A bit of etymology -------------------
"Droit" and "droite" can be adverbs or adjectives. Both terms come from the Latin directus, "without curvature, direct, at a right angle," hence the figurative sense "without detour, direct, fair." Directus derives from dirigere, not to be confused with another verb dirigere, from which regere derives (régir, "to govern"; roi, "king").
The word "droite" has kept both Latin values in French, including the figurative senses "fair, exact, true," then "fair, honest and loyal," and finally, around 1140, "following correct reasoning, not straying from reason and common sense" — a sense still current today.
As early as 1080, "droit" and "droite" concretely described something without deviation from one end to the other, without curvature — which, when applied to a person, implied a notion of verticality. It specifically denoted a path with no detours, holding a constant direction, that took the shortest route. Here we clearly find the mathematical notion of a straight line (or going straight ahead). In several uses, they are synonymous with "vertical," as with the jupe droite (straight skirt), or the piano droit (upright piano) as opposed to the grand piano, whose strings are horizontal.