We study on the geometric properties of the quadratic rational $Bézier$ curves and approximations using them. We find necessary and sufficient conditions for the curvature of a quadratic rational $Bézier$ curve to be monotone, to have a unique local minimum, to have a unique local maximum and to have both extrema, and we also visualize them in figures. We characterize the best approximation of a regular plane curve by a quadratic rational $Bézier$ curve with possible contact order at both end points and prove its uniqueness. We also present a Remes type algorithm to obtain the best approximation. We apply our characterization to the degree reduction of cubic rational $Bézier$ curve to quadratic one and also to the cubic offset approximation, and present the numerical results. For the circular arc of angle 0＜α＜π we present the explicit form of the best $GC^3$ quartic approximation and the best $GC^2$ quartic approximations of various types, and give the explicit form of the Hausdorff distance between the circular arc and the approximate $Bézier$ curves for each case. We also show the existence of the $GC^4$ quintic approximations to the arc, and find the explicit form of the best $GC^3$ quintic approximation in certain constraints and their distances from the arc. All approximations we construct in this thesis have the optimal order of approximation, twice of the degree of approximate $Bézier$ curves.