In this paper, we classify space-time curves up to Galilean group of transformations with Cartans method of equivalence. As an aim, we elicit invariats from action of special Galilean group on space-time curves, that are, in fact, conservation laws i
n physics. We also state a necessary and sufficient condition for equivalent Galilean motions.
