Does the tangent match necessarily take away 2 degrees of freedom? It seems that as long as the line connecting the endpoint and the next point has the correct gradient, the tangents should line up and that only takes away one degree of freedom.
@xTheBHox Really terrific question: what you mention is the difference between C^1 continuity (tangents are equal in both direction and magnitude) and G^1 continuity (tangens only agree in direction, but not magnitude). The former generally looks a bit nicer, but it’s true that in some cases you can get away with just geometric continuity rather than traditional analytic continuity.
Are these shape functions used in Finite elements?? Seem to be similar
@nihar There are some finite elements based on Bernstein or Bezier bases, yes.