Vertical line fit using polyfit

First of all, this happens due to the method of fitting that you are using. When doing polyfit, you are using the least-squares method on Y distance from the line.

_{(source: une.edu.au)}

Obviously, it will not work for vertical lines. By the way, even when you have something close to vertical lines, you might get numerically unstable results.

There are 2 solutions:

1. Swap x and y, as you said, if you know that the line is almost vertical. Afterwards, compute the inverse linear function.
2. Use least-squares on perpendicular distance from the line, instead of vertical (See image below) (more explanation in here)

_{(from MathWorld - A Wolfram Web Resource: wolfram.com)}

Polyfit uses linear ordinary least-squares approximation and will not allow repeated abscissa as the resulting Vandermonde matrix will be rank deficient. I would suggest trying to find something of a more statistical nature. If you wish to research Andreys method it usually goes by the names Total least squares or Orthogonal distance regression http://en.wikipedia.org/wiki/Total_least_squares

I would tentatively also put forward the possibility of detecting when you have simultaneous x values, then rotating your data about the origin, fitting the line and then transform the line back. I could not say how poorly this would perform and only you could decide if it was an option based on your accuracy requirements.