Intersection between two Arcs? (arc = distance between pair of angles)

Question

I\'m trying to find a way to calculate the intersection between two arcs. I need to use this to determine how much of an Arc is visually on the right half of a circle, and h

Answer 1

This test can be resumed with a one-line test. Even if a good answer is already posted, let me present mine.

Let assume that the first arc is A:(a0,a1) and the second arc is B:(b0,b1). I assume that the angle values are unique, i.e. in the range [0°,360°[, [0,2*pi[ or ]-pi,pi] (the range itself is not important, we will see why). I will take the range ]-pi,pi] as the range of all angles.

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To explain in details the approach, I first design a test for interval intersection in R. Thus, we have here a1>=a0 and b1>=b0. Following the same notations for real intervals, I compute the following quantity:

S = (b0-a1)*(b1-a0)

If S>0, the two segments are not overlapping, else their intersection is not empty. It is indeed easy to see why this formula works. If S>0, we have two cases:

• b0>a1 implies that b1>a0, so there is no intersection: a0=.

• b1 implies that b0, so there is no intersection: b0=.

So we have a single mathematical expression which performs well in R.

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Now I expand it over the circular domain ]-pi,pi]. The hypotheses a0 and b0 are not true anymore: for example, an arc can go from pi/2 to -pi/2, it is the left hemicircle. So I compute the following quantity:

S = (b0-a1)*(b1-a0)*H(a1-a0)*H(b1-b0)

where H is the step function defined by H(x)=-1 if x<0 else H(x)=1

Again, if S>0, there is no intersection between the arcs A and B. There are 16 cases to explore, and I will not do this here ... but it is easy to make them on a sheet :).

Remark: The value of S is not important, just the signs of the terms. The beauty of this formula is that it is independant from the range you have taken. Also, you can rewrite it as a logical test:

T := (b0>a1)^(b1>a0)^(a1>=a0)^(b1>=b0)

where ^ is logical XOR

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EDIT

Alas, there is an obvious failure case in this formula ... So I correct it here. I realize that htere is a case where the intersection of the two arcs can be two arcs, for example when -pi.

The solution to correct this is to introduce a second test: if the sum of the angles is above 2*pi, the arcs intersect for sure.

So the formula turns out to be:

T := (a1+b1-a0-b0+2*pi*((b1a1)^(b1>a0)^(a1>=a0)^(b1>=b0))

Ok, it is way less elegant than the previous one, but it is now correct.