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...et]] family) of a [[minimal surface]] is a one-parameter family of minimal surfaces which share the same [[Weierstrass–Enneper parameterization|Weierstrass dat ...nimal Surfaces in Euclidean Space by Examples, in Global Theory of Minimal Surfaces: ...

...ld Scientific, 1992 (p. 239-240)</ref> It is a family of surfaces with one planar [[End (topology)|end]] and one [[Enneper surface]]-like self-intersecting e ...1680/11/1/26 Erhan Güler and Ömer Kişi, ''(ζ−m, ζm)-Type Algebraic Minimal Surfaces in Three-Dimensional Euclidean Space''] ...

...al Surfaces as Isotropic Curves in '''C'''³: Associated minimal surfaces and the Björling's problem. MIT BA Thesis. 2004 http://ocw.mit.edu/courses/ ...urnal |author=W.H. Meeks III |title=The classification of complete minimal surfaces in '''R'''³ with total curvature greater than <math>-8\pi</math> ...

More generally, every planar graph of minimum degree at least three either has an edge of total degree a Generalizations of the theorem are also known for [[graph embedding]]s onto surfaces with higher [[Genus (mathematics)|genus]].{{r|z}} ...

...|0-8247-9046-4}}</ref> The following derivations apply only to ideal solid surfaces; they are only valid for the state in which the interfaces are not moving a ===Simplification to planar geometry, Young's relation=== ...

...sually understood to be a family of [[doughnut]]-like [[surface (geometry)|surfaces]] (technically, a [[topology|topological]] [[torus (mathematics)|torus]]) w ...}} scaled and horizontally shifted. If {{math|1=''v'' = 0}}, the curve is planar and spans the intervals: ...

...such as [[computer-aided design]] and [[computer-aided manufacturing]], '''surfaces''' are one way of representing objects. The other ways are [[Wire-frame mo ==Open and closed surfaces== ...

{{Short description|Every triangle-free planar graph is 3-colorable}} [[File:Groetsch-3-coloring.svg|thumb|A 3-coloring of a triangle-free planar graph]] ...

...[[Dirichlet eigenvalue|first eigenvalue of the Laplace operator on compact surfaces]]).<ref>{{MathGenealogy|id=85148|name =Jürg Peter Buser}}</ref> As a post-d Buser is known for his construction of curved isospectral surfaces (published in 1986 and 1988). His 1988 construction led to a negative solut ...

...f of Bernstein's theorem by deducing it from the fact that there is no non-planar area-minimizing cone in '''R'''³. {{harvtxt|De Giorgi|1965}} showed that if there is no non-planar area-minimizing cone in '''R'''^{''n''&minus;1} then the analogue o ...

...4|doi=10.1016/0012-365X(91)90166-Y|doi-access=free}}</ref> showed that any planar graph has a special (3,2)-coloring in which each color class is a linear fo ...hor-link= Dan Archdeacon |title=A note on defective colorings of graphs in surfaces|journal=Journal of Graph Theory|date=1987|volume=11|issue=4|pages=517–519|d ...

...mate forces between [[colloid|colloidal particles]], as forces between two planar bodies are often much easier to calculate. The Derjaguin approximation expr where ''W''(''h'') is the interaction energy per unit area between the two planar walls and ''R''_eff the effective radius. When the two bodies are ...

| contribution = Planar conformal mappings of piecewise flat surfaces ...a specified subset of pairs of circles (corresponding to the edges of a [[planar graph]] ) have a given inversive distance with respect to each other. This ...

If both curves are contained in a common plane, the translation surface is planar (part of a plane). This case is generally ignored. Translation surfaces are popular in [[descriptive geometry]]<ref>H. Brauner: ''Lehrbuch der Kons ...

...he angles of the spherical triangle exceed the corresponding angles of the planar triangle by approximately one third of the [[spherical excess]] (the spheri ...ntre). The area of such a small triangle is well approximated by that of a planar equilateral triangle with the same sides: <math>\tfrac12a^2\sin\tfrac\pi3</ ...

{{Short description|Planar graph drawn by relaxing springs}} ...s always crossing-free, and more strongly that every face of the resulting planar embedding is convex.<ref>{{Citation ...

...ion of the mold's [[parting line]] also affects castability, because a non-planar parting line also increases tooling complexity. ...s can be very economical for part designs that require intricate contoured surfaces, thickness variations, and internal features.<ref name=ravi3>Ravi, p. 3.</r ...

...ion''' is a means to construct [[Function (mathematics)|functions]] over a planar domain in such a way that they match a given function on the boundary. This ...the [[Coons patch]], invented in 1967. <ref name="coons">Steven A. Coons, Surfaces for computer-aided design of space forms, Technical Report MAC-TR-41, Proje ...

...ed problem on [[graph coloring]] in mathematics. It is an extension of the planar map coloring problem (solved by the [[four color theorem]]), and was posed ...ccording to the [[four color theorem]], the resulting planar graph (or any planar graph) can be colored using at most four different colors, no matter how ma ...

...rallel surface is placed on top and the glue spreads out (Middle), and the surfaces are lifted, air, a less viscous fluid, attempts to invade the space where t == Derivation for a planar interface == ...