Geometric characteristics of planar quintic Pythagorean-hodograph curves

Interpolation of G1 Hermite data by C1 cubic-like sparse Pythagorean hodograph splines

“…Conditions (11) mean that the angles \(q 1 q 0 , d 0 ) and \(d 1 , q 1 q 0 ) both belong to ]0, ⇡ 2 [. They ensure the existence of a unique point q 0,1 common to the two lines passing by q 0 , q 1 with directions d 0 , d 1 , respectively.…”

“…. , r + 1, such that any two consecutive pairs (q k , d k ), (q k+1 , d k+1 ) are tight in the sense of (11) (for short, we will say that the data are tight). Repeating the process provides us with infinitely many G 1 cubic-like sparse PH-spline curves interpolating the data.…”

“…This is motivated by the fact that the advantages / inconveniences presented by the interpolating splines described in the previous subsection by comparison with the cubic case, are similar to those described in Section 2, namely: more flexibility but no symmetry preservation. To clarify the question of symmetry, take three pairs of G 1 data (q k , d k ), k = 0, 1, 2, satisfying two by two the tightness conditions (11), and assume the two triangles [q 0 , q 0,1 , q 1 ] and [q 1 , q 1,2 , q 2 ] to be symmetric of each other, as in Fig. 7.…”

“…Pythagorean hodograph planar curves (for short, PH curves) form a remarkable family of parametric polynomial curves with many useful features and characteristics. They have been extensively studied during the last three decades, see [12], [39,15,37,2,21,42,10,9,23,8,11,13,14], and other references therein. In particular, it is well known that, among all cubic planar parametric curves, cubic PH curves can be characterised by geometric properties of their control polygons: equal interior angles and constant ratios between consecutive segments [17].…”