6 ms·

If a point is the intersection of two lines, and a line is the intersection of two planes, then what structures intersect to give a plane? Two 3D spaces? And so

by bawana 6y ago

If a point is the intersection of two lines, and a line is the intersection of two planes, then what structures intersect to give a plane? Two 3D spaces? And so the intersection of 2 4D spaces should give a 3D space? How do I express these ideas with GA?

jbay808 6y ago
Total non expert here but your fascinating question got me thinking out loud. Two 3D volumes in a 3D space obviously intersect to give another 3D volume, unless they're tangent. So at first glance I'd be tempted to say no, but... Two overlapping polygons in a 2D space also define another 2D polygon. Two overlapping line segments in a 1D space define another line segment. You only get a reduced-dimension object at the intersection if they're intersecting in a higher dimensional space. Otherwise they can't intersect "at an angle". So I'd say purely by following the pattern that it must be that two 3D spaces intersecting "at an angle" within a 4D space define a plane by their intersection.
- bawana 6y ago
  That's why I ask how to formally describe this problem. It's too 'squishy' to say two 3d spaces intersecting at an angle in a 4d space. I thought perhaps I could use time to untangle this visualization-defining a solid as the result of taking a plane and sweeping it through space. So, envisioning the 'growth' of a solid from a moving plane and imagining the intersection of the growing solid with the another growing solid from a different plane.
  - Kliment 6y ago
    It's not too squishy. The important thing to understand here is that a plane is an infinite object in a given subspace - just like a line is. Two infinite objects intersecting give you an infinite object in the subspaces they share, and a zero-sized object in the subspaces they don't share. The shared subspace of two planes in 2D is obviously the same plane. The shared subspace of two planes in 3D is obviously either zero (two parallel nonincident planes), a line (the intersection that they share) or a plane (two incident planes). So you get a null, a one-dimensional subspace, or a two-dimensional subspace, depending on how many shared subspaces you have between the objects. So a pair of three-dimensional infinite objects, in four-dimensional space, may either be disjoint, resulting in a null, incident, resulting in an infinite volume, or intersecting, resulting in an infinite plane. The source of confusion here is that we're mixing up (bounded) volumetric shapes, line segments, and planar polygons, with their infinite counterparts - the lines, points, planes, and infinite volumes. An infinite object has no shape - it is infinite in some dimensions and zero in all others - this is what gives us those clear well-defined intersection objects, which are either zero or infinite. A bounded object has a shape - it has a boundary that is not infinite. When you intersect such objects within their shared subspaces (two line segments on the same line, two polygons on the same plane, two volumes in the same space) you get either a null, or an object of the same type. This is obvious when you make one of the objects infinite - an intersection of a line segment with its line gives you the same line segment. So the reason you are having difficulty imagining two somethings that will intersect in a plane is that a plane is an infinite object, so only two infinite objects can intersect in a plane. So you need two infinite volumes, that pass through the same 4D space, but are not the same volume. These volumes share a two-dimensional subspace - the area of 4D space they intersect in is a plane. It's infinite in two dimensions, and zero-sized in two others.
    - jbay808 6y ago
      Ah, I was starting with bounded objects on purpose to build intuition, but I tried to be clear in my language about when I was talking about bounded vs unbounded objects. I was aware that the question was about unbounded objects. I thought it would be easier this way because it's hard to visualize intersecting 3D spaces correctly, so I wanted to show that picturing them as bounded volumes leads to the wrong conclusion by way of analogy to lower dimensional objects. I left out the null intersection case for brevity. Thanks for the more thorough explanation!
AnHonestComment 6y ago
We need to clean up your model: Not just any two lines intersect at a point, they need to be non-parallel. Parallel lines intersect either nowhere or everywhere. The same is true of planes: they must be non-parallel to intersect in a line, otherwise it’s a nowhere-or-everywhere situation. But how do we define this? We look at the intersection of spans defining each object — which overlap in one of three ways, nowhere or everywhere (parallel), or at a one-dimension smaller object. In this model, every pair of lines or planes meeting is a linear transform of a few basic situations: Two lines are the same as the origin defined by x-axis and y-axis meeting. Two planes are the same as the xy-plane meeting the xz-plane on the x-axis. Two spaces are the same as the xyz-space and the xyt-space meeting in the xy-plane (at t,z=0). In general, two n-spaces meeting in n+1 space intersect at a n-1 space, or are “parallel”. An intuition is that since each object is missing one dimension from the parent space, either they’re missing the same dimension (parallel) or their intersection will be missing two dimensions. A further intuition is that we have some space, and define each smaller space with a constraint, eg t=0. Then the intersection of subspaces has three cases: 1. The have the same constraint. 2. They have incompatible constraints, eg t=0 and t=1. 3. The intersection satisfies both constraints. The first two are cases of “parallel”.
wnoise 6y ago
A point is the intersection of two "generic" lines in an ambient 2-d space. But you can have coincident lines which intersect in a line, or parallel non-coincident lines which have no intersection. Further, in 3-d, the majority of lines are "skew", and neither parallel nor intersecting. Similarly in an ambient 3-d space, 2 generic planes intersect to give a line. But the planes can also be coincident or parallel. And in higher ambient spaces can intersect at only a point, rather than a line, or even fail to intersect in a non-parallel way. In 4-d space, the intersection of two generic 3-d spaces does indeed give a plane, with exactly similar caveats. The standard GA doesn't directly represent general lines or planes, however. The elements are the equivalent of "vectors" rather than "points", and always go through 0. The obvious way to handle these are parameterizing the lines and surfaces, but you're essentially working with equations for the surfaces, and keeping track of the variables. The slick way of handling it is with _projective_ geometric algebra, and intersections turn into "meets". The meet of two parallel lines (planes) is now a "point (line) at infinity", and of a (line, plane) with itself is the line (plane) again. Skew lines have a meet of 0 (not the point 0, the number 0).
- bawana 6y ago
  thanks again internet. some unknown intelligence, ironically named, wnoise, is confirming my intuition. That a plane is the result of two intersecting 3d volumes in a 4d space. And 'our' 3d space (where the meat lives) is the intersection of 2 4d volumes in a 5d space. But dear Wnoise, enlighten me about ' _projective_ geometric algebra, and intersections turn into "meets".' Where can I find the book, 'Projective_ geometric algebra' for dummies?
  - wnoise 6y ago
    I don't have a book recommendation handy. It's "just" a combination of two neat techniques. Get books on each of those and understand them each, and you have the combination. GA has been well discussed here, so, the other half: Projectivization is a fairly standard trick even for normal geometry. It's adding an additional dimension, which is in most contexts just set to 1. (Projective actually just means treating all points on a ray as equivalent; this loses the dimension you just gained It lets rotations and translations be treated in a nearly uniform manner, and lets you do rotations around points that aren't the origin. It's used all over the place in much graphics code (usually under the name homogeneous coördinates). The last section of https://en.wikipedia.org/wiki/Homogeneous_coordinates https://en.wikipedia.org/wiki/Homogeneous_coordinates discusses this briefly. For online resources for the combination: another commenter has recommended https://bivector.net/ https://bivector.net/ , and it looks okay, with sections specifically on 2 and 3 dimensional projective geometric algebra. There is also the nice C++ header library klein: https://www.jeremyong.com/klein/ https://www.jeremyong.com/klein/

