Track transition curve

[ValeryOchkov]

Where is my or Mathcad error? May be I must have not a & b, but a, b & c?

See please the attach.

Answer please as Prime 6 or pic or pdf

[Werner_E]

You simply wish for far too much!
You think that your cubic depends on two (independent) parameters, but you two parameters a and b are NOT independent and in reality your cubic only depends on one parameter and therefore the solution block cannot find a solution.

You cannot demand the starting and end points, as well as the curvature in both points (zero and r0) and use a simple cubic as transition curve.
You could, for example, use the x-coordinate of the starting point on the straight line as the second parameter in the solution blocks.

Maybe you have more luck using a clothoide instead of a cubic as transition curve.

[ValeryOchkov]

I have a solution where the origin is at the vertex of a cubic parabola and I would like to transfer the coordinates to the center of the circle.

[Werner_E]

A word of caution:

Your equations in the solve block ensure that the transition curve and the circle have a common point and the same curvature there, but they do NOT ensure that the first derivatives would match!

Here an example using a polynomial of order 4 which looks like a good solution but actually there is a slight change in direction in the point where the quartic meets the circle even though the curvatures are equal!

It can be seen it with the naked eye in the next picture which also shows a polynomial of order 4 fulfilling your constraints -  common point and matching curvatures, nonetheless a huge kink / direction change  😉

Prime 6 sheet attached

[ValeryOchkov]

More information in attach

[Werner_E]

Yes, railway transitions curves must consider much more than pure geometry and first and second derivative.

I am just referring to the Prime sheet you posted and there you simply were demanding too much.

If you wan to use just a cubic parabola, you can fix  the point on the straight line (with horizontal tangent and zero curvature) as you did when you defined your function y. But because as I had shown that function is just dependent on one single parameter, but you need three constraints for the point on the circle (position, tangent, curvature) you need two additional free and independent parameters. So you may for example fix the radius of the circle but not its exact position or the position where the transition curve meets the circle.

See attachment where I made the two coordinates of the circle center variable