Scale 2D coordinates and keep their relative euclidean distances intact?

Question

I have a set of points like: pointA(3302.34,9392.32), pointB(34322.32,11102.03), etc.

I need to scale these so each x- and y-coordinate is in the range (0.0 - 1.0). I tried doing this by first finding the largest x value in the data set (maximum_x_value), and the largest y value in the set (minimum_y_value). I then did the following:

pointA.x = (pointA.x - minimum_x_value) / (maximum_x_value - minimum_x_value)
pointA.y = (pointA.y - minimum_y_value) / (maximum_y_value - minimum_y_value)

This changes the relative distances(?), and therefore makes the data useless for my purposes. Is there a way to scale these coordinates while keeping their relative distances the intact?

Answer 1

You need to scale the x values and the y values by the same amount! I would suggest scaling by the larger of the two ranges (either x or y). In pseudocode, you'd have something like

scale = max(maximum_x_value - minimum_x_value,
            maximum_y_value - minimum_y_value)

Then all the distances between points will be scaled by scale, which is what I presume you're asking for, so if point p_1 was twice as far from point p_2 as from p_3 before rescaling, it will be twice as far after rescaling as well. You should be able to prove this to yourself pretty easily using the Pythagorean theorem.