3 ms·

"Build a hyperplane (analog of a "line" in higher dimensions) that passes through C and is perpendicular to the line segment connecting A and B." Isn't that ju

"Build a hyperplane (analog of a "line" in higher dimensions) that passes through C and is perpendicular to the line segment connecting A and B." Isn't that just a fancy way of saying a line from O (a tuple of zeroes) to C to infinity?

dgacmu 3y ago
No, it's more restricted than that. Think of 2d planes in 3 space: There are an infinite number of planes that you can construct on a line from A to C (they're all rotations of a plane extending out from that line).
- Solvency 3y ago
  Okay I'm trying to map this visually in my brain. So, in 3 space, say I extrude a random plane out from a line segment. I keep going, now I imagine I've extruded every possible plane approaching infinity. This eventually forms... a cylinder, no? So a hyperplane is what, a cylinder? Or is it shorthand for "every possible plane..."
  - deleted 3y ago
    [deleted]
  - DougBTX 3y ago
    No, rotate by 90 degrees. The plane is normal to the line, so the line doesn’t lie in the plane.
  - wongarsu 3y ago
    In an n-dimensional space, an infinite n-1-dimensional thing is a hyperplane. Or put more simply: it's something that divides the space in two. In 1 dimension that's a point, in 2 dimensions it's a line, in 3 dimensions it's a plane (hence the name), for more dimensions we just call it a hyperplane. Now there are various ways to define which hyperplane you speak about. The easiest to reason about is to give a couple points that are on the hyperplane (1 point defines a point, a line can be defined by any two points on that line, a plane by three points on that plane etc.). A slightly weirder but equally useful way is to say "it's orthogonal to this line, and intersects this point". That's easy to reason about in three dimensions, but apparently works in higher dimensions as well.
medium_spicy 3y ago
No, this is only true if that hyperplane contains the origin; imagine an infinite number of hyperplanes that contain A and B; there are an infinite number of such planes for dimensions higher than 2. Now imagine the same, but connecting O and C; most of those AB hyperplanes are not orthogonal to those OC hyperplanes, it’s only coincidence if they are, though for dimensions higher than 2 you can always find a point C that happens to lie along the orthogonal line from the AB plane to the origin.

dgacmu 3y ago

No, it's more restricted than that. Think of 2d planes in 3 space: There are an infinite number of planes that you can construct on a line from A to C (they're all rotations of a plane extending out from that line).

Solvency 3y ago
Okay I'm trying to map this visually in my brain. So, in 3 space, say I extrude a random plane out from a line segment. I keep going, now I imagine I've extruded every possible plane approaching infinity. This eventually forms... a cylinder, no? So a hyperplane is what, a cylinder? Or is it shorthand for "every possible plane..."
- deleted 3y ago
  [deleted]
- DougBTX 3y ago
  No, rotate by 90 degrees. The plane is normal to the line, so the line doesn’t lie in the plane.
- wongarsu 3y ago
  In an n-dimensional space, an infinite n-1-dimensional thing is a hyperplane. Or put more simply: it's something that divides the space in two. In 1 dimension that's a point, in 2 dimensions it's a line, in 3 dimensions it's a plane (hence the name), for more dimensions we just call it a hyperplane. Now there are various ways to define which hyperplane you speak about. The easiest to reason about is to give a couple points that are on the hyperplane (1 point defines a point, a line can be defined by any two points on that line, a plane by three points on that plane etc.). A slightly weirder but equally useful way is to say "it's orthogonal to this line, and intersects this point". That's easy to reason about in three dimensions, but apparently works in higher dimensions as well.

medium_spicy 3y ago

No, this is only true if that hyperplane contains the origin; imagine an infinite number of hyperplanes that contain A and B; there are an infinite number of such planes for dimensions higher than 2. Now imagine the same, but connecting O and C; most of those AB hyperplanes are not orthogonal to those OC hyperplanes, it’s only coincidence if they are, though for dimensions higher than 2 you can always find a point C that happens to lie along the orthogonal line from the AB plane to the origin.