How to go from one straight line to another in a twice differentiable way? By a [url=https://en.wikipedia.org/wiki/Euler_spiral]clothoid[/url] or Euler spiral.

You can visualize the osculating circle, the curvature is represented by a color: red for a large curvature, green for a small one. You can also adjust the circular part of constant curvature to modify the angle between the half-lines.