{
  "id": "1901.06885",
  "version": "v1",
  "published": "2019-01-21T11:33:25.000Z",
  "updated": "2019-01-21T11:33:25.000Z",
  "title": "B-spline-like bases for $C^2$ cubics on the Powell-Sabin 12-split",
  "authors": [
    "Tom Lyche",
    "Georg Muntingh"
  ],
  "categories": [
    "math.NA",
    "cs.CG"
  ],
  "abstract": "For spaces of constant, linear, and quadratic splines of maximal smoothness on the Powell-Sabin 12-split of a triangle, the so-called S-bases were recently introduced. These are simplex spline bases with B-spline-like properties on the 12-split of a single triangle, which are tied together across triangles in a B\\'ezier-like manner. In this paper we give a formal definition of an S-basis in terms of certain basic properties. We proceed to investigate the existence of S-bases for the aforementioned spaces and additionally the cubic case, resulting in an exhaustive list. From their nature as simplex splines, we derive simple differentiation and recurrence formulas to other S-bases. We establish a Marsden identity that gives rise to various quasi-interpolants and domain points forming an intuitive control net, in terms of which conditions for $C^0$-, $C^1$-, and $C^2$-smoothness are derived.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-21T11:33:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "41A15",
      "65D07",
      "65D17"
    ],
    "keywords": [
      "b-spline-like bases",
      "powell-sabin",
      "simplex spline bases",
      "maximal smoothness",
      "basic properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}