On curves in affine space

In this thesis, the definition of the unique parameter called the affine arc length parameter is given for the α curve when the plane curve α:I→R^2 has no inflection point. Using the affine arc length parameter, an affine tangent vector and an affine normal vector of the α curve are obtained. By taking the derivatives of these vectors, the invariant curvature of the α curve is obtained. A definition of a family of affine distance functions is given. It has been shown that results completely similar to the Euclidean theory are obtained. In addition, affine arc length parameter, affine curvatures and affine framework are given for α:I→R^3 space curves, proceeding similarly to plane curve case and Euclidean space curve theory. A family of affine distance functions has also been defined. Then, the characterizations of the affine space curve with constant affine curvature and Bertrand curves were investigated.