Configuration space

A configuration space parametrizes collections of distinct points in a given space, often recording their positions while excluding collisions. Its points represent whole configurations rather than individual locations.

Cartesian product: The set of ordered tuples formed by choosing one element from each of several sets. Ordered configurations begin in the product of one copy of the underlying space for each point.
Braid group: The group of equivalence classes of braids under concatenation. In the plane, loops of configurations correspond to braids traced by moving points.
Robotics: The science and engineering of machines that sense, plan, and act in physical environments. A robot's configuration space encodes its joint positions and helps identify feasible motions.
Configuration space of the line: The space of distinct point configurations on the real line, with a chosen ordering convention. Its components reflect the possible left-to-right orderings of labeled points.
Diagonal subset: A subset of a product in which two or more coordinates are equal. Collision configurations lie on diagonals that are removed from ordered configuration spaces.
Fundamental group: A group encoding loops in a space up to continuous deformation. It detects whether point motions can be continuously undone without collisions.
Motion planning: The problem of finding a valid path between states while avoiding obstacles or constraints. Collision-free paths are paths through the allowable region of a configuration space.
Configuration space of the circle: The space of distinct point configurations on a circle. Cyclic ordering and motion around the circle produce topology unlike the line's.
Quotient space: A topological space formed by identifying points according to an equivalence relation. Forgetting point labels identifies configurations related by permutations.
Fadell–Neuwirth fibration: A fibration relating configuration spaces by forgetting one or more points. Forgetting a point connects configuration spaces with different numbers of particles.

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