Curve (physics)

In Physics, a curve is a set of elements which are ordered (thus one-dimensional, and not self-intersecting); and with distance values measured pairwise between its elements allowing the identification of certain subsets as topological neighborhood, and of the whole curve as a connected topological space.

The elements of a curve may be ordered

• either by the distance values as well; provided that for every element P there are two elements, A and Z, with d( Z, A ) > d( Z, P ) ≥ d( P, A ), such that for any two elements J and K with d( Z, A ) > d( Z, J ) ≥ d( J, A ) and d( Z, A ) > d( Z, K ) ≥ d( K, A ) holds: if d( K, A ) > d( J, A ) then d( Z, J ) > d( Z, K )
• or in reference to measures other than distance values; for instance in case of a trajectory: the duration measured pairwise between its elements
• or directly in reference to the observational contents of the elements; e.g. for a worldline.

Accordingly, for any two distinct elements of a curve, one can distintinguish the remaining elements into those between, and those not between the given two; and one distinguishes

• closed curves, for which each element is between any other two. (Also: for any three elements, A, P, and Z, every fourth element Q either belongs to the set of elements between A and Z which contains P as well; or else Q belongs to the set of elements between A and Z which does not contain P.); and
• open curves, each element of which is not between at least one particular pair of elements. In particular:
  • open curves with two ends which have precisely two elements (its two ends, separately) not between any pair of elements,
  • infinite open curves with one end, which have precisely one element (its one end) not between any pair of elements, or
  • unbounded infinite open curves, each element of which is between certain pairs of elements (and not between certain other pairs of elements).

The connectedness of the curve is then established if for any two elements A and Q and for each element B between A and Q there exists an element N between B and Q (where A does not belong to the set between B and Q which also contains N) such that for each element P between N and Q (where B does not belong to the set between N and Q which also contains P) holds that d( P, Q ) < d( B, N ) < d( B, Q ).

In topology applicable to physics, a (simple) curve C is correspondigly either of the following topological spaces with at most two boundary elements (ends):

• a simple closed curve, i.e. without ends: if for any three elements of C there are at least two distinct closed sets which have exactly these three elements as boundary; and for any two such closed sets, their complements relative to C are not disjoint; or
• a simple open curve, with two ends: if C is a closed subset, with nonempty interior, of a simple closed curve; or
• a simple open curve, with one end: i.e. obtained by removing one of the ends from a simple closed curve with two ends; or
• a simple open curve, without ends: i.e. obtained by removing both ends from a simple closed curve with two ends.