Best fit line with known offsets.

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Hello all it's my first post on these forums so if I'm breaking any ettiqute feel free to let me know.

So it's been a few years since I took a linear algebra class in college but I remember using Total Least squares to find a best fit line with the errors in both the x and y variables to give you a line with errors that were perpendicular to the created line.

I'd like program something similar to this with the difference being that not all of my data points actually land on said line. Some of them are intentional perpendicular offsets on both sides of the line that have known distances.

I've searched the internet for a similar problem but I've yet to find anything similar. Maybe I don't know the right terms to search for.

Does this sound like a possible problem? Any nudge in right direction would be helpful.

Thanks in advance.

Calvin