On control polygons of quartic Pythagorean–hodograph curves

Pythagorean hodograph spline spirals that match G3 Hermite data from circles

“…(13) and (14) only depend on the ratio α 0 /α 1 if α 1 ̸ = 0, or α 1 /α 0 if α 0 ̸ = 0, and that considering parameters α 0 = α 1 = 1 in relations (10) yields a PH quartic curve which coincides with a degree elevated T-cubic [23].…”

“…A direct analysis shows that for any value α 0 ≥ 0 and α 1 ≥ 0, α 0 and α 1 being not simultaneously zero, we have [23] L 0L2 ≤L 2 1 ≤ 4 3L 0L2 .…”

“…We consider PH quartic curves as introduced in [23]. Thanks to two auxiliary control points and two parameters (see Proposition 3), we analyze the geometric properties of these PH quartics for the purpose of our construction, as specified in Section 2.5:…”

“…with constant geometric parameters λ i , γ i , µ i defined by relations (26) and (28) and with functions d i (u), x i (u, v) defined by relations (21), (23) and (24), for i = 0, 1.…”

Geometric characteristics of a class of cubic curves with rational offsets

“…While in general the algebraic structure of the polynomial curves with rational offsets is not simply transferred to intuitive constraints on the control polygon geometry, Farouki and Sakkalis provided an elegant geometric characterization for cubic PH curves [9], which are two constraints on the lengths of legs and two interior angles of the Bézier control polygons. Wang and Fang derived the geometric characterization of quartic PH curves also in terms of Bézier control polygons [30]. This thus provides a geometric approach for constructing PH quartics.…”