On double Pythagorean-Hodograph curves of degree seven

This paper presents a comprehensive and constructive analysis of degree double Pythagorean-Hodograph (DPH) curves for the three distinct structural classes. A unified framework is introduced for their construction, particularly useful for interpolation tasks involving prescribed boundary data. The approach explicitly identifies the degrees of freedom available in each class and distinguishes between helical and non-helical curve types. Furthermore, the structure of a specific rational curve in the complex plane that via the normalized Hopf map generates the tangent indicatrix is revealed for all degree DPH curves. This confirms known results for helical curves and extends the interpretation to non-helical cases. The practical applicability of the derived curves is demonstrated through a numerical interpolation example, which also validates the stated number of degrees of freedom. • Complete analysis of degree seven DPH curves • Determination of helical/non-helical property • Determination of number of degrees of freedom for construction • Numerical interpolation examples with the derived curves