Admissible curvature continuous areas for fair curves using G2 Hermite PH quintic polynomial

In this paper we derive admissible curvature continuous areas for monotonically increasing curvature continuous smooth curve by using a single Pythagorean hodograph (PH) quintic polynomial of G2 contact matching Hermite end conditions. Curves with monotonically increasing or decreasing curvatures are considered highly smooth (fair) and are very useful in geometric design. Making the design by using smooth curves is a fascinating problem of computing with significant physical and esthetic applications especially in high speed transportation and robotics. First we derive sufficient conditions for curvature continuity on a single PH quintic polynomial with given Hermite end conditions then we find the admissible area for the smooth curve with respect to the curvatures at its endpoints