After how much distance does the x,y coordinate system become noticeably distorted?

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I'm working with maritime sensors that give information of moving objects it detects, in azimuth, range and elevation, and we convert it to x,y coordinates for our application to be able to process it.

We found out that most naval systems consider the earth's center as the point of reference when doing position calculations, but since we needed x,y, we are considering that when our software starts, we note the position of our boat with reference to the earth's center and convert it to x,y coordinates, where the current position of our boat would be 0,0 (the origin).

The boat can move at upto 25 knots, so when we have traveled a good distance, we would convert all x,y coordinates in the software to radius, azimuth and elevation with respect to the earth's center and then convert it to current x,y position of the boat (with the current position being the origin of the x,y grid). Only problem would be, the ambiguity in position of all the moving objects that the sensor was tracking.

Two questions:1. Is this the best way to do it or is there a better way? We have to work with x,y coordinates.2. How many kilometers would a boat have to move so that the x,y coordinate system would become erroneous due to the earth's curvature?



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