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The double sphere is the degenerate quartic surfaceobtained by squaring the left-hand side of the equation of a usual sphere

A bubble is a minimal-energy surface of the type that is formed by soap film. The simplest bubble is a single sphere, illustrated above (courtesy of J. M. Sullivan). More complicated forms occur when multiple bubbles are joined together. The simplest example is the double bubble, and beautiful configurations can form when three or more bubbles are conjoined (Sullivan).An outstanding problem involving bubbles is the determination of the arrangements of bubbles with the smallest surface area which enclose and separate given volumes in space.

A superegg is a solid described by the equation(1)The special case gives a spheroid.Special cases of volume are given by(2)(3)

Let a vault consist of two equal half-cylinders of radius which intersect at right angles so that the lines of their intersections (the "groins") terminate in the polyhedron vertices of a square. Two vaults placed bottom-to-top form a Steinmetz solid on two cylinders.Solving the equations(1)(2)simultaneously gives(3)(4)One quarter of the vault can therefore be described by the parametricequations(5)(6)(7)The surface area of the vault is therefore givenby(8)where is the length of a cross section at height and is the angle a point on the center of this line makes with the origin. But , so(9)and(10)(11)(12)The volume of the vault is(13)(14)The geometric centroid is(15)

To find the volume of a solid of revolution by adding up a sequence of thin cylindrical shells, consider a region bounded above by , below by , on the left by the line , and on the right by the line . When the region is rotated about the z-axis, the resulting volume is given byThe following table gives the volumes of various solidsof revolution computed using the method of cylinders.solidvolumecone0conical frustum0cylinder0oblate spheroidprolate spheroidspheretorusspherical segmenttorispherical dome0To find the volume of a solid of revolution by adding up a sequence of thin flat washers, consider a region bounded on the left by , on the right by , on the bottom by the line , and on the top by the line . When the region is rotated about the z-axis, the resulting volume isThe following table gives the volumes of various solids of revolution computed using the method of washers.solidvolumebarrel (elliptical)0barrel (parabolic)0cone0conical frustum0cylinder0oblate..

Gabriel's horn, also called Torricelli's trumpet, is the surface of revolution of the function about the x-axis for . It is therefore given by parametric equations(1)(2)(3)The surprising thing about this surface is that it (taking for convenience here) has finite volume(4)(5)(6)but infinite surface area,since(7)(8)(9)(10)(11)(12)This leads to the paradoxical consequence that while Gabriel's horn can be filled up with cubic units of paint, an infinite number of square units of paint are needed to cover its surface!The coefficients of the first fundamental formare,(13)(14)(15)and of the second fundamental form are(16)(17)(18)The Gaussian and meancurvatures are(19)(20)The Gaussian curvature can be expressed implicitly as(21)

The funnel surface is a regular surface and surface of revolution defined by the Cartesian equation(1)and the parametric equations(2)(3)(4)for and . The coefficients of the first fundamental form are(5)(6)(7)the coefficients of the second fundamentalform are(8)(9)(10)the area element is(11)and the Gaussian and mean curvatures are(12)(13)The Gaussian curvature can be given implicitly as(14)Both the surface area and volumeof the solid are infinite.

The surface of revolution of the parabola which is the shape used in the reflectors of automobile headlights (Steinhaus 1999, p. 242; Hilbert and Cohn-Vossen 1999). It is a quadratic surface which can be specified by the Cartesian equation(1)The paraboloid which has radius at height is then given parametrically by(2)(3)(4)where , .The coefficients of the first fundamental formare given by(5)(6)(7)and the second fundamental form coefficientsare(8)(9)(10)The area element is then(11)giving surface area(12)(13)The Gaussian curvature is given by(14)and the mean curvature(15)The volume of the paraboloid of height is then(16)(17)The weighted mean of over the paraboloid is(18)(19)The geometric centroid is then given by(20)(Beyer 1987).

The first theorem of Pappus states that the surface area of a surface of revolution generated by the revolution of a curve about an external axis is equal to the product of the arc length of the generating curve and the distance traveled by the curve's geometric centroid ,(Kern and Bland 1948, pp. 110-111). The following table summarizes the surface areas calculated using Pappus's centroid theorem for various surfaces of revolution.solidgenerating curveconeinclined line segmentcylinderparallel line segmentspheresemicircleSimilarly, the second theorem of Pappus states that the volume of a solid of revolution generated by the revolution of a lamina about an external axis is equal to the product of the area of the lamina and the distance traveled by the lamina's geometric centroid ,(Kern and Bland 1948, pp. 110-111). The following table summarizes the surface areas and volumes calculated using Pappus's centroid theorem..

