We say that a curve has geometric continuity if the curve, its unit tangent vector and its curvatures relative to a Frenet frame are continuous. Here we study locally supported basis functions in spaces of piecewise polynomials which can be used for the Bézier representation of geometrically continuous curves. For spaces with identical connection conditions at each knot, we study the dependence of the basis functions on the parameters defining the connection conditions, and give a method for the computation of these functions. We also investigate the effect of the various parameters on the shape of the Bézier curve. For connection conditions prescribed by a totally positive matrix, we obtain conditions for the solvability of the cardinal interpolation problem and give a new proof of the uniqueness and non-negativity of the basis functions.