Planar EPH curves: from algebraic and geometric characterizations to constructive strategies

EPH curves are the exponential-polynomial counterparts of the well-known polynomial PH (Pythagorean-Hodograph) curves and, unlike the latter, include a free parameter that offers greater flexibility and proves to be very promising for applications. While PH curves, known for several years, have been already well characterized both algebraically and geometrically, EPH curves still lack significant results in terms of characterization and construction. The first part of this thesis is thus devoted to providing algebraic and geometric characterizations for planar regular EPH curves defined in exponential-polynomial spaces of order 4 and 6, as they generalize the cubic and the quintic polynomial case, respectively. Mimicking the polynomial case, the proposed algebraic characterizations consist of one or more polynomial equations written in terms of the edges of the Bézier control polygon as complex numbers, while the geometric characterizations are based on similarities between quadrilaterals or triangles, as well as on relationships between edges and interior angles of the Bézier control polygon itself. All these characterizations also involve the shape parameter of the EPH curve, and by taking the limit as it approaches zero, the well-known results for polynomial PH curves are recovered. The second part of this thesis is then focused on planar EPH curves of order 6 and on strategies for their geometric construction. More precisely, after fixing two control edges, it is shown how the remaining ones can be determined by solving a low-degree complex polynomial equation. Since an unconstrained modification of the control points rarely preserves the PH nature of the curve, the final issue addressed in this work concerns the adjustment of the Bézier control points of EPH curves of order 6. Specifically, a structured analysis is presented to determine the appropriate displacements, i.e. those that ensure the resulting curve remains an EPH curve.