# how can one calculate the minimum and maximum distance between two given circular arcs?

how can one calculate the minimum and maximum distance between two given circular arcs?

I know everything of each arc: startangle, endangle, center, radius of arc. The only thing I don't know how to calculate is the min and max distance.

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you can set it up as a variational problem – yoyo Dec 29 '11 at 19:04
I think it would be better if we knew the equations of the arcs. However if they have a point of intersection, the of course the minimum distance between them will be zero. – smanoos Dec 29 '11 at 19:06
Are the arcs coplanar? – Mark Bennet Dec 29 '11 at 19:18
@MarkBennet yes. – Mig Dec 29 '11 at 19:34

There are four cases to consider:

1) Endpoints of both arcs

2) An endpoint of one and an interior point of the other, which is on the line through that endpoint and the centre of the other arc.

3) Interior points of both arcs, which are on the line through the centres of the two arcs.

4) Intersections of the two arcs (thanks for pointing that out, Lopsy)

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"4) Intersections of the two arcs" too, surely? – Lopsy Dec 29 '11 at 20:14
@Lopsy - that looks right But other than that Robert gave the answer I was just about to post. – Mark Bennet Dec 29 '11 at 20:29
Thanks @robert-israel – Mig Jan 2 '12 at 20:06

If you know the center of the circular arcs (not meaning its midpoint but a point from which the distance to all the points on the arc is the same, just to avoid confusion), then you can easily calculate the minimum distance. Just use the distance formula to find the distance between the centers and subtract (add alternatively) the radii. This will yield the min. distance. The only problem might be that in some cases the arcs will not face each other directly and if that is the case, you would have to proceed differently. You would have to set a distance function that would express distance between one arc and second arc. This can be done by finding the min. distance between arc and the other arc's center and its radius and plugging it into the cosine rule. Minimizing this will give you the angle at which is the point closest to the other arc.

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