Consider a sphere, an infinitely long cylinder, and a plane of infinite length and width (a, b and c below). Imagine that you can hover above each one in your own personal helicopter. In which case do you have the most freedom to move about without your view of the object changing? In other words, for each case consider if there are directions that you can move in without the objects distance or orientation, relative to you, changing.

A plane of infinite length and width.

Explanation:

For the case of the sphere, we should consider spherical coordinates.

You can move around the sphere without changing your distance from the center of the sphere, however you can alter your azimuthal angle and polar angle, since they are symmetric in spherical coordinate system.

r --> Cannot change

Φ --> Free to change

θ --> Free to change

For the case of the infinite cylinder, we should consider cylindrical coordinates.

You can change your height and angular coordinate, but you cannot change your distance from the axis of the cylinder.

r --> Cannot change

θ --> Free to change

z --> Free to change

For the case of the infinite plane, we should consider cartesian coordinates.

Since the length and width of the plane is infinite, we cannot recognize whether we are getting closer or further away.

x --> Free to move

y --> Free to move

z --> Free to move

Therefore, in the case of infinite plane you have the most freedom to move about without your view of the object changing.

answer; how far out of place which the object finished relative to the starting location;