Affine differential invariants of a curve on the plane

In this thesis, arc length and curvature which are affine differential invariants of a curve in the plane are investigated.In Chapter 1, the notions Vector Spaces and Linear Transformations, Matrices, Determinants and Systems of Linear Equations, Jordan Canonical Form, Affine Space, Group Action, Orbits and Invariants, Affine Group and Subgroups, Elementary Differential Geometry which form a basis for this thesis are given. In Chapter 2, we make elementary calculations on Lie transformation groups and associate that group theory with affine differential geometry. Then we introduce group operators, infinitesimal operators and by using these notions we obtain one parameter group of affine transformations on the plane. Finally, by using operators, affine arc length and affine curvature of a curve in the plane are calculated and all these ideas are applied to affine group and subgroups. Key Words: Affine differential invariants, Affine arc length, Affine curvature, r-parameter Lie group