Continuity of composite Bézier curves

Question

The composite curve S with pieces

where c0 = (−1, 1), c1 = (−1, 0), c2 = (0, 0), and d0 = (0, 0), d1 = (1, 0), d2 = (2, 1). What is the order of continuity of s at (0, 0)?

Answer 1

Calculate the first derivatives of $\mathbf{p}$ and $\mathbf{q}$ at $t=1$. You get $$ \mathbf{p}'(1) = 2(\mathbf{c}_2 - \mathbf{c}_1) = (2,0) $$ $$ \mathbf{q}'(1) = 2(\mathbf{d}_1 - \mathbf{d}_0) = (2,0) $$ The first derivatives agree, so the composite curve is $C_1$.

Do the same sort of calculation with second derivatives, and you will find that they do not agree, so the curve is not $C_2$.