Where Do Two Perpendicular Lines Intersect?

In Euclidean geometry, two perpendicular lines intersect at a single point called the intersection. If the two lines are y = ax + b and y = cx + d, then their intersection has x coordinate (d-b)/(a-c) and y coordinate [a(d-b)/(a-c) + b].

Outside of Euclidean geometry, perpendicular lines may intersect at multiple points, but they always intersect at least one point. In spherical geometry, which is the geometry concerning the surface of a sphere, perpendicular lines intersect at two points. These points are directly across from each other and called antipodes. An example is the north and south poles of a sphere.