One of the three standard tori given by the parametricequations(1)(2)(3)corresponding to the torus with .It has coefficients of the first fundamentalform given by(4)(5)(6)and of the second fundamental form givenby(7)(8)(9)The area element is(10)and the surface area and volumeare(11)(12)The geometric centroid is at , and the moment of inertia tensor for a solid torus is given by(13)for a uniform density torus of mass .The inversion of a horn torus is a horn cyclide. The above figures show a horn torus (left), a cutaway (middle), and a cross section of the horn torus through the -plane (right).

Two planes that do not intersect are said to be parallel. Two planes specified in Hessian normal form are parallel iff or (Gellert et al. 1989, p. 541).Two planes that are not parallel always intersect in a line.

The dihedral angle is the angle between two planes. The dihedral angle between the planes(1)(2)which have normal vectors and is simply given via the dot product of the normals,(3)(4)The dihedral angle is therefore trivial to compute via equation (3) if the two planes are specified in Hessian normal form(5)for planes (Gellert et al. 1989, p. 541).The dihedral angle between planes in a general tetrahedron is closely connected with the face areas via a generalization of the law of cosines.

A cylindric section is the intersection of a plane with a right circular cylinder. It is a circle (if the plane is at a right angle to the axis), an ellipse, or, if the plane is parallel to the axis, a single line (if the plane is tangent to the cylinder), pair of parallel lines bounding an infinite rectangle (if the plane cuts the cylinder), or no intersection at all (if the plane misses the cylinder entirely; Hilbert and Cohn-Vossen 1999, pp. 7-8).

A spheric section is the curve formed by the intersection of a plane with a sphere. Excluding the degenerate cases of the plane tangent to the sphere or the plane not intersecting the sphere, all spheric sections are circles.A spheric section that does not contain a diameter of the sphere is known as a small circle, while a spheric section containing a diameter is known as a great circle.

Let , ..., be four planes in general position through a point and let be a point on the line . Let denote the plane . Then the four planes , , , all pass through one point . Similarly, let , ..., be five planes in general position through . Then the five points , , , , and all lie in one plane. And so on.

The set of all planes through a line. The line is sometimes called the axis of the sheaf, and the sheaf itself is sometimes called a pencil (Altshiller-Court 1979, p. 12; Gellert et al. 1989, p. 542).The equation of a sheaf of planes specified in Hessiannormal form is

Let be a Hilbert space and a closed subspace of . Corresponding to any vector , there is a unique vector such thatfor all . Furthermore, a necessary and sufficient condition that be the unique minimizing vector is that be orthogonal to (Luenberger 1997, p. 51).This theorem can be viewed as a formalization of the result that the closest point on a plane to a point not on the plane can be found by dropping a perpendicular.

Geometric objects lying in a common plane are said to be coplanar. Three noncollinear points determine a plane and so are trivially coplanar. Four points are coplanar iff the volume of the tetrahedron defined by them is 0,Coplanarity is equivalent to the statement that the pair of lines determined by the four points are not skew, and can be equivalently stated in vector form asAn arbitrary number of points , ..., can be tested for coplanarity by finding the point-plane distances of the points , ..., from the plane determined by and checking if they are all zero. If so, the points are all coplanar.A set of vectors is coplanar if the nullity of the linear mapping defined by has dimension 1, the matrix rank of (or equivalently, the number of its singular values) is (Abbott 2004).Parallel lines in three-dimensional space are coplanar,but skew lines are not...

It is especially convenient to specify planes in so-calledHessian normal form. This is obtained from the general equation of a plane(1)by defining the components of the unit normal vector ,(2)(3)(4)and the constant(5)Then the Hessian normal form of the plane is(6)and is the distance of the plane from the origin (Gellert et al. 1989, pp. 540-541). Here, the sign of determines the side of the plane on which the origin is located. If , it is in the half-space determined by the direction of , and if , it is in the other half-space.The point-plane distance from a point to a plane (6) is given by the simple equation(7)(Gellert et al. 1989, p. 541). If the point is in the half-space determined by the direction of , then ; if it is in the other half-space, then .

