In the differential geometry of surfaces, the Monge patch designates the parameterization of a surface by its height over a flat reference plane. It is also called Monge parameterization or Monge form.
In physical theory of surface and interface roughness, and especially in the study of shape conformations of membranes, it is usually called the Monge gauge, or less frequently the Monge representation.
Details
If the reference plane is the Cartesian xy plane, then in the Monge gauge the surface under study is fully characterized by its height z=u(x,y).8 Typically, the reference plane represents the average surface so that the first moment of the height is zero, <u>=0.
The Monge gauge has two obvious limitations: If the average surface is not plane, then the Monge gauge only makes sense on length scales smaller than the curvature of the average surface. And the Monge gauge fails completely if the surface is so strongly bent that there are overhangs (points x,y corresponding to more than one z).
Origin of the term
The term refers to Gaspard Monge and his work in differential geometry. "Monge form" was found in a textbook from 1947,9 "Monge patch" in one from 1966.10 The first use of "Monge gauge" seems to be in a 1989 physics paper by Golubović and Lubensky.11
References
O'Neill, B (1966). Elementary Differential Geometry. Orlando: Academic Press. ↩
Gray, A (1993). Modern differential geometry of curves and surfaces. Boca Raton: CRC Press. ISBN 978-0849378720. 978-0849378720 ↩
Bloch, ED (1997). A First Course in Geometric Topology and Differential Geometry. Boston: Birkhäuser. ISBN 978-0817681210. 978-0817681210 ↩
Hsiung, C-C (1981). A First Course in Differential Geometry. New York: Wiley Interscience. ISBN 978-0471079538. 978-0471079538 ↩
Eisenhart, LP (1947). An introduction to Differential Geometry. Princeton Univ Press. /wiki/Luther_P._Eisenhart ↩
E.g. Deserno, Markus (2015). "Fluid lipid membranes: From differential geometry to curvature stresses". Chemistry and physics of lipids. 185. Elsevier: 11–45. doi:10.1016/J.CHEMPHYSLIP.2014.05.001. Sect 2.7 /wiki/Doi_(identifier) ↩
Ungar, LH; et al. (1985). "Cellular interface morphologies in directional solidification. III. The effects of heat transfer and solid diffusivity". Phys Rev B. 31 (9): 5923--5930. doi:10.1103/PhysRevB.31.5923. /wiki/Doi_(identifier) ↩
See any of the differential geometry textbooks cited above. ↩
Eisenhart, LP (1947). An introduction to Differential Geometry. Princeton Univ Press. /wiki/Luther_P._Eisenhart ↩
O'Neill, B (1966). Elementary Differential Geometry. Orlando: Academic Press. ↩
Golubović; Lubensky (1989). "Smectic elastic constants of lamellar fluid membrane phases: Crumpling effects". Phys Rev B. 39 (16): 12110--12133. doi:10.1103/PhysRevB.39.12110. on page 9 and in footnote 29) /wiki/Doi_(identifier) ↩