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I have two separated arcs ($AB$ and $CD$) and I want to calculate a new arc ($BP$) so that:

  • The arc $BP$'s start tangent must be the same of the arc $AB$'s end tangent.
  • The arc $BP$'s end tangent must be the same to the tangent of a point lying on the arc $CD$.
  • The arc $AB$'s end tangent is fixed.
  • The new arc ($BP$) must not intersect with arc $CD$

Is there a way to calculate the green arc ($BP$)?

Thanks.

The problem:

The problem

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  • $\begingroup$ Your final point is somewhat vague. What do you mean, not intersect? $\endgroup$ – user2277550 Sep 23 '16 at 13:04
  • $\begingroup$ Please have a look at this photo: i.imgur.com/E5c8HcB.png $\endgroup$ – TesX Sep 23 '16 at 13:09
  • $\begingroup$ Yes, I guess that is taken care of by my answer. Where do you have a problem with it? $\endgroup$ – user2277550 Sep 23 '16 at 13:12
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There are an infinite number of curves of that kind. One way to do that would be to consider the curve as $$ y^2= mx^2+nx $$ $$ 2y \frac{dy}{dx}= 2mx+n $$

From the second equation your can get 2 simultaneous equations in m and n for the two points B and C.

Hence for the 2 points B and C you get two equations.

The resulting curve would be that of a hyperbola.

As for the final property of non intersection, you would need to know the character of the given curves.

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