Calculate the area intersected by a sphere and a rectangular prism

guv · Feb 11, 2024

Think of a 3D rectilinear grid made of these rectangular cells, some of the cells will intersect with the sphere. I am trying to compute each intersecting area and the total sum. Ideally the total sum of the intersecting area should be close to ##4 \pi r^2##. I have not found any literature discussing this problem. Part of the problem is that the polygon on the sphere as a result of the intersection is not necessarily made of geodesics. Moreover, part of the rectangular prism can embed inside the sphere causing irregular intersecting shape. I am looking for ideas how to get this done correctly. Has anyone seen any problem. I would imagine this might be useful in computer science, but obviously people there are not too concerned with accuracy of the result. Thanks!

Mark44 · Feb 11, 2024

guv said:

Homework Statement: I am trying to figure out the area on the surface of a sphere as a result of the sphere intersecting a much smaller rectangular prism.
Relevant Equations: N/A

Think of a 3D rectilinear grid made of these rectangular cells, some of the cells will intersect with the sphere. I am trying to compute each intersecting area and the total sum. Ideally the total sum of the intersecting area should be close to ##4 \pi r^2##. I have not found any literature discussing this problem. Part of the problem is that the polygon on the sphere as a result of the intersection is not necessarily made of geodesics. Moreover, part of the rectangular prism can embed inside the sphere causing irregular intersecting shape. I am looking for ideas how to get this done correctly. Has anyone seen any problem. I would imagine this might be useful in computer science, but obviously people there are not too concerned with accuracy of the result. Thanks!

If the polyhedron is inside the sphere the only possible points of intersections will be at points or edges on the polyhedron where two or more polygons meet. At least that's what I come up with based on your description.

WWGD · Feb 11, 2024

How are these figures given? Do you have their equations, or are they given in a different format?

Frabjous · Feb 11, 2024

guv said:

Has anyone seen any problem. I would imagine this might be useful in computer science, but obviously people there are not too concerned with accuracy of the result.

They call it interface reconstruction and are very concerned with accuracy vs. computational cost.

guv · Feb 12, 2024

Take a look at the pictures attached, I am guessing 'interface construction' is the algorithm that results in what we see in the pictures. How about area calculation? Is there any literature on how to find the intersection and area in the general case? The intersection looks like a simpler problem, the area seems difficult.

Calculate the area intersected by a sphere and a rectangular prism

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