Involute of Cubic Pythagorean-Hodograph Curve

Since the mode of Pythagorean-Hodograph curve’s derivate is a polynomial, its involute should be a rational polynomial. We study the expression form of PH cubic’s involute, discovering that its control points and weights are all constructed by corresponding PH cubic’s geometry property. Moreover, we prove that cubic PH curves have neither cusp nor inflection; there are two points coincidence in the six control points of cubic PH curves’ involute; the control polygon of cubic PH curve has two edges which are perspectively vertical with two edges in the control polygon of corresponding involute, and we compute the length of the two edge in the involute control polygon; So if we give the cubic PH curve, then we can decide five control points of the six, and we give the vector about the remaining point and another point. When the cubic PH curve is a symmetry curve, its two involute are symmetry, other cases are not.