We follow the Farin-Boehm construction, see "Geometric Continuity of Parametric Curves: Constructions of Geometrically Continuous Splines" by Brian A. Barsky and Tony D. DeRose. Without loss of generality we set \(\Delta_F = [0,1]\) and \(\Delta_G = [1,2]\). By the theorem on the characterization of Geometric continuity we have \[ \begin{align} G'(1) &= \beta_1 F'(1)\\ G''(1) &= \beta_1^2 F''(1) + \beta_2 F'(1)\\ \end{align} \] where \(\beta_1\) and \(\beta_2\) are the \(\beta\)-constraints or also called shape parameters. We use the expresions of the derivative of Bézier curves in terms of the control points for bouth curves and obtain: \[ \begin{align} G^0-\text{continuity: } &\mathbf{b}_n = \mathbf{c}_0\\ G^1-\text{continuity: } &\beta_1(\mathbf{b}_n - \mathbf{b}_{n-1}) = \mathbf{c}_1 - \mathbf{c}_0\\ G^2-\text{continuity: } &\beta_1^2 (n-1)(\mathbf{b}_n - 2\mathbf{b}_{n-1} + \mathbf{b}_{n-2}) + \beta_2(\mathbf{b}_n - \mathbf{b}_{n-1}) = (n-1) (\mathbf{c}_2 - 2\mathbf{c}_{1} + \mathbf{c}_{0}) \end{align} \] Then, given the left curve \(F\) we construct the \(C^2\) continuity at \(u = 1\) by setting the first three coefficients \(\mathbf{c}_0, \mathbf{c}_1, \mathbf{c}_2\) of \(G\) as follows:
Select and drag control points in order to interact with the graphic representation. Use the sliders to change the values of the beta constraints, and observe how the curve behaves.