A plane is a two-dimensional doubly ruled surface spanned by two linearly independent vectors. The generalization of the plane to higher dimensions is called a hyperplane. The angle between two intersecting planes is known as the dihedral angle.The equation of a plane with nonzero normal vector through the point is(1)where . Plugging in gives the general equation of a plane,(2)where(3)A plane specified in this form therefore has -, -, and -intercepts at(4)(5)(6)and lies at a distance(7)from the origin.It is especially convenient to specify planes in so-called Hessian normal form. This is obtained from (◇) by defining the components of the unit normal vector (8)(9)(10)and the constant(11)Then the Hessian normal form of the plane is(12)(Gellert et al. 1989, p. 540), the (signed) distance to a point is(13)and the distance from the origin is simply(14)(Gellert et al. 1989, p. 541).In intercept form, a plane passing through..

A set of planes sharing a point in common. For planes specified in Hessian normal form, a bundle of planes can therefore be specified aswhere are free real parameters. This can be made more symmetrical by introducing homogeneous parameters and to obtain(Gellert et al. 1989, p. 543).

A double bubble is pair of bubbles which intersect and are separated by a membrane bounded by the intersection. The usual double bubble is illustrated in the left figure above. A more exotic configuration in which one bubble is torus-shaped and the other is shaped like a dumbbell is illustrated at right (illustrations courtesy of J. M. Sullivan).In the plane, the analog of the double bubble consists of three circular arcs meeting in two points. It has been proved that the configuration of arcs meeting at equal angles) has the minimum perimeter for enclosing two equal areas (Alfaro et al. 1993, Morgan 1995).It had been conjectured that two equal partial spheres sharing a boundary of a flat disk separate two volumes of air using a total surface area that is less than any other boundary. This equal-volume case was proved by Hass et al. (1995), who reduced the problem to a set of integrals which they carried out on an ordinary PC. Frank Morgan,..

A projective plane, sometimes called a twisted sphere (Henle 1994, p. 110), is a surface without boundary derived from a usual plane by addition of a line at infinity. Just as a straight line in projective geometry contains a single point at infinity at which the endpoints meet, a plane in projective geometry contains a single line at infinity at which the edges of the plane meet. A projective plane can be constructed by gluing both pairs of opposite edges of a rectangle together giving both pairs a half-twist. It is a one-sided surface, but cannot be realized in three-dimensional space without crossing itself.A finite projective plane of order is formally defined as a set of points with the properties that: 1. Any two points determine a line,2. Any two lines determine a point,3. Every point has lines on it, and 4. Every line contains points. (Note that some of these properties are redundant.) A projective plane is therefore a symmetric (, , 1)..

A minimal surface that contains lemniscatesas geodesics which is given by the parametric equations(1)(2)(3)(4)where is an incomplete elliptic integral of the first kind and is a complex number. A given lemniscate is the intersection of the surface with the -plane. The surface is periodic in the direction of the axis with period(5)where is a complete elliptic integral of the first kind.

The Costa surface is a complete minimal embedded surface of finite topology (i.e., it has no boundary and does not intersect itself). It has genus 1 with three punctures (Schwalbe and Wagon 1999). Until this surface was discovered by Costa (1984), the only other known complete minimal embeddable surfaces in with no self-intersections were the plane (genus 0), catenoid (genus 0 with two punctures), and helicoid (genus 0 with two punctures), and it was conjectured that these were the only such surfaces.Rather amazingly, the Costa surface belongs to the dihedral group of symmetries.The Costa minimal surface appears on the cover of Osserman (1986; left figure) as well as on the cover of volume 2, number 2 of La Gaceta de la Real Sociedad Matemática Española (1999; right figure).It has also been constructed as a snow sculpture (Ferguson et al. 1999, Wagon1999).On Feb. 20, 2008, a large stone sculpture by Helaman Ferguson was..

A minimal surface discovered by L. P. M. Jorge and W. Meeks III in 1983 with Enneper-Weierstrass parameterization(1)(2)(Dickson 1990). Explicitly, it is given by(3)(4)(5)The coefficients of the first fundamental formare given by(6)(7)(8)and the coefficients of the second fundamentalform by(9)(10)(11)The area element is(12)The Gaussian and meancurvatures are given by(13)(14)

A surface which is simultaneously complete and minimal. There have been a large number of fundamental breakthroughs in the study of such surfaces in recent years, and they remain the focus of intensive current research.Until the Costa minimal surface was discovered in 1982, the only other known complete minimal embeddable surfaces in with no self-intersections were the plane, catenoid, and helicoid. The plane is genus 0 and the catenoid and the helicoid are genus 0 with two punctures, but the Costa minimal surface is genus 1 with three punctures (Schwalbe and Wagon 1999).

