I have two Arcs that intersect each other.

One Arc should be translated (Arc B) while the other must stay fixed in place (Arc A).
How can I find the minimum vector that solves the intersection of Arc B in one specific direction?

In my case, for example, I can move Arc B only upwards (+Y).

Is there a mathematical function that works for every possible case (eg. same / opposite orientation, A above / below B, etc...)?

The Problem


PS: Sorry, I am not an expert in mathematics...

PPS: The desired result
The desired result

  • $\begingroup$ What do you mean by"vector that solves the intersection"? $\endgroup$ – Qwerty Jun 7 '16 at 14:17
  • $\begingroup$ And even experts post questions here, so no need to feel guilty. $\endgroup$ – Qwerty Jun 7 '16 at 14:17
  • $\begingroup$ @Qwerty For example v(0, +15). In this case Arc B should move up by 15 units in Y to not intersect anymore with Arc A. $\endgroup$ – TesX Jun 7 '16 at 14:19
  • $\begingroup$ Well moving up is not the only solution.It could move sideways or any other direction to a certain distance to stop being intersected by Arc A $\endgroup$ – Qwerty Jun 7 '16 at 14:22
  • $\begingroup$ @Qwerty In my case, though, I only need one specific direction (eg. up). $\endgroup$ – TesX Jun 7 '16 at 14:23

Find the max value of the y coordinates of Arc A. Keep increasing all Y coordinates of Arc B until the minimum Y value of Arc B exceeds The max y value of Arc A.

If your A is above B then put min in place of max and vice versa..

  • $\begingroup$ That wouldn't work when Arc B is flipped upside down though... i.imgur.com/H94pDOQ.png $\endgroup$ – TesX Jun 7 '16 at 14:58
  • $\begingroup$ @TesX this will work.. Upvote if you like my answer $\endgroup$ – Qwerty Jun 7 '16 at 15:01

Eliminate x from equations of these two circles.

$$ ( x- H)^2 + ( y- K)^2 = R_A^2; \, ( x- h)^2 + ( y- k)^2 = R_B^2; \, $$

Equate the discrimnant of obtained quadratic equation in $y$ to zero.

You will get $K$ in terms of $ h,k, R_A, H, R_B $

Can you take it up further? If not I could hint further.

  • $\begingroup$ I'm sorry but how can I eliminate x from those equations? -.- $\endgroup$ – TesX Jun 7 '16 at 15:42

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