Abstract: Bézier curve is required to interpolate the end points of given curve segment with GC1 continuity. One-sided approximation error is given and the choice of optimal interpolation points is studied, the optimal error estimate is obtained. Based on the above results for curve segments, the approximation of a hyperboloid surface patch using bicubic Bézier polynomials both for one sheet and two sheets is worked out. Its approximation accuracy is of sixth order. Furthermore the adjacent approximation surface patches have the same tangent plane at their common boundaries.