Intersecting lines, planes, cells, etc...

Higher-dimensional geometry (previously "Polyshapes").

Intersecting lines, planes, cells, etc...

Postby BClaw » Fri Mar 12, 2004 11:34 pm

Hey, where can I find the formulae for determining if and where two lines intersect, two planes, and so on? I fake my way through the math on a lot of this stuff, so I'm not sure how to figure this stuff out.

Thanks!

Also, where are some good resources for figuring out what the vertices coordinates are for the polytopes? Which points are connected by lines, which points comprise a face, and all that. I can do the 5-cell, 8-cell, and 16-cell, but the others just blow my mind!
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Postby Aale de Winkel » Tue Apr 06, 2004 11:59 am

Simply do the math: given the lines
line_1: P1[sub]μ[/sub] + λV1[sub]μ[/sub]
line_2: P2[sub]μ[/sub] + νV2[sub]μ[/sub]
If those line are in the same plane and are not parallel, then for a given λ and ν these form the intersecting point ie:
P1[sub]μ[/sub] + λV1[sub]μ[/sub] = P2[sub]μ[/sub] + νV2[sub]μ[/sub]
for all values of μ which range from 1 to the concidered dimension.
If no solution the lines are simply not co-planar or parallel.

Look for any school-book on vector algebra. Simular for intesection of line / plane (= point) or plane / plane (= line). You simply need to construct valid equations for the objects, and equate those equation which give you valid variable values if conditions are met.
things can get quite tricky with the various objects around, intersecting a plane with a sphere on has the three possibilities (nothing, point, circle), a general formulae should reflect these properly.
.
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