We present a shape-preserving subdivision scheme with a tension parameter that generalizes the four-point
Deslauriers-Dubuc scheme and the cubic B-spline. Whereas many shape-preserving schemes are non-linear and
non-uniform, the proposed scheme is linear and stationary. The refinement rule has the same support length
as the four-point scheme and provides fourth-order accuracy. The scheme is nearly interpolant such that by
sacrificing the interpolating property, it attains an improved smoothness, that is C2, while the interpolatory
four-point scheme is C1. In addition, we show that the proposed scheme preserves monotonicity and convexity
under some mild conditions. Some numerical examples are presented to illustrate the performance of the
proposed scheme.