A minimal embedded surface discovered in 1992 consisting of a helicoid with a hole and handle (Science News 1992). It has the same topology as a punctured sphere with a handle, and is only the second complete embedded minimal surface of finite topology and infinite total curvature discovered (the helicoid being the first).A three-ended minimal surface of genus1 is sometimes also called Hoffman's minimal surface (Peterson 1988).

A periodic minimal surface constructed by Schwarzusing the following two principles: 1. If part of the boundary of a minimal surface is a straight line, then the reflection across the line, when added to the original surface, makes another minimal surface. 2. If a minimal surface meets a plane at right angles, then the mirror image of the plane, when added to the original surface, also makes a minimal surface.

A minimal surface and double algebraic surface of 15th order and fifth class which can be given by parametric equations(1)(2)(3)The coefficients of the first fundamental formof this parameterization are given by(4)(5)(6)and the coefficients of the second fundamentalform are(7)(8)(9)giving area element(10)and Gaussian and meancurvatures are(11)(12)A slightly different version of this surface (also algebraic of order 15 but with slightly different coefficients) can also be obtained from the Enneper-Weierstrass parameterization with(13)(14)which gives a parameterization of the form(15)(16)(17)The coefficients of the first fundamental formof this parameterization are given by(18)(19)(20)and the coefficients of the second fundamentalform are(21)(22)(23)giving area element(24)and Gaussian and meancurvatures are(25)(26)Henneberg's minimal surface is a nonorientable surface defined over the unit disk. It is..

Scherk's two minimal surfaces were discovered by Scherk in 1834. They were the first new surfaces discovered since Meusnier in 1776. Beautiful images of wood sculptures of Scherk surfaces are illustrated by Séquin.Scherk's first surface is doubly periodic and is defined by the implicit equation(1)(Osserman 1986, Wells 1991, von Seggern 1993). It has been observed to form in layers of block copolymers (Peterson 1988).Scherk's second surface is the surface generated by Enneper-Weierstrassparameterization with(2)(3)It can be written parametrically as(4)(5)(6)(7)(8)(9)for , and . With this parametrization, the coefficients of the first fundamental form are(10)(11)(12)and of the second fundamental form are(13)(14)(15)The Gaussian and mean curvatures are(16)(17)

The (circular) helicoid is the minimal surface having a (circular) helix as its boundary. It is the only ruled minimal surface other than the plane (Catalan 1842, do Carmo 1986). For many years, the helicoid remained the only known example of a complete embedded minimal surface of finite topology with infinite curvature. However, in 1992 a second example, known as Hoffman's minimal surface and consisting of a helicoid with a hole, was discovered (Sci. News 1992). The helicoid is the only non-rotary surface which can glide along itself (Steinhaus 1999, p. 231).The equation of a helicoid in cylindricalcoordinates is(1)In Cartesian coordinates, it is(2)It can be given in parametric form by(3)(4)(5)which has an obvious generalization to the elliptic helicoid. Writing instead of gives a cone instead of a helicoid.The first fundamental form coefficientsof the helicoid are given by(6)(7)(8)and the second fundamental form coefficientsare(9)(10)(11)giving..

A catenary of revolution. The catenoid and plane are the only surfaces of revolution which are also minimal surfaces. The catenoid can be given by the parametric equations(1)(2)(3)where .The line element is(4)The first fundamental form has coefficients(5)(6)(7)and the second fundamental form has coefficients(8)(9)(10)The principal curvatures are(11)(12)The mean curvature of the catenoid is(13)and the Gaussian curvature is(14)The helicoid can be continuously deformed into a catenoid with by the transformation(15)(16)(17)where corresponds to a helicoid and to a catenoid.This deformation is illustrated on the cover of issue 2, volume 2 of The MathematicaJournal.

The gyroid, illustrated above, is an infinitely connected periodic minimal surface containing no straight lines (Osserman 1986) that was discovered by Schoen (1970). Große-Brauckmann and Wohlgemuth (1996) proved that the gyroid is embedded.The gyroid is the only known embedded triply periodic minimal surface with triple junctions. In addition, unlike the five triply periodic minimal surfaces studied by Anderson et al. (1990), the gyroid does not have any reflectional symmetries (Große-Brauckmann 1997).The image above shows a metal print of the gyroid created by digital sculptor BathshebaGrossman (https://www.bathsheba.com/).

