Difference between revisions of "Fréchet surface"
Ulf Rehmann (talk | contribs) m (moved Frechet surface to Fréchet surface over redirect: accented title) |
(Tex done) |
||
Line 1: | Line 1: | ||
− | A generalization of the concept of a surface in a Euclidean space to the case of an arbitrary metric space | + | A generalization of the concept of a surface in a Euclidean space to the case of an arbitrary metric space $A$. Let $M^2$ be a compact two-dimensional manifold (either closed or with a boundary). The points of $M^2$ play the role of parameter. Continuous mappings $f : M^2 \rightarrow A$ are called parametrized surfaces. Two parametrized surfaces are regarded as equivalent if |
+ | $$ | ||
+ | \rho(f_1,f_2) \equiv \inf_\sigma \max_{m \in M^2} d(f_1(x),f_2(\sigma(x))) = 0 | ||
+ | $$ | ||
+ | where $d$ is the distance in $A$ and the $\sigma$ are all possible homeomorphisms of $M^2$ onto itself. A class of equivalent parametrized surfaces is called a Fréchet surface (see [[#References|[1]]]), and each of the parametrized surfaces in this class is called a parametrization of the Fréchet surface. Many properties of parametrized surfaces are properties of the Fréchet surface, and not of its concrete parametrization. For two Fréchet surfaces, the value of $\rho(f_1,f_2)$ is independent of the choice of the parametrizations $f_1$ and $f_2$; it is called the Fréchet distance between the Fréchet surfaces. If one changes the domain $M^2$ of the parameter in the definition of a Fréchet surface to a circle or a closed interval, one obtains the definition of a Fréchet curve (see [[#References|[2]]]). | ||
− | <table | + | ====References==== |
+ | <table> | ||
+ | <TR><TD valign="top">[1]</TD> <TD valign="top"> M. Fréchet, ''Ann. Soc. Polon. Math.'' , '''3''' (1924) pp. 4–19</TD></TR> | ||
+ | <TR><TD valign="top">[2]</TD> <TD valign="top"> M. Fréchet, "Sur quelques points du calcul fonctionnel" ''Rend. Circolo Mat. Palermo'' , '''74''' (1906) pp. 1–74</TD></TR> | ||
+ | </table> | ||
− | + | {{TEX|done}} | |
− | |||
− | |||
− |
Latest revision as of 20:15, 20 December 2016
A generalization of the concept of a surface in a Euclidean space to the case of an arbitrary metric space $A$. Let $M^2$ be a compact two-dimensional manifold (either closed or with a boundary). The points of $M^2$ play the role of parameter. Continuous mappings $f : M^2 \rightarrow A$ are called parametrized surfaces. Two parametrized surfaces are regarded as equivalent if $$ \rho(f_1,f_2) \equiv \inf_\sigma \max_{m \in M^2} d(f_1(x),f_2(\sigma(x))) = 0 $$ where $d$ is the distance in $A$ and the $\sigma$ are all possible homeomorphisms of $M^2$ onto itself. A class of equivalent parametrized surfaces is called a Fréchet surface (see [1]), and each of the parametrized surfaces in this class is called a parametrization of the Fréchet surface. Many properties of parametrized surfaces are properties of the Fréchet surface, and not of its concrete parametrization. For two Fréchet surfaces, the value of $\rho(f_1,f_2)$ is independent of the choice of the parametrizations $f_1$ and $f_2$; it is called the Fréchet distance between the Fréchet surfaces. If one changes the domain $M^2$ of the parameter in the definition of a Fréchet surface to a circle or a closed interval, one obtains the definition of a Fréchet curve (see [2]).
References
[1] | M. Fréchet, Ann. Soc. Polon. Math. , 3 (1924) pp. 4–19 |
[2] | M. Fréchet, "Sur quelques points du calcul fonctionnel" Rend. Circolo Mat. Palermo , 74 (1906) pp. 1–74 |
Fréchet surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fr%C3%A9chet_surface&oldid=40064