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This paper presents a new interpolation method that enables the construction of C2 cubic polynomial spline curves without solving a global system of equations, while providing slackness/continuity control and convexity preserving ability. The basic idea is to blend a cubic B-spline curve with a singularly parametrized sequence of connected line segments. A global slackness parameter controls the tautness, specifically the distance between the interpolating curve and the linear interpolant. The order of continuity at each knot is controlled via multiple knot insertions so that cusps and straight-line segments can be conveniently prescribed. In addition, a method for selecting local slackness values to produce G1 convexity preserving curve is presented. With the low-degree polynomials and direct computation of control vertices, this local method is computationally simple and is useful for interactive shape design and computer graphics applications.