In this work we present the blossoming principle: for every polynomial map F of degree n between affine spaces there exists a symmetric, multiaffine map f in n variables, which agrees with F on the diagonal. We call f the (multiaffine) blossom of F. With it we can represent each polynomial curve or surface as a Bezier curve or triangular Bezier patch. The control points of said curve or patch are determined by the values of the blossom f on a chosen affine frame. Utilising the blossoming principle, we also describe splines of Bezier curves and splines of triangular Bezier patches and their smoothness. We show that the conditions for a spline to satisfy a particular order of continuity can be expressed as equalities between the blossoms of both curves or surfaces on a specific collection of arguments.
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