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We have a horn-shaped 3d body, which is represented as a list of vertices and faces. Each face is a triangle represented by 3 vertices. The body is positioned along the Z-axis (height). We would like to make several cuts at certain heights. Each cut (a plane perpendicular to the Z- axis) may create one or more cross-sections with the body (the body may split to several branches). The question is how to find those cross-sections.

Another question: how to find (efficiently) the maximum width of the body, i.e. 2 points (not necessarily from the list of vertices) on the surface of the body with the same z-coordinate, and maximum distance between them?

Thank you

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This isn't the right forum for this question. – Peter Shor Dec 28 '10 at 14:27
up vote 0 down vote accepted

These are applied computational geometry questions, not really mathematics research questions.

Maybe try asking on stackoverflow or on a blender or CAD forum or something.

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The above should be a comment, not an answer. – Daniel Moskovich Dec 28 '10 at 15:41
But the poster didn't have enough reputation for a comment. – Peter Shor Dec 28 '10 at 19:02

how to find those cross-sections?

  • Find/write a function which intersects a triangle and a plane (e.g. Dave Eberly's wild magic library). Provided that you need to intersect only with plane perpendicular to the Z axis, you may optimize it.
  • Intersect all triangles with the plane. You will obtain a list of segments (some of them may be of null length) - let's call it the raw list.
  • Make a list of lists (I will name them section-lists), like this: Take one segment. Search through the others which segment has a vertex in common with yours. Make sure to makle the comparison within a tolerance. Whenever you move one segment to a section-list, remove it from the raw list. When you complete section-list by adding to it a complete polygon, start a new section-list and fill it starting with the next segment (repeat).
  • Now you have a list of polygons which are the cross-section. The cross-section may be disconnected, that is, you have more elements in your list.
  • You can try to optimize it by BSP.

how to find (efficiently) the maximum width of the body

  • It is enough to intersect with planes having Z=Z of the vertices in your mesh.
  • To optimize, calculate just the AABB's for each Z, and make a list sorted descending by the diagonal of the AABB.
  • Take the first element, and find the maximum diameter (say "current_width") (it is enough to do this for the vertices of the cross-section polygon)
  • From the diagonal-sorted list, keep only those with the diagonal greater than the current_width (you just need an iterator, will call it "bookmark")
  • Move to the next element of the list, find the new current_width, and if it is greater, replace the old one and remove again from the list the elements with smaller diagonal (or update the bookmark).
  • Do this until you reach the end of the list (or the bookmark)
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