by bawana 6y ago

jbay808 6y ago

Total non expert here but your fascinating question got me thinking out loud. Two 3D volumes in a 3D space obviously intersect to give another 3D volume, unless they're tangent. So at first glance I'd be tempted to say no, but... Two overlapping polygons in a 2D space also define another 2D polygon. Two overlapping line segments in a 1D space define another line segment. You only get a reduced-dimension object at the intersection if they're intersecting in a higher dimensional space. Otherwise they can't intersect "at an angle". So I'd say purely by following the pattern that it must be that two 3D spaces intersecting "at an angle" within a 4D space define a plane by their intersection.

bawana 6y ago
That's why I ask how to formally describe this problem. It's too 'squishy' to say two 3d spaces intersecting at an angle in a 4d space. I thought perhaps I could use time to untangle this visualization-defining a solid as the result of taking a plane and sweeping it through space. So, envisioning the 'growth' of a solid from a moving plane and imagining the intersection of the growing solid with the another growing solid from a different plane.
- Kliment 6y ago
  It's not too squishy. The important thing to understand here is that a plane is an infinite object in a given subspace - just like a line is. Two infinite objects intersecting give you an infinite object in the subspaces they share, and a zero-sized object in the subspaces they don't share. The shared subspace of two planes in 2D is obviously the same plane. The shared subspace of two planes in 3D is obviously either zero (two parallel nonincident planes), a line (the intersection that they share) or a plane (two incident planes). So you get a null, a one-dimensional subspace, or a two-dimensional subspace, depending on how many shared subspaces you have between the objects. So a pair of three-dimensional infinite objects, in four-dimensional space, may either be disjoint, resulting in a null, incident, resulting in an infinite volume, or intersecting, resulting in an infinite plane. The source of confusion here is that we're mixing up (bounded) volumetric shapes, line segments, and planar polygons, with their infinite counterparts - the lines, points, planes, and infinite volumes. An infinite object has no shape - it is infinite in some dimensions and zero in all others - this is what gives us those clear well-defined intersection objects, which are either zero or infinite. A bounded object has a shape - it has a boundary that is not infinite. When you intersect such objects within their shared subspaces (two line segments on the same line, two polygons on the same plane, two volumes in the same space) you get either a null, or an object of the same type. This is obvious when you make one of the objects infinite - an intersection of a line segment with its line gives you the same line segment. So the reason you are having difficulty imagining two somethings that will intersect in a plane is that a plane is an infinite object, so only two infinite objects can intersect in a plane. So you need two infinite volumes, that pass through the same 4D space, but are not the same volume. These volumes share a two-dimensional subspace - the area of 4D space they intersect in is a plane. It's infinite in two dimensions, and zero-sized in two others.
  - jbay808 6y ago
    Ah, I was starting with bounded objects on purpose to build intuition, but I tried to be clear in my language about when I was talking about bounded vs unbounded objects. I was aware that the question was about unbounded objects. I thought it would be easier this way because it's hard to visualize intersecting 3D spaces correctly, so I wanted to show that picturing them as bounded volumes leads to the wrong conclusion by way of analogy to lower dimensional objects. I left out the null intersection case for brevity. Thanks for the more thorough explanation!

