From plane to hyperplane in high dimension

In summary, the conversation discusses the concept of building objects in a 3D or 4D Euclidean space with given points. The question asks for the name of the object that can be built with 3 points in 4D space and where to find more information on finding the distance from a point to an object in n-dimensional space. One suggestion is to look into the concept of an affine subspace, but finding the distance may not be as simple as in the case of a line or hyperplane.
  • #1
uwowizard
14
0
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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  • #2
Well, through n points goes exactly one (n-1)-dimensional affine subspace. Maybe that's what you're looking for?
 
  • #3
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?
 

1. What is a "hyperplane" in high dimension?

A hyperplane in high dimension is a geometric object that extends the concept of a plane in three-dimensional space to higher dimensions. It is a flat subspace of a higher-dimensional space, defined by one less dimension than the space itself. In other words, a hyperplane in n-dimensional space is a (n-1)-dimensional subspace.

2. How does the concept of a hyperplane relate to high dimensional data?

The concept of a hyperplane is important in high dimensional data analysis because it allows us to visualize and understand relationships between variables in a dataset with a large number of dimensions. Hyperplanes can be used to separate and classify data points, and they are often used in algorithms for dimensionality reduction and data clustering.

3. What is the significance of going from a plane to a hyperplane in high dimension?

In high dimensional data analysis, going from a plane to a hyperplane allows us to better understand the complex relationships between variables in a dataset with a large number of dimensions. It also allows us to identify patterns and make predictions about the data, which can be useful in various fields such as machine learning, statistics, and data mining.

4. How do scientists use hyperplanes in their research?

Scientists use hyperplanes in a variety of ways in their research, depending on their specific field of study. Some examples include using hyperplanes to visualize and analyze complex datasets, to classify data points into different categories, and to reduce the dimensionality of data in order to make it more manageable for further analysis.

5. Are there any limitations to using hyperplanes in high dimensional data analysis?

While hyperplanes are a useful tool for understanding high dimensional data, there are some limitations to their use. For example, in datasets with a very large number of dimensions, it may be difficult to accurately visualize and interpret hyperplanes. Additionally, hyperplanes may not always be able to fully capture the complexity of relationships between variables in a dataset, and other techniques may be needed for a more comprehensive analysis.

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