Sphere world

A sphere world is a mathematical concept used in robotic motion planning. Essentially, if the environment is represented as a sphere, and the robot and obstacles within the environment are represented as spheres, then it is possible to construct navigation functions which create paths from a start position to a goal position. Although the real world is rarely composed of spheres, sphere worlds are still useful for motion planning, as they are topologically equivalent to star worlds.

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Sphere-world

The idea of a sphere-world was constructed by Henri Poincaré who, while pursuing his argument for conventionalism (see philosophy of space and time), offered a thought experiment about a sphere with strange properties.

The concept

Poincaré asks us to imagine a sphere of radius R. The temperature of the sphere decreases from its maximum at the center to absolute zero at its extremity such that a body’s temperature at a distance r from the center is proportional to .

In addition, all bodies have the same coefficient of dilatation so every body shrinks and expands in similar proportion as they move about the sphere. To finish the story, Poincaré states that the index of refraction will also vary with the distance r, in inverse proportion to .

How will this world look to inhabitants of this sphere?

In many ways it will look normal. Bodies will remain intact upon transfer from place to place, as well as seeming to remain the same size (the Spherians would shrink along with them). The geometry, on the other hand, would seem quite different. Supposing the inhabitants were to view rods believed to be rigid, or measure distance with light rays. They would find that a geodesic is not a straight line, and that the ratio of a circle’s circumference to its radius is greater than .