๐ Good question! The three directions are all real coordinates in the 3D visualization, but mathematically they represent the real component, imaginary component, and the parameter/time direction.
Railways got there first. Talbot had the integrals by 1880 and named it the railway transition spiral: curvature builds with distance rather than jumping. An aeroplane rolling into a turn draws the same curve, and in wind the ground track stretches downwind while the bank holds.
