Helix splines as an example of affine Tchebycheffian splines

C1 Smooth Triangular Surface Patch Constructed by C-curves

“…(1) - (4) . C-curves can provide exact representations of circles (the most important conics in engineering), cones and ellipses that the cubic curves can not.…”

“…In order to avoid the inconveniences of the rational B-splines model, finding new basis in new spaces seems to be the only way (6) . This is resolved by Zhang (1) - (3) , Pottmann and Wagner (4) , who investigated curves generated by the space span {sint,cost,t,1}. These authors called such curves Ccurves and helix splines, respectively.…”

“…C-curves are developed by using the basis {sint,cost,t,1} by Zhang(1) -(3) , Pottmann and Wagner (4) , and it overcomes some shortcomings of the non-uniform rational B-splines (NURBS) model. For example, it can represent exactly transcendental curves such as the helix and the cycloid.…”

C-Bézier Curves and Surfaces

“…Professor Pottmann got many useful results from the general theory of Chebyshevian space [3,4]. He gives more underlying theories, mainly discusses C-Bézier curves, and calls them helix splines.…”

Piecewise Smooth Spaces in Duality: Application to Blossoming

“…The first one, used by H.-P. Seidel for geometrically continuous polynomial splines [27], then by R. Kulkarni, P.-J. Laurent, and M.-L. Mazure for Q-splines [10,18], and later by H. Pottmann [22,23,29] for splines based on a given Chebyshev space (see also M.-L. Mazure and H. Pottmann [21], M.-L. Mazure [14,16,17], and M.-L. Mazure and P.-J. Laurent [20]), relies on geometric properties: the blossom is defined by means of intersections of convenient osculating flats.…”