Surfaces

A sphere (from Ancient Greek , ) is a surface analogous to the circle, a curve. In solid geometry, a sphere is the set of points that are all at the same distance from a given point in three-dimensional space. That given point is the center of the sphere, and the distance is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians.

thumb|upright=1.2|A right circular cone and an oblique circular cone thumb|A double cone, not infinitely extended

A cylinder () has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.

two-dimensional manifold, and, as such, may be an "abstract surface" not embedded in any Euclidean space

two-dimensional surface with only one side and only one edge

{| class=wikitable align=right |- align=center |150pxHyperboloid of one sheet |160pxconical surface in between |150pxHyperboloid of two sheets |} In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation.

thumb|right|Paraboloid of revolution 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.

type of non-orientable surface

in geometry, an object that is perpendicular to a given object, vector perpendicular to a curve or surface

generalization of a multiple integral to (possibly)-curved surfaces

infinite surface of revolution with infinite surface area enclosing a finite volume, which contributed to 17th century debate on the nature of infinity

right|thumb|350px|A helicoid with α = 1, −1 ≤ ρ ≤ 1 and − ≤ θ ≤ .

topological property

surface through every point of which runs a straight line which equally is on the surface

surface of constant scalar potential

In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean space, an affine space or a projective space. Hypersurfaces share, with surfaces in a three-dimensional space, the property of being defined by a single implicit equation, at least locally (near every point), and sometimes globally.

product of the principal curvatures of a surface

In geometry, a pseudosphere is a surface in \mathbb{R}^3. It is the most famous example of a pseudospherical surface. A pseudospherical surface is a surface piecewise smoothly immersed in \mathbb{R}^3 with constant negative Gaussian curvature. A "pseudospherical surface of radius " is a surface in \mathbb{R}^3 having curvature −1/R2 at each point. Its name comes from the analogy with the sphere of radius , which is a surface of curvature 1/R2. Examples include the tractroid, Dini's surfaces, breather surfaces, and the Kuen surface.

"Remarkable theorem" about Gauss' curvature as an invariant

right|thumb|A torus is an orientable surface alt=Animation of a flat disk walking on the surface of a Möbius strip, flipping with each revolution.|thumb|The Möbius strip is a non-orientable surface. Note how the disk flips with every loop. right|thumb|The Roman surface is non-orientable.

term for the surface of the Earth

molecular model visualizing atoms as intersecting spheres with hidden bonds

300px|thumb|Right circular conoid:

small, local deviations of a surface from a perfectly flat ideal; defined by the three characteristics of lay, surface roughness, and waviness

in differential geometry, an extrinsic measure of curvature of a surface

species of mathematical spline used in computer graphics, computer-aided design, and finite element modeling, is defined by a set of control points

a compact non-orientable two-dimensional manifold

at a given point of a surface, one of the two eigenvalues of the shape operator at the point

thumb|right|The surface of an apple has various perceptible characteristics, such as curvature, smoothness, texture, color, and shininess; observing these characteristics by sight or touch allows the object to be identified. thumb|right|Water droplet lying on a [[damask. Surface tension is high enough to prevent it passing through the textile.]] thumb|right|The Sun, like all stars, appears from a distance to have a distinct surface, but on closer approach has no set surface. A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object. It is the porti

union of all the straight lines that pass through a fixed point and intersect a fixed space curve

stochastically generated naturalistic terrain

mathematical concept

thumb|100px|right

three dimensions self-intersecting surface

closed surface in three-dimensional space through which the flux of a vector field is calculated; usually the gravitational field, the electric field, or magnetic field

in differential geometry, a function that maps each point in a surface to its normal direction

three-dimensional self-intersecting surface, an immersion of the real projective plane

area of all the sides of the object, excluding the area of its base and top

The superformula is a generalization of the superellipse and was proposed by Johan Gielis in 2003. Gielis suggested that the formula can be used to describe many complex shapes and curves that are found in nature. Gielis has filed a patent application related to the synthesis of patterns generated by the superformula, which expired effective 2020-05-10.

mathematical idealization of the surface of a 3D body

surface whose boundary is a knot or a link

nonlinear hyperbolic partial differential equation in 1 + 1 dimensions

right|thumb|Brass superegg by Piet Hein (Denmark)|Piet Hein. thumb|Piet Hein Superegg

scientific model in solid-state physics; electron gas that is free to move in two dimensions, but tightly confined in the third

material and chemical engineering of solid surfaces

a special surface in three dimensions

surface in the Euclidean space

geometric inversion of a standard torus, cylinder or double cone

concept in differential geometry

where the solid (or liquid) material of the outer crust on certain types of astronomical objects contacts the atmosphere or outer space

division of a surface into triangles

half-way model of an inverted sphere

conic section which describes the local shape of a surface

Construction for minimal surfaces

ruled surface made of lines orthogonal to an axis

locally spherical point on a mathematical surface

Bézier surface created by control point interpolation