Forest of stars
Encyclopedia
A forest of stars is a set of star world
s whose adjacency matrix
is a tree. This means that no intersecting star worlds create a cycle, or hole, in the overall space. If an object or space can be represented by a forest of stars, it can be mapped onto a sphere-world
by mapping each star world onto the boundary of its parent star world in the adjacency tree. The root of an adjacency tree can be picked arbitrarily.
All star worlds in a forest of stars must have intersections that are also star worlds with respect to their center point.
Forests of stars are used in robot navigation to create navigation functions such as artificial potential functions. A forest of stars is used to represent robots or obstacles that have shapes which can be approximated by the union of separate stars.
is a space whose boundary is a sphere of the same dimension as the space. A star world is any world whose boundary can be mapped onto the boundary of a sphere world. Since a forest of stars is the union of a number of star worlds, the forest can be recursively mapped onto a single sphere world, and then navigation techniques for sphere worlds can be used.
Star world
This article is about the robotic concept. For the TV channel, see STAR World.Star worlds are mathematical concepts used in robot navigation. Essentially, a star world is any n-dimensional shape that can be mapped to a sphere world of the same dimension...
s whose adjacency matrix
Adjacency matrix
In mathematics and computer science, an adjacency matrix is a means of representing which vertices of a graph are adjacent to which other vertices...
is a tree. This means that no intersecting star worlds create a cycle, or hole, in the overall space. If an object or space can be represented by a forest of stars, it can be mapped onto a sphere-world
Sphere-world
The idea of a sphere-world was constructed by Henri Poincaré who, while pursuing his argument for conventionalism , offered a thought experiment about a sphere with strange properties....
by mapping each star world onto the boundary of its parent star world in the adjacency tree. The root of an adjacency tree can be picked arbitrarily.
All star worlds in a forest of stars must have intersections that are also star worlds with respect to their center point.
Forests of stars are used in robot navigation to create navigation functions such as artificial potential functions. A forest of stars is used to represent robots or obstacles that have shapes which can be approximated by the union of separate stars.
Relation to sphere worlds
A sphere worldSphere 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...
is a space whose boundary is a sphere of the same dimension as the space. A star world is any world whose boundary can be mapped onto the boundary of a sphere world. Since a forest of stars is the union of a number of star worlds, the forest can be recursively mapped onto a single sphere world, and then navigation techniques for sphere worlds can be used.