A minimal surface given by the parametricequations(1)(2)(3)(Gray 1997), or(4)(5)(6)where(7)(do Carmo 1986).The first fundamental form has coefficients(8)(9)(10)and the second fundamental form has coefficients(11)(12)(13)The principal curvatures are(14)(15)the mean curvature is(16)and the Gaussian curvature is(17)

If the Gauss map of a complete minimal surface omits a neighborhood of the sphere, then the surface is a plane. This was proven by Osserman (1959). Xavier (1981) subsequently generalized the result as follows. If the Gauss map of a complete minimal surface omits points, then the surface is a plane.

Calculus of variations can be used to find the curve from a point to a point which, when revolved around the x-axis, yields a surface of smallest surface area (i.e., the minimal surface). This is equivalent to finding the minimal surface passing through two circular wire frames. The area element is(1)so the surface area is(2)and the quantity we are minimizing is(3)This equation has , so we can use the Beltrami identity(4)to obtain(5)(6)(7)(8)(9)(10)(11)(12)which is called a catenary, and the surface generated by rotating it is called a catenoid. The two constants and are determined from the two implicit equations(13)(14)which cannot be solved analytically.The general case is somewhat more complicated than this solution suggests. To see this, consider the minimal surface between two rings of equal radius . Without loss of generality, take the origin at the midpoint of the two rings. Then the two endpoints are located at and , and(15)But ,..

Gray (1997) defines Bour's minimal curve over complex by(1)(2)(3)and then derives a family of minimal surfaces.The order-three Bour surface resembles a cross-cap and is given using Enneper-Weierstrass parameterization by(4)(5)or explicitly by the parametric equations(6)(7)(8)(Maeder 1997). It is an algebraic surface oforder 16.The coefficients of the first fundamental formare given by(9)(10)(11)and the coefficients of the second fundamentalform by(12)(13)(14)The area element is(15)The Gaussian and meancurvatures are given by(16)(17)

Minimal surfaces are defined as surfaces with zero mean curvature. A minimal surface parametrized as therefore satisfies Lagrange's equation,(1)(Gray 1997, p. 399).Finding a minimal surface of a boundary with specified constraints is a problem in the calculus of variations and is sometimes known as Plateau's problem. Minimal surfaces may also be characterized as surfaces of minimal surface area for given boundary conditions. A plane is a trivial minimal surface, and the first nontrivial examples (the catenoid and helicoid) were found by Meusnier in 1776 (Meusnier 1785). The problem of finding the minimum bounding surface of a skew quadrilateral was solved by Schwarz in 1890 (Schwarz 1972).Note that while a sphere is a "minimal surface" in the sense that it minimizes the surface area-to-volume ratio, it does not qualify as a minimal surface in the sense used by mathematicians.Euler proved that a minimal surface is planar..

A self-intersecting minimal surface which can be generated using the Enneper-Weierstrass parameterization with(1)(2)Letting and taking the real part give(3)(4)(5)(6)(7)(8)where and . Eliminating and then gives the implicit form(9)so Enneper's minimal surface is algebraic of order 9.The coefficients of the first fundamental formare(10)(11)(12)the second fundamental form coefficientsare(13)(14)(15)and the Gaussian and meancurvatures are(16)(17)Letting gives the figure above, with parametrization(18)(19)(20)(do Carmo 1986, Gray 1997). In this parameterization, the coefficients of the first fundamental form are(21)(22)(23)the second fundamental form coefficientsare(24)(25)(26)the area element is(27)and the Gaussian and meancurvatures are(28)(29)

If a minimal surface is given by the equation and has continuous first and second partial derivatives for all real and , then is a plane.

A developable surface is a ruled surface having Gaussian curvature everywhere. Developable surfaces therefore include the cone, cylinder, elliptic cone, hyperbolic cylinder, and plane.A developable surface has the property that it can be made out of sheet metal, since such a surface must be obtainable by transformation from a plane (which has Gaussian curvature 0) and every point on such a surface lies on at least one straight line.

