Cubic polynomial A-splines: G1, G2 and inflection point control

We present three results concerning the construction of algebraic splines whose segments are singular cubic and admit a polynomial parameterization. We use the implicit expression of the cubic in barycentric coordinates. We choose each segment of the spline from a monoparametric family of cubic segments that interpolate an additional intermediate point. These splines are of class G1 (continuous tangent line). We also give, sufficient conditions for the spline to be of class G2; moreover, in certain cases, the curvature value in the connection points can be prescribed. Also, we offer a method for the design of concavity changing splines.