I have a bit of a hard time understanding your original question.
What are your initial points?
What is a “flat path”?
Can you draw this original “linear” path first before you apply your transformation?
Could you provide a screenshot of what you mean with that?
Please rephrase your challenge as it will be easier to answer.
If I word what you are trying to do, you want to do a vectorial addition of your initial linear path with a sinusoidal path which will result in a curve path. And you want this path to join at the end.
Probably this post relates to your effort? Wave displaced circle
I have modified your initial code to make it dynamic. You can move your mouse along the x and y. Along the x, it adjust the amplitude of the sine wave. Along the Y, it adjust the steps to draw the curve. I use the blue color to draw your base line for the wave, red and green to plot your referential wave and yellow to be your initial line (?). This is the output:
Kf