A ruled surface is a surface that can be swept out by moving a line in space. It therefore has a parameterization of the form(1)where is called the ruled surface directrix (also called the base curve) and is the director curve. The straight lines themselves are called rulings. The rulings of a ruled surface are asymptotic curves. Furthermore, the Gaussian curvature on a ruled regular surface is everywhere nonpositive.Examples of ruled surfaces include the elliptic hyperboloidof one sheet (a doubly ruled surface)(2)the hyperbolic paraboloid (a doublyruled surface)(3)Plücker's conoid(4)and the Möbius strip(5)(Gray 1993).The only ruled minimal surfaces are the planeand helicoid (Catalan 1842, do Carmo 1986).

A surface in 3-space can be parameterized by two variables (or coordinates) and such that(1)(2)(3)If a surface is parameterized as above, then the tangentvectors(4)(5)are useful in computing the surface area and surfaceintegral.

A surface of revolution is a surface generated by rotating a two-dimensional curve about an axis. The resulting surface therefore always has azimuthal symmetry. Examples of surfaces of revolution include the apple surface, cone (excluding the base), conical frustum (excluding the ends), cylinder (excluding the ends), Darwin-de Sitter spheroid, Gabriel's horn, hyperboloid, lemon surface, oblate spheroid, paraboloid, prolate spheroid, pseudosphere, sphere, spheroid, and torus (and its generalization, the toroid).The area element of the surface of revolution obtained by rotating the curve from to about the x-axis is(1)(2)so the surface area is(3)(4)(Apostol 1969, p. 286; Kaplan 1992, p. 251; Anton 1999, p. 380). If the curve is instead specified parametrically by , the surface area obtained by rotating the curve about the x-axis for if in this interval is given by(5)Similarly, the area of the surface of revolution..

Let and be nonnegative and continuous functions on the closed interval , then the solid of revolution obtained by rotating the curves and about the -axis from to and taking the region enclosed between them has volume given by

Let be a plane region bounded above by a continuous curve , below by the -axis, and on the left and right by and , then the volume of the solid of revolution obtained by rotating about the -axis is given by

Let be a nonnegative and continuous function on the closed interval , then the solid of revolution obtained by rotating the curve about the -axis from to has volume given by

The surface of revolution generated by the external catenary between a fixed point and its conjugate on the envelope of the catenary through the fixed point is equal in area to the surface of revolution generated by its two Lindelof tangents which cross the axis of rotation at the point and are calculable from the positions of the points and catenary.

A barrel solid of revolution composed of parallel circular top and bottom with a common axis and a side formed by a smooth curve symmetrical about the midplane.The term also has a technical meaning in functional analysis. In particular, a subset of a topological linear space is a barrel if it is absorbing, closed, and absolutely convex (Taylor and Lay 1980, p. 111). (A subset of a topological linear space is absorbing if for each there is an such that is in if for each such that . A subset of a topological linear space is absolutely convex if for each and in , is in if .)When buying supplies for his second wedding, the great astronomer Johannes Kepler became unhappy about the inexact methods used by the merchants to estimate the liquid contents of a wine barrel. Kepler therefore investigated the properties of nearly 100 solids of revolution generated by rotation of conic sections about non-principal axes (Kepler, MacDonnell, Shechter, Tikhomirov..

There are nine possible types of isolated singularities on a cubic surface, eight of them rational double points. Each type of isolated singularity has an associated normal form and Coxeter-Dynkin diagram (, , , , , , , and ).The eight types of rational double points (the type being the one excluded) can occur in only 20 combinations on a cubic surface (of which Fischer 1986a gives 19): , , , , , , , , , , , , , (), , , , , and (Looijenga 1978, Bruce and Wall 1979, Fischer 1986a).In particular, on a cubic surface, precisely those configurations of rational double points occur for which the disjoint union of the Coxeter-Dynkin diagram is a subgraph of the Coxeter-Dynkin diagram . Also, a surface specializes to a more complicated one precisely when its graph is contained in the graph of the other one (Fischer 1986a)...

Let be a regular surface and a unit tangent vector to , and let be the plane determined by and the normal to the surface . Then the normal section of is defined as the intersection of and .

A ruled surface parameterization is called noncylindrical if is nowhere . A noncylindrical ruled surface always has a parameterization of the formwhere and , where is called the striction curve of and the director curve.

