Source: Wikipedia. Pages: 123. Chapters: Sphere, Möbius strip, Klein bottle, Surface, Torus, Spheroid, Genus, Ellipsoid, Plane, Roman surface, Boy's surface, Quadric, Steiner surface, Pseudosphere, Hyperboloid, Paraboloid, Surface normal, Bézier triangle, Bézier surface, Differential geometry of surfaces, Riemannian connection on a surface, Orientability, Real projective plane, Introduction to systolic geometry, Sine¿Gordon equation, Gauss¿Codazzi equations, Parametric surface, Gaussian curvature, Computer representation of surfaces, Cone, Carathéodory conjecture, Gaussian surface, Ruled surface, Principal curvature, Freeform surface modelling, Nielsen¿Thurston classification, Cylinder, Seifert surface, Mean curvature, Dupin cyclide, Bäcklund transform, Systoles of surfaces, Surface of revolution, Superformula, First fundamental form, Theorema Egregium, Morin surface, Gauss map, Archard equation, Cross-cap, Willmore energy, Conical surface, Standard torus, Developable surface, Prolate spheroid, PDE surface, Filling area conjecture, Helicoid, Oblate spheroid, Euler's theorem, Umbilical point, Seashell surface, Plücker¿s conoid, Spring, Pinched torus, Whitney umbrella, Saddle surface, Dini's surface, Willmore conjecture, Asymptotic curve, Catalan surface, Unduloid, Monkey saddle, Index ellipsoid, Circular surface, Focaloid, Right conoid, Nadirashvili surface, Focal surface, Prüfer manifold, Asperity, Weierstrass¿Enneper parameterization, Dupin indicatrix, Triple torus, Wallis¿s conical edge, Biharmonic Bézier surface, Zoll surface, Homoeoid, Equipotential surface, Breather surface, List of surfaces, Channel surface, Ridge, Liouville surface, Bonnet theorem, Tangent developable, Klein surface, Bicone, Bryant surface, Nodoid. Excerpt: In mathematics, the differential geometry of surfaces deals with smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: extrinsically, relating to their embedding in Euclidean space and intrinsically, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss (1825-1827), who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space. Surfaces naturally arise as graphs of functions of a pair of variables, and sometimes appear in parametric form or as loci associated to space curves. An important role in their study has been played by Lie groups (in the spirit of the Erlangen program), namely the symmetry groups of the Euclidean plane, the sphere and the hyperbolic plane. These Lie groups can be used to describe surfaces of constant Gaussian curvature; they also provide an essential ingredient in the modern approach to intrinsic differential geometry through connections. On the other hand extrinsic properties relying on an embedding of a surface in Euclidean space have also been extensively studied. This is well illustrated by the non-linear Euler-Lagrange equations in the calculus of variations: although Euler developed the one variable equations to understand geodesics, defined independently of an embedding, one of Lagrange's main applications of the two variable equations was to minimal surfaces, a concept that can only be defined in terms of an embedding. Polyhedra in the Eu...