What is the shape of the path followed by an object on a hyperbolic trajectory relative to the celestial body it is escaping?

Hyperbolic. The path followed by an object on a hyperbolic trajectory relative to the celestial body it is escaping is in the shape of a hyperbola. This open curve resembles a stretched ellipse, with two distinct branches that extend to infinity. The hyperbolic path is a result of the object having excess kinetic energy compared to the gravitational potential energy binding it to the celestial body. As the object moves away from the body, its velocity remains greater than the escape velocity, allowing it to continue on an escape trajectory and never return to the vicinity of the celestial body. Hyperbolic trajectories are studied in celestial mechanics, astrophysics, and aerospace engineering to understand the motion of objects in space and to plan interplanetary missions that require a spacecraft to escape the gravitational influence of a planet or moon.