Conformally squishing a hemisphere into a square (or if you like, squishing a 90°–90°–90° triangle on the sphere into a 45°–45°–90° triangle on the plane) cause

Conformally squishing a hemisphere into a square (or if you like, squishing a 90°–90°–90° triangle on the sphere into a 45°–45°–90° triangle on the plane) causes very extreme distortion near the corners. Can be fun from an aesthetic perspective, but it’s not the most practical for most uses as a working map.