Finding the Point of Intersection Between Two Line Segments and Level of Error

I am working on calculating the intersection of two lines. Each line segment is represented as two geographic coordinate pairs. Additionally, each line segment is less than 1km in distance.

My question pertains to whether it's feasible to use the standard mathematical calculation for determining the intersection point between two lines (using the formula for the equation of a line while assuming the line segments are on a two-dimensional plane).

Does anyone know what kind error I am dealing with if I were to take that approach?

Obviously, it will also depend on where these line segments are situated as distortion increases as we approach the poles.

I'm open to suggestions for other approaches as well (perhaps converting to a different coordinate system?). I am writing this in Python (not ArcPy).

• Could you please explain what you mean by "margin of error"? This has a standard statistical meaning that might apply if you conceive of the four input points (endpoints of the segments) as being measured with (known) error. Since you don't mention that, what are you trying to ask? Perhaps the error made between a Euclidean calculation and a spherical or spheroidal calculation? (That would simply be an "error" with no "margin" in it.) And what do you mean by "situation due to the natural level of error"? – whuber Apr 7 '17 at 18:53
• Hi, sorry for the confusion. I have edited the question to address your questions. I basically just want to know the level of error I am dealing with (i.e. how many metres a calculated point of intersection could be off from the true intersection point), and if there are other methods of calculating that would avoid such error or at least limit it. – Savannah Ostrowski Apr 7 '17 at 19:09
• As you note, it does depend on the projection. At almost all points covered by almost any projection, though, the error made in using a Euclidean formula for segments less than 1 km apart will be negligible for all practical purposes: sub-centimeter and smaller. For analyses of closely related questions please see gis.stackexchange.com/a/151982, gis.stackexchange.com/a/18740, and gis.stackexchange.com/a/156150. In your case, since errors affecting one segment often will cancel those in the other segment, you can expect even higher accuracy. – whuber Apr 7 '17 at 19:14
• Can you clarify how those errors affecting each segment will cancel? Thanks so much! – Savannah Ostrowski Apr 10 '17 at 21:06
• The relative error in almost any projection in almost any small area will change very little, and therefore any additive error will disappear in all differences and any multiplicative error will disappear in all ratios in the calculation. – whuber Apr 10 '17 at 21:14