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This is very simple, but sometimes confusing. Here I have a sentence:

Having rectified edges, the fixing of boundary is more stable and geometrically acceptable.

I know a phrase can be added in front of a sentence (and subject), but I want to know what is the best way to place this type of subject supportive phrase.

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  1. Your proposed initial phrase is semantically ambiguous. It is unclear whether you intend having to have a possessive sense ('because [something] has edges which are rectified') or acts as the head of a perfect construction ('because [somebody] has rectified the edges').

  2. In either case, an initial participle phrase like this is obliged to modify the subject of the following main clause, which acts as the 'somebody' or 'something' in my paraphrases. I'm not sure what you mean by 'fixing of boundary', but I doubt that it is either the 'possessor' or the agent of the rectification.

I think what you probably mean is something like

Once the edges have been rectified, the [?fixed boundary?] is more stable and geometrically acceptable.

or possibly

Once the edges have been rectified, the boundary may be fixed in a more stable and geometrically acceptable position.

If you really want to use the participle construction, it should be like this, with the particpial modifying the subject:

Having rectified the edges, we may fix the boundary in a more stable and geometrically acceptable position.

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