You're looking for a map where a straightish line gives you the direct path between two points? That's completely impossible. Flying from LA to Tokyo will take you over Alaska. Try to work that one out on a map.
(Yeah, you can make a map where straight lines work, but only for a chosen point of origin or antipodes around that point.)
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.
0. http://www.spoon-tamago.com/wp-content/uploads/2016/10/autha...