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Calculating these regions can be solved completely geometrically, by iterating through all of the circle intersections, and reducing the results into intersected and subtracted areas. However, this can be done iteratively with only a little geometry and a simple algorithm.
The points of intersection and arcs
One of the few geometric steps of this algorithm is to calculate the point of intersection of every pair of intersecting circles. This is already a well solved problem
. With the intersection points calculated and reference to the intersecting circles, we are able to divide the circles into segments.
For each of these segments we should then calculate the midpoint
of the arc and assign it an identifier, as an incrementing numerical value.
Fig 1. Arrangement of three intersecting circles. Circles labelled cⁿ, the points of intersection labelled vⁿ and arcs labelled aⁿ.
With all of the intersecting vectors calculated, it provides us with a way to navigate from vector to vector and start to create a path of arcs, that define the intersecting regions. We can then use a set of three conditions that will allow us to form only the valid regions and also ensure that the algorithm is as efficient as possible by not unnecessarily traversing arcs that will lead to an invalid region.
The conditions for a valid region
1. An arc cannot be a continuation of the previously traversed arc
Fig 2. Showing the next possible traversals after a traversal of v1 to v4 (a2).
A continuation of arcs means traversing around the same circle twice. This implies either traversing back to the previous vector or traversing across the intersection of another region, and thus forming an invalid region.
For example, as shown in Fig 2 above, taking the arc a2 (traversing from v1 to v4) the connecting arcs are a2, a3, a10 and a11. This condition allows us to exclude a2 and a3 as next possible traversals.
2. The same two arcs cannot have been traversed before
Intersecting regions consist of 2 or more arcs, and each intersecting region can be identified as unique from as few as 2 of it’s arcs, in other words no 2 arcs can be traversed, in any direction twice. Each arc can be identified by the assigned numerical value (from the initialisation phase), and can be represented in binary form as a Bitset. Each valid traversal of an intersecting region can then be stored in a map as a bitwise OR of the two Bitsets. This is then used to exit early out of traversals that have been traversed before.
Fig 3. A valid region with the traversal path of v1 to v4 (a2), v4 to v5 (a10) and v5 to v1 (a5).
Fig 4. Showing the next possible traversals after a traversal of v5 to v4 (a1).
For example, taking a traversal of v1 to v4 (a2) and v4 to v5 (a10) from the complete valid region show in Fig 3, this would give us a Bitset of ‘10000000100` which is the bitwise OR of ‘100` (4) and ‘10000000000` (10). In a later traversal, shown in Fig 4, of v5 to v4 (a10), the next connecting arcs are again a2, a3, a10 and a11. This condition allows us to exclude the arc a2, because even though the traversal of v5 to v4 to v1 is different to the original traversal of v1 to v4 to v5,the Bitsets remain the same and it would lead to the identical region shown in Fig 3.
3. The midpoint of an arc must exist inside or outside of the non-intersecting circles of the endpoints of all previously traversed arcs
Each traversable arc has 3 related circles. 1 is the circle belonging to the circumference that is being traversed and the 2 other circles that formed the 2 endpoints of the arc. Note that this is not the case of 2 intersecting circles, these 3 related circles would only reference 2 circles but the relationships still apply.
For every valid arc traversal, the midpoints of the arcs can be used to determine the space to which all subsequent midpoints must all exist inside. If the midpoint lies inside the area of either of the non-intersecting circles then all future arc midpoints must also exist inside the area of that circle. Likewise, if the midpoint lies outside the area of either of the non-intersecting circles then all future arc midpoints must also exist outside the area of that circle.
Fig 5. Showing the next possible traversals after a traversal of v1 to v4 (a2) and v4 to v6 (a11).
Using the same arrangement of circles and, as shown in Fig 5, starting with a traversal of v1 to v4 (a2), we can say that the midpoint of a2 exists within the circles of c2 and outside of the circle c3. The previous two rules allow us to have the traversal v4 to v6 (a11) as a following valid traversal. Using this condition, we can rule out a7 because the midpoint exists inside the circle c3, which the first arc (a2) midpoint does not. Leaving us with only a6 as a valid traversal, and for this specific example connect us back to our starting vector (v1), thus completing a valid region.
If we iterate through all of the intersecting vectors using the method described above we can build up a complete set of the intersecting and subtracted areas. To test out this theory, I built an online application for creating artwork by filling in the intersecting regions of circles, which can be found here