From plane to hyperplane in high dimension

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SUMMARY

The discussion centers on the geometric concepts of affine subspaces in N-dimensional space. Specifically, it establishes that given N points in N-dimensional space, one can construct a unique (N-1)-dimensional affine subspace. The inquiry also addresses the challenge of calculating the distance from a point to such an affine subspace, highlighting its complexity compared to simpler cases like lines and hyperplanes. Participants suggest exploring the mathematical foundations of affine subspaces and distance calculations for further understanding.

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  • Understanding of Euclidean geometry and affine spaces
  • Familiarity with N-dimensional space concepts
  • Knowledge of distance metrics in geometry
  • Basic linear algebra principles
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  • Research "affine subspaces in N-dimensional geometry"
  • Study "distance from a point to an affine subspace"
  • Explore "linear algebra applications in geometry"
  • Investigate "properties of hyperplanes in higher dimensions"
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uwowizard
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Hello All,

It's rather a simple question for advanced people.

Consider a 3D Euclidean space: if one is given 2 points, a line can be build that goes through the points; for 3 points -- there is a plane.

For 4D space: a line goes through 2 points and a hyperplane through 4 points.

The question -- what's the name for the object that can be build with given 3 points in 4D space? And more generally, for (N-i) points, 1<i<N-2, in N-dimensional space?

Where can I look for more info on this topic.

Thanks in advance!
 
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Well, through n points goes exactly one (n-1)-dimensional affine subspace. Maybe that's what you're looking for?
 
Thank you! I'm looking for a way to find distance from a point to an object (line, hyperplane, etc) that goes through n points.. I've just done a search on a distance from a point to an affine subspace and it seems that it's not a trivial topic (at least not as simple as in the case of a line or hyperplane).

Would there be a suggestion in which direction to go?
 

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