Minimal surface

Minimal surface

In mathematics, a minimal surface is a surface with a mean curvature of zero.These include, but are not limited to, surfaces of minimum area subject to various constraints....

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• Costa surface
• Catenoid
  Catenoid
  A catenoid is a three-dimensional surface made by rotating a catenary curve about its directrix. Not counting the plane, it is the first minimal surface to be discovered. It was found and proved to be minimal by Leonhard Euler in 1744. Early work on the subject was published also by Jean Baptiste...
  
  
• Helicoid
  Helicoid
  The helicoid, after the plane and the catenoid, is the third minimal surface to be known. It was first discovered by Jean Baptiste Meusnier in 1776. Its name derives from its similarity to the helix: for every point on the helicoid there is a helix contained in the helicoid which passes through...
  
  
• Riemann's surface
• Saddle tower
• Gyroid
  Gyroid
  "Gyroid" redirects here. For the creature, see Animal Crossing .A gyroid is a certain infinitely connected triply periodic minimal surface discovered by Alan Schoen in 1970.The gyroid has space group Iad...
  
  
• Catalan surface
  Catalan surface
  In geometry, a Catalan surface, named after the Belgian mathematician Eugène Charles Catalan, is a ruled surface all of whose rulings are parallel to a fixed plane. The vector equation of a Catalan surface is given by...
  
  
• Enneper surface
  Enneper surface
  In mathematics, in the fields of differential geometry and algebraic geometry, the Enneper surface is a surface that can be described parametrically by:It was introduced by Alfred Enneper in connection with minimal surface theory....
  
  

Non-orientable

Orientability

In mathematics, orientability is a property of surfaces in Euclidean space measuring whether or not it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a "clockwise" direction of loops in the...

 surfaces

• Klein bottle
  Klein bottle
  In mathematics, the Klein bottle is a non-orientable surface, informally, a surface in which notions of left and right cannot be consistently defined. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a surface with boundary, a...
  
  

• Real projective plane
  Real projective plane
  In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in our usual three-dimensional space without intersecting itself...
  
  
  • Cross-cap
    Cross-cap
    In mathematics, a cross-cap is a two-dimensional surface that is a model of a Möbius strip with a single self intersection. This self intersection precludes the cross-cap from being topologically equivalent to a Möbius strip...
    
    
  • Roman surface
    Roman surface
    The Roman surface is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry...
    
    
  • Boy's surface
    Boy's surface
    In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901...
    
    

Quadric

Quadric

In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial...

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• Sphere
  Sphere
  A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
  
  
• Spheroid
  Spheroid
  A spheroid, or ellipsoid of revolution 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....
  
  
  • Oblate spheroid
• Cone (geometry)
  Cone (geometry)
  A cone is an n-dimensional geometric shape that tapers smoothly from a base to a point called the apex or vertex. Formally, it is the solid figure formed by the locus of all straight line segments that join the apex to the base...
  
  
• Ellipsoid
• Hyperboloid of one sheet
• Hyperboloid of two sheets
• hyperbolic paraboloid (a ruled surface)
• Paraboloid
  Paraboloid
  In mathematics, a paraboloid is a quadric surface of special kind. There are two kinds of paraboloids: elliptic and hyperbolic. The elliptic paraboloid is shaped like an oval cup and can have a maximum or minimum point....
  
  

Algebraic surface

Algebraic surface

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 and so of dimension four as a smooth manifold.The theory of algebraic surfaces is much more complicated than that...

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• Cayley cubic
• Barth sextic
• Clebsch cubic
• Monkey saddle
  Monkey saddle
  In mathematics, the monkey saddle is the surface defined by the equation z = x^3 - 3xy^2. \, It belongs to the class of saddle surfaces and its name derives from the observation that a saddle for a monkey requires three depressions: two for the legs, and one for the tail...
  
   (saddle-like surface for 3 legs.)
• Torus
  Torus
  In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
  
  
• Dupin cyclide (inversion of a torus)
• Whitney umbrella
  Whitney umbrella
  right|frame|240px|Section of the surfaceIn mathematics, the Whitney umbrella is a self-intersecting surface placed in three dimensions...