AnHonestComment 6y ago

We need to clean up your model: Not just any two lines intersect at a point, they need to be non-parallel. Parallel lines intersect either nowhere or everywhere. The same is true of planes: they must be non-parallel to intersect in a line, otherwise it’s a nowhere-or-everywhere situation. But how do we define this? We look at the intersection of spans defining each object — which overlap in one of three ways, nowhere or everywhere (parallel), or at a one-dimension smaller object. In this model, every pair of lines or planes meeting is a linear transform of a few basic situations: Two lines are the same as the origin defined by x-axis and y-axis meeting. Two planes are the same as the xy-plane meeting the xz-plane on the x-axis. Two spaces are the same as the xyz-space and the xyt-space meeting in the xy-plane (at t,z=0). In general, two n-spaces meeting in n+1 space intersect at a n-1 space, or are “parallel”. An intuition is that since each object is missing one dimension from the parent space, either they’re missing the same dimension (parallel) or their intersection will be missing two dimensions. A further intuition is that we have some space, and define each smaller space with a constraint, eg t=0. Then the intersection of subspaces has three cases: 1. The have the same constraint. 2. They have incompatible constraints, eg t=0 and t=1. 3. The intersection satisfies both constraints. The first two are cases of “parallel”.

wnoise 6y ago

A point is the intersection of two "generic" lines in an ambient 2-d space. But you can have coincident lines which intersect in a line, or parallel non-coincident lines which have no intersection. Further, in 3-d, the majority of lines are "skew", and neither parallel nor intersecting. Similarly in an ambient 3-d space, 2 generic planes intersect to give a line. But the planes can also be coincident or parallel. And in higher ambient spaces can intersect at only a point, rather than a line, or even fail to intersect in a non-parallel way. In 4-d space, the intersection of two generic 3-d spaces does indeed give a plane, with exactly similar caveats. The standard GA doesn't directly represent general lines or planes, however. The elements are the equivalent of "vectors" rather than "points", and always go through 0. The obvious way to handle these are parameterizing the lines and surfaces, but you're essentially working with equations for the surfaces, and keeping track of the variables. The slick way of handling it is with _projective_ geometric algebra, and intersections turn into "meets". The meet of two parallel lines (planes) is now a "point (line) at infinity", and of a (line, plane) with itself is the line (plane) again. Skew lines have a meet of 0 (not the point 0, the number 0).

bawana 6y ago
thanks again internet. some unknown intelligence, ironically named, wnoise, is confirming my intuition. That a plane is the result of two intersecting 3d volumes in a 4d space. And 'our' 3d space (where the meat lives) is the intersection of 2 4d volumes in a 5d space. But dear Wnoise, enlighten me about ' _projective_ geometric algebra, and intersections turn into "meets".' Where can I find the book, 'Projective_ geometric algebra' for dummies?
- wnoise 6y ago
  I don't have a book recommendation handy. It's "just" a combination of two neat techniques. Get books on each of those and understand them each, and you have the combination. GA has been well discussed here, so, the other half: Projectivization is a fairly standard trick even for normal geometry. It's adding an additional dimension, which is in most contexts just set to 1. (Projective actually just means treating all points on a ray as equivalent; this loses the dimension you just gained It lets rotations and translations be treated in a nearly uniform manner, and lets you do rotations around points that aren't the origin. It's used all over the place in much graphics code (usually under the name homogeneous coördinates). The last section of https://en.wikipedia.org/wiki/Homogeneous_coordinates https://en.wikipedia.org/wiki/Homogeneous_coordinates discusses this briefly. For online resources for the combination: another commenter has recommended https://bivector.net/ https://bivector.net/ , and it looks okay, with sections specifically on 2 and 3 dimensional projective geometric algebra. There is also the nice C++ header library klein: https://www.jeremyong.com/klein/ https://www.jeremyong.com/klein/