  It is important to understand that what one means by a  “curve” is the map (the parametric description) itself,  and not the set of its image points, the path. Consequently, two curves are mathematically considered to be different curves if  they are given by different maps (different parameter descriptions),  even if their image set, that is, their path, is the  same. If we change a curve’s parameter description we change the curve but not its image set (its path), the points it passes through. A path is therefore sometimes defined as an equivalence class of  curves under arbitrary parameter transformations. Hence, projective  geometry may be defined as an equivalence class of affine geometries. 