The C1 continuity you refer to is relative to the parametrization t=0..1 per segment. This is useless in practice. For e.g. animation purposes, you need to rep

The C1 continuity you refer to is relative to the parametrization t=0..1 per segment. This is useless in practice. For e.g. animation purposes, you need to reparametrize to arc length which destroys the apparent C1 continuity.