I am currently reading the paper G.Salton et al, A Vector Space Model for Automatic Indexing, November 1975, Volume 18

It is about indexing documents. Representing them in vectors to find similarities.

But this particular part of the paper:

Paragraph 3 in Document Space Configuration

Instead of identifying each document as by complete vector, originating at the 0-point in the coordinate system, the relative distance between the vectors is preserved by normalizing all vector lengths to one, and considering the projection of the vectors onto the envelope of the space represented by the unit sphere. In that case, each document may be depicted by a single point whose position is specified by the area where the corresponding document vector touches the envelope of the space"

I do not understand how they are going to find the projection of a vector on a unit sphere. Can someone please explain?

Am I missing something?

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    $\begingroup$ I'm voting to close this question as off-topic because it's a bout geometry not statistics. $\endgroup$ – Aksakal Dec 27 '17 at 18:33
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    $\begingroup$ @Aksakal On that basis most questions here ought to be closed because they ultimately are answered using some kind of mathematics! In the present case the issue seems to be one of statistical terminology: what the heck do these authors mean by "envelope of the space" and "where the corresponding document vector touches the envelope"? Those are awfully cumbersome ways to refer to normalizing all vectors to unit length. One would suppose something more complicated is intended--but what? $\endgroup$ – whuber Dec 27 '17 at 18:57
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    $\begingroup$ What @whuber says. The OP's question "how they are going to find the projection of a vector on a unit sphere" is answered easily enough (standardize by the Euclidean length of the vector), but the rest of the quoted paragraph is rather incomprehensible. $\endgroup$ – Stephan Kolassa Dec 27 '17 at 20:25

First, build a vector, where each coordinate represents a feature (property) of a book. Then draw the line from the origin through the point where this vector is pointing to. Next, find where this line intersects with a unit sphere. That place is a projection.


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