An Explicit Basis for $C^1 $ Quartic Bivariate Splines

Homology of Smooth Splines: Generic Triangulations and a Conjecture of Strang

“…This was proved for m > 5 by Morgan and Scott [17] and for m -4 by Alfeld, Piper and Schumaker [3] for any 2-manifold A.…”

“…This gives the conclusion of Lemma 5.7 for A. Morgan and Scott [17] have shown that for any planar embedding of a 2-manifold A, and for m > 5, the dimension of Sm(A) is equal to the value given in Theorem 5 plus the number of quadrilaterals in the complex which are triangulated by crossing straight lines. Alfeld, Piper and Schumaker [3] have shown this to hold for m = 4 as well, while Schumaker [19] has shown this to be a lower bound for m > 2. It is easy to see that for a single quadrilateral triangulated with a single interior vertex, the nongeneric positions are exactly those in which the four interior edges lie on two straight lines (aside from trivial degeneracies in which all the vertices of some triangle lie on some line).…”

“…Along with Piper [4, 5], they constructed explicit bases for these spaces. Alfeld, Piper and Schumaker [3] also extended the Morgan-Scott C1 results to m = 4.…”

“…In the past few years, there has been much progress on this problem [1][2][3][4][5][6]. Alfeld and Schumaker [6] extended the Morgan-Scott result for all r by showing that Schumaker's lower bound is in fact the dimension of Sm (A) when m > 4r + 1.…”

“…In fact, [17] and [3] prove an equality in which the right-hand side of (4.6) is augmented by a correction term that counts the number of degree 4 vertices whose edges lie on only two straight lines. A similar term, involving a measure of the local deficiency in slope over all vertices, appears in the inequality of Schumaker for each r. By (4.4), these correction terms are captured somehow by the spaces Hi(Wm).…”