Then why are you labeling it "completely impossible"? Here's a perfectly sensible straightish-line path between LAX and NRT, using a polar-aspect orthographic projection:
Yes, if you cherry pick the right projection for a particular route you'll get something like that, which is exactly what I said. But plot Johannesburg to Sydney on the same map.
What I said is impossible is to have a map where all routes between pairs of points look sensible. It gets worse if you optimize for other factors, like making maps equal-area, conformal, rectilinear, etc.
http://www.gcmap.com/mapui?P=LAX-NRT&MS=wls&MP=p&DU=mi