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.
- 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).
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.