Abstract: In this paper, we study the construction methods of cubic Pythagorean Hodograph(PH) C-curves under three geometric constraints, respectively. The key idea is to utilize the complex representation of planar parametric curves and the geometric characteristics of PH C-curves. Firstly, constructing a PH C-curve for any given G¹ Hermite condition is revisited and converted into solving a real quadratic equation in real variables. Secondly, for a given one-endpoint C¹ constraint, the PH C-curves can be constructed by solving a complex quadratic equation in complex variables. Finally, constructing PH C-curves for three planar points can be transformed into solving a complex quadratic equation. This paper considers the existence of solutions in each of the three cases and provides detailed algorithms. We construct several numerical examples of planar curve interpolation using the proposed methods, the results illustrate that the proposed methods can be applied in geometric modeling with PH C-curves.