In
Physics, a
surface is a set of elements with pairwise measured
distance d, satisfying the following properties:
- For any two distinct elements A and B there are elements X and Y such that
- d( Y, A ) < d( X, A ) < d( B, A ),
- d( Y, X ) < d( X, A ),
- d( Y, X ) < d( X, B ),
- d( Y, B ) < d( X, B ),
- d( Y, B ) < d( B, A ); and
- for any four distinct elements A, B, C, and Q, which satisfy d( Q, A ) < d( C, B ) < d( C, A ) < d( B, A ) < d( C, A ) + d( C, B ), holds
- either Vol( A, B, C, Q ) = 0, or
- there exist elements J and K with d( K, A ) < d( J, A ) < d( C, A ) < d( J, A ) + d( K, A )
- such that for all elements T with d( T, A ) < d( K, A ) holds
- Vol( A, J, K, T ) / Area( A, J, K ) ≤ ½ Vol( A, B, C, Q ) / Area( A, B, C ).
Here Vol(), as a function of four elements, denotes the Volume of the corresponding 3-simplex (i.e. tetrahedron), expressed in terms of the six distance values (pairwise) between these four elements by Tartaglia's formula; and Area() , as a function of three elements, denotes the Area of the corresponding 2-simplex (i.e. triangle), expressed in terms of the three distance values (measured pairwise) between these three elements by Heron's formula.
In topology as applicable to physics, a surface is a topological space which satisfies for any three elements:
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