A Monge patch is a patch of the form(1)where is an open set in and is a differentiable function. The coefficients of the first fundamental form are given by(2)(3)(4)and the second fundamental form by(5)(6)(7)For a Monge patch, the Gaussian curvature andmean curvature are(8)(9)

A ruled surface is called a right conoid if it can be generated by moving a straight line intersecting a fixed straight line such that the lines are always perpendicular (Kreyszig 1991, p. 87). Taking the perpendicular plane as the -plane and the line to be the x-axis gives the parametric equations(1)(2)(3)(Gray 1997). Taking and gives the helicoid.

Plücker's conoid is a ruled surface sometimes also called the cylindroid, conical wedge, or conocuneus of Wallis. von Seggern (1993, p. 288) gives the general functional form as(1)whereas Fischer (1986) and Gray (1997) give(2)A polar parameterization therefore gives(3)(4)(5)The cylindroid is the inversion of the cross-cap (Pinkall1986).A generalization of Plücker's conoid to folds is given by(6)(7)(8)(Gray 1997), which is a slight variation of the form called the "conical wedge" by von Seggern (1993, p. 302).The coefficients of the first fundamental formof the generalized Plücker's conoid are(9)(10)(11)and of the second fundamental form are(12)(13)(14)The Gaussian and meancurvatures are given by(15)(16)

A hyperboloid is a quadratic surface which may be one- or two-sheeted. The one-sheeted hyperboloid is a surface of revolution obtained by rotating a hyperbola about the perpendicular bisector to the line between the foci, while the two-sheeted hyperboloid is a surface of revolution obtained by rotating a hyperbola about the line joining the foci (Hilbert and Cohn-Vossen 1991, p. 11).

A generalization of an ordinary two-dimensional surface embedded in three-dimensional space to an -dimensional surface embedded in -dimensional space. A hypersurface is therefore the set of solutions to a single equationand so it has codimension one. For instance, the -dimension hypersphere corresponds to the equation .

The term "poweroid" has at least two meanings. Sheffer sequences are sometimes called poweroids (Steffensen 1941, Shiu 1982, Di Bucchianico and Loeb 2000). Jackway and Deriche (1996) and Jackway (2002) use the term to mean a function of the formThe case corresponds to the paraboloid and the case is sometimes called a quartoid (Jackway and Deriche 1996).

The surface with parametric equations(1)(2)(3)where is the torsion.The coefficients of the first fundamental formare(4)(5)(6)and those of the second fundamental formare(7)(8)(9)The Gaussian curvature is a somewhat complicated,but the mean curvature is given by(10)

The right conoid surface given by the parametricequations(1)(2)(3)

The surface generated by a twisted curve when rotated about a fixed axis and, at the same time, displaced parallel to so that the velocity of displacement is always proportional to the angular velocity of rotation.

A smooth two-dimensional surface given by embedding the projectiveplane into projective 5-space by the homogeneous parametric equationsThe surface can be projected smoothly into four-space, but all three-dimensional projections have singularities (Coffman). The projections of these surfaces in three dimensions are called Steiner surfaces. The volume of the Veronese surface is .

The envelope of the plane(1)where is the speed of propagation of a wave in the direction (i.e., , , and are the direction cosines) is known as the wave surface of a given medium (Love 1944, p. 299).In an isotropic medium, is independent of , , and and is given by(2)where is the medium density and and are the so-called Lamé constants of the solid. The wave surface is then two-sheeted and both sheets are spheres (Love 1944, p. 299).In the case of anisotropy, the surface in consists of three sheets corresponding to the values of that are roots of(3)where the are functions of , , and in terms of the coefficients of the strain-energy function (Christoffel 1877, Love 1944, p. 299). Green (1839) showed that the wave surface for the most general case of an elastic solid that allows the propagation of purely transverse plane waves consists of a sphere and the two sheets that are the envelope of the plane (1) subject to the condition(4)where..

A surface is -embeddable if it can be placed in -space without self-intersections, but cannot be similarly placed in any for . A surface so embedded is said to be an embedded surface. The Costa minimal surface and gyroid are embeddable in , but the Klein bottle is not (the commonly depicted representation requires the surface to pass through itself).There is particular interest in surfaces which are minimal, complete, and embedded.

A regular surface is called orientable if each tangent space has a complex structure such that is a continuous function.

