There are three issues that must be considered in your question. Normally the projection used for paper chart plotting considers the maximum error that printing technology normally handles. This is, considering the thermal paper deformations due to heating sources/sinks and the mechanical tolerances at printing, this means like
0.1mm tolerances are allowed. On a normal
1:25000 chart this means an absolute error of 2.5m in the field. The counterpart issue to be considered is the display screen resolution. At resolutions of 4096 pixels by 4096 pixels, you have a relative error of
1/4096 you cannot fix, as a limitation of the display screen. This can make the error far greater in absolute units as you grow larger in map extension, which makes the projection absolutely effective by burying the errors inside the other errors.
Normal proyections do diverge normally from the point of construction (or the line in case of UTM, Lambert or Straight mercator) I consider each one.
Normally, in land places, the most commonly accepted (now) for its precission is UTM, which is not continous, as it consists in mapping the earth surface as a eliptical cilinder, with axis perpendicular to the rotating axis of the earth. In this cilinder, only a six degree huse in longitude is considered and the plane is tangent to a complete meridian. The maximum apart is 6 degree in longitude which makes it to deviate from the true point on less than 0.05% (and by that reason it gets scaled by 0.9995 to get half the deviation all the plane extension) Normally, a mapping of this class completely fits in the resolution of a normal display (even at resolutions of 4000x4000 pixels) without making any point to divert by one pixel apart from the surface of reference. This makes for a 0.03% max at aprox. 350km apart from the meridian.
Lambert, which uses a parallel of tangency, even having more diverting errors, is also far below the limit for a 4000x4000 pixel screen, even from departing from the parallel of tangency by more than 1000km. This makes an effective exact screen for maps with more than 2000km (N/S) coverage.
Finally, polar stereographic is normally used only for high latitudes (near the poles) and is not considered in most of the globe. Due to the elliptical surface of the earth is normally not used for representation at one point, due to the different curvature radii of the earth surface in the meridian and parallel directions.
Gnomonic projection has also been mentioned, for the properties that all the maximum circles map to straight lines (this makes normal straight approximations to map to straight lines) has similar deviations (like double, by not being a tangent line projection, but a tangent point projection it deviates more, but errors aprox. double the ones you have in tangential line projections)
It should be considered also that in case of a radar system, probably the most exact projection to achieve the best results should be a local UTM projection (with the central meridian passing through the radar rotating axis) as then you don't have any deviation from the north pole due to meridian convergence. It is remarkable that with this remarks, probably the differences between astronomical (the ones that you measure to the geoid) and geodesic coordinates begin to be significative at ranges well over
On other side, radars make errors and you get on the screen normally points referenced by a distance and azimuth. As it has been said on other responses to this question. Distance is subject to meteorological perturbations that make errors to completely overrun the ones mentioned previously. And with angles the problem is even greater, as you can be affected by refraction issues in radar. This makes that some point acquisition be affected by errors far larger than the ones made in the projection (except in the case the projection is badly calculated) you can trust completely the used projection.
Only in the case your projection is mistakenly selected or calculated, that you have to worry about the errors introduced by (at any range) for radar positioning. If the projecting software is well configured, the errors will be several orders of magnitude below the measurement ones and you can consider the earth surface as almost flat. (you have to make only the azimuth corrections due to meridian convergence anyway, as the projections are conformant only locally, but everithing you'll see in the screen will be exact as measured on the screen with the screen resolution capabilities)
As a software engineer, the most probable chartographic projection you have will be lambert conical or UTM mercator, independently on how then the coordinates are represented, as both combine the exactitude of the results with the conformant of data grids. But I recommend to read the radar screen manual to get the final response. UTM is official in the majority or countries in the world, but historically, lambert has been used for long time.