This is a list of

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...

s in mathematics. They are divided into

minimal surfaces In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that ...

, ruled surfaces, non-orientable surfaces,

quadrics In mathematics, a quadric or quadric surface is a generalization of conic sections (ellipses, parabolas, and hyperbolas). In three-dimensional space, quadrics include ellipsoids, paraboloids, and hyperboloids. More generally, a quadric hyper ...

, pseudospherical surfaces,

algebraic surfaces In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of dim ...

, and other types of surfaces.

Minimal surfaces

* Catalan's minimal surface * Costa's minimal surface *

Catenoid In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis (a surface of revolution). It is a minimal surface, meaning that it occupies the least area when bounded by a closed space. It was formally describ ...

* Enneper surface * Gyroid *

Helicoid The helicoid, also known as helical surface, is a smooth Surface (differential geometry), surface embedded in three-dimensional space. It is the surface traced by an infinite line that is simultaneously being rotated and lifted along its Rotation ...

* Lidinoid * Riemann's minimal surface * Saddle tower * Scherk surface * Schwarz minimal surface * Triply periodic minimal surface

Ruled surfaces

* Catalan surface * Right conoid *

Conical surface In geometry, a conical surface is an unbounded surface in three-dimensional space formed from the union of infinite lines that pass through a fixed point and a space curve. Definitions A (''general'') conical surface is the unbounded surface ...

* Developable rollers (

sphericon In solid geometry, the sphericon is a solid that has a continuous developable surface with two Congruence (geometry), congruent, semicircle, semi-circular edges, and four Vertex (geometry), vertices that define a square. It is a member of a spe ...

, oloid) *

Hyperboloid of one sheet In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface (mathematics), surface generated by rotating a hyperbola around one of its Hyperbola#Equation, principal axes. A hyperboloid is the surface obtained ...

(doubly ruled) *

Hyperbolic paraboloid In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry. Every pla ...

(doubly ruled) * Rational normal scroll *

Regulus Regulus is the brightest object in the constellation Leo (constellation), Leo and one of the List of brightest stars, brightest stars in the night sky. It has the Bayer designation designated α Leonis, which is Latinisation of names, ...

Non-orientable surfaces

Klein bottle In mathematics, the Klein bottle () is an example of a Orientability, non-orientable Surface (topology), surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the ...

Real projective plane In mathematics, the real projective plane, denoted or , is a two-dimensional projective space, similar to the familiar Euclidean plane in many respects but without the concepts of distance, circles, angle measure, or parallelism. It is the sett ...

** Cross-cap **

Roman surface In mathematics, the Roman surface or Steiner surface is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry. This mapping is not an immersion of the projective plane; ...

Boy's surface In geometry, Boy's surface is an immersion of the real projective plane in three-dimensional space. It was discovered in 1901 by the German mathematician Werner Boy, who had been tasked by his doctoral thesis advisor David Hilbert to prove th ...

Quadrics

Sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

Spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface (mathematics), surface obtained by Surface of revolution, rotating an ellipse about one of its principal axes; in other words, an ellipsoid with t ...

Oblate spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circu ...

Prolate spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has cir ...

Ellipsoid An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface; that is, a Surface (mathemat ...

Cone (geometry) In geometry, a cone is a three-dimensional figure that tapers smoothly from a flat base (typically a circle) to a point not contained in the base, called the '' apex'' or '' vertex''. A cone is formed by a set of line segments, half-lines ...

* Hyperboloid of two sheets *

(a ruled surface) *

Paraboloid In geometry, a paraboloid is a quadric surface that has exactly one axial symmetry, axis of symmetry and no central symmetry, center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar p ...

Pseudospherical surfaces

* Dini's surface *

Pseudosphere In geometry, a pseudosphere is a surface with constant negative Gaussian curvature. A pseudosphere of radius is a surface in \mathbb^3 having Gaussian curvature, curvature −1/''R''2 at each point. Its name comes from the analogy with the sphere ...

Algebraic surfaces

{{Main, List of algebraic surfaces BarthSextic

* Cayley cubic * Barth sextic * Clebsch cubic * Monkey saddle (saddle-like surface for 3 legs.) *

Torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...

Dupin cyclide In mathematics, a Dupin cyclide or cyclide of Dupin is any Inversive geometry, geometric inversion of a standard torus, Cylinder (geometry), cylinder or cone, double cone. In particular, these latter are themselves examples of Dupin cyclides. They ...

(inversion of a torus) * Whitney umbrella

Miscellaneous surfaces

* Cantor tree surface * Homoeoid *

Jacob's ladder surface In mathematics, Jacob's ladder is a two-dimensional manifold, surface with infinite Genus (mathematics), genus and two End (topology), ends. It was named after Jacob's ladder by Étienne , because the surface can be constructed as the boundary ...

* Loch Ness monster surface * Morin surface * Seashell surface * Superegg *

Supertoroid In geometry and computer graphics, a supertoroid or supertorus is usually understood to be a family of doughnut-like surfaces (technically, a topological torus) whose shape is defined by mathematical formulas similar to those that define the sup ...

Wallis's conical edge In geometry, Wallis's conical edge is a ruled surface given by the parametric equations : x=v\cos u,\quad y=v\sin u,\quad z=c\sqrt where , and are constants. Wallis's conical edge is also a kind of right conoid. It is named after the English math ...

* Whitney umbrella