New approach for constructing edge B-spline-like basis functions for $C^1$ and $C^2$ splines over triangulationsKnez, Marjetka (Avtor) Šteblaj, Matija (Avtor) Bernstein-Bézier form▫$C^r$▫ splines over triangulationsB-spline-like basiscontrol trianglessubdivisionSplines over triangulations provide a flexible and efficient way to approximate bivariate functions defined over domains, partitioned by triangles. One of the important challenges is the construction of locally supported non-negative basis functions that form a partition of unity, i.e., B-spline-like basis functions. Due to smoothness conditions that depend on the geometry of the triangulation, spline basis functions are typically divided into three groups: triangle, vertex, and edge functions. While the construction of triangle and vertex basis splines is well developed, the computation of non-negative edge basis splines of general degree that form a partition of unity has so far been addressed only for $C^1$ continuous splines, using a completely algebraic approach, with no explicit geometric interpretation ([7]). In this paper a novel approach is presented that extends the control-triangle-based idea known for vertex basis splines and can be geometrically explained and generalized to splines of a higher order of smoothness. The construction is developed on pairs of adjacent triangles forming a strictly convex quadrilateral. Since each smoothness order requires a separate analysis, a detailed study is provided for the $C^1$ and $C^2$ continuous splines of degrees $d \ge 2$ and $d \ge 4$, respectively. Numerical examples are provided to confirm the effectiveness and applicability of the derived construction.20272026-07-01 12:07:19Neznano30714UDK: 519.6ISSN pri članku: 0377-0427DOI: 10.1016/j.cam.2026.117914COBISS_ID: 283120643sl