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The Beta-spline introduced recently by Barsky is a generalization of the uniform cubic B-spline: parametric discontinuities are introduced in such a way as to preserve continuity of the unit tangent and curvature vectors at joints (geometric continuity) while providing bias and tension parameters, independent of the position of control vertices, by which the shape of a curve or surface can be manipulated. Using a restricted form of quintic Hermite interpolation, it is possible to allow distinct bias and tension parameters at each joint without destroying geometric continuity. This provides a new means of obtaining local control of bias and tension in piecewise polynomial curves and surfaces.