Question

Suppose I have the following:

• A region defined by minimum and maximum latitude and longitude (commonly a ‘lat-long rect’, though it’s not actually rectangular except in certain projections).
• A circle, defined by a center lat/long and a radius

How can I determine:

1. Whether the two shapes overlap?
2. Whether the circle is entirely contained within the rect?

I’m looking for a complete formula/algorithm, rather than a lesson in the math, per-se.

Answer 1

• Yes, if the box corners contain the circle-center.
• Yes, if any of the box corners are within radius of circle-center.
• Yes, if the box contains the longitude of circle-center and the longitude intersection of the box-latitude closest to circle-center-latitude is within radius of circle-center.
• Yes, if the box contains the latitude of circle-center and the point at radius distance from circle-center on shortest-intersection-bearing is “beyond” the closest box-longitude; where shortest-intersection-bearing is determined by finding the initial bearing from circle-center to a point at latitude zero and a longitude that is pi/2 “beyond” the closest box-longitude.
• No, otherwise.

Assumptions:

• You can find the initial-bearing of a minimum course from point A to point B.
• You can find the distance between two points.

The first check is trivial. The second check just requires finding the four distances. The third check just requires finding the distance from circle-center to (closest-box-latitude, circle-center-longitude).

The fourth check requires finding the longitude line of the bounding box that is closest to the circle-center. Then find the center of the great circle on which that longitude line rests that is furthest from circle-center. Find the initial-bearing from circle-center to the great-circle-center. Find the point circle-radius from circle-center on that bearing. If that point is on the other side of the closest-longitude-line from circle-center, then the circle and bounding box intersect on that side.

It seems to me that there should be a flaw in this, but I haven’t been able to find it.

The real problem that I can’t seem to solve is to find the bounding-box that perfectly contains the circle (for circles that don’t contain a pole). The bearing to the latitude min/max appears to be a function of the latitude of circle-center and circle-radius/(sphere circumference/4). Near the equator, it falls to pi/2 (east) or 3*pi/2 (west). As the center approaches the pole and the radius approaches sphere-circumference/4, the bearing approach zero (north) or pi (south).