The corkscrew surface, sometimes also called the twisted sphere (Gray 1997, p. 477), is a surface obtained by extending a sphere along a diameter and then twisting. It can be specified parametrically as(1)(2)(3)The coefficients of the first fundamental formare(4)(5)(6)and those of the second fundamental formare(7)(8)(9)The Gaussian and meancurvatures are(10)(11)

The word "surface" is an important term in mathematics and is used in many ways. The most common and straightforward use of the word is to denote a two-dimensional submanifold of three-dimensional Euclidean space. Surfaces can range from the very complicated (e.g., fractals such as the Mandelbrot set) to the very simple (such as the plane). More generally, the word "surface" can be used to denote an -dimensional submanifold of an -dimensional manifold, or in general, any codimension-1 subobject in an object (like a Banach space or an infinite-dimensional manifold).Even simple surfaces can display surprisingly counterintuitive properties. For example, the surface of revolution of around the x-axis for (called Gabriel's horn) has finite volume but infinite surface area...

Let be a link in and let there be a disk in the link complement . Then a surface such that intersects exactly in its boundary and its boundary does not bound another disk on is called a compressible surface (Adams 1994, p. 86).

A surface generated by the parametricequations(1)(2)(3)The above image uses and .The coefficients of the first fundamental formare(4)(5)(6)the coefficients of the second fundamentalform are(7)(8)(9)the area element is(10)and the Gaussian and meancurvatures are given by(11)(12)

A surface such as the Möbius strip or Klein bottle (Gray 1997, pp. 322-323) on which there exists a closed path such that the directrix is reversed when moved around this path. The real projective plane is also a nonorientable surface, as are the Boy surface, cross-cap, and Roman surface, all of which are homeomorphic to the real projective plane (Pinkall 1986).There is a general method for constructing nonorientable surfaces which proceeds as follows (Banchoff 1984, Pinkall 1986). Choose three homogeneous polynomials of positive even degree and consider the map(1)Then restricting , , and to the surface of a sphere by writing(2)(3)(4)and restricting to and to defines a map of the real projective plane to .In three dimensions, there is no unbounded nonorientable surface which does not intersect itself (Kuiper 1961, Pinkall 1986)...

A capsule is a term coined here for a stadium of revolution, i.e., a cylinder with two hemispherical caps on either end.The capsule is implemented in the Wolfram Language as CapsuleShape[x1, y1, z1, x2, y2, z2, r].A capsule with cap radius and cylinder height has volumeand surface area

A surface parameterized in variables and is called smooth if the tangent vectors in the and directions satisfywhere is a cross product.

A surface given by the parametric equations(1)(2)(3)The coefficients of the coefficients of the firstfundamental form are(4)(5)(6)and the second fundamental form coefficientsare(7)(8)(9)giving area element(10)and Gaussian and meancurvatures(11)(12)

An unduloid, also called an onduloid, is a surface of revolution with constant nonzero mean curvature. It is a roulette obtained from the path described by the foci of a conic section when rolled on a line. This curve then generates an unduloid when revolved about the line. These curves are special cases of the shapes assumed by soap film spanning the gap between prescribed boundaries. The unduloid of a parabola gives a catenoid.

A constant-curvature surface which can be given parametrically by(1)(2)(3)where(4)(5)(6)with and (Reckziegel 1986).The coefficients of the first fundamental formare(7)(8)(9)and the coefficients of the second fundamentalform are(10)(11)(12)The Sievert surface has Gaussian and meancurvatures given by(13)(14)

A surface of constant Gaussian curvature thatcan be given parametrically by(1)(2)(3)where(4)(5)(6)and , and . The value of is restricted to(7)(Reckziegel 1986), and the values correspond to the ends of the cleft in the surface. The surface illustrated above corresponds to .The coefficients of the first fundamental formare given by(8)(9)(10)with coefficients of the second fundamental form given by similar, but rather complicated, expressions. The Gaussian curvature is(11)with the mean curvature given by a rather complicatedexpression.

The pseudosphere is the constant negative-Gaussian curvature surface of revolution generated by a tractrix about its asymptote. It is sometimes also called the tractroid, tractricoid, antisphere, or tractrisoid (Steinhaus 1999, p. 251). The Cartesian parametric equations are(1)(2)(3)for and .It can be written in implicit Cartesian form as(4)Other parametrizations include(5)(6)(7)for and (Gray et al. 2006, p. 480) and(8)(9)(10)for and , where(11)(12)(Gray et al. 2006, p. 477).In the first parametrization, the coefficients of the firstfundamental form are(13)(14)(15)the second fundamental form coefficientsