In 3D Euclidean space, the surface equidistant from a point is a sphere, with constant curvature.

In (2+1)-dimensional Minkowski spacetime, the equidistant surface is a (1+1)-dimensional de Sitter spacetime, which also has constant curvature.

The image shows the spacelike (blue), timelike (black) and lightlike (white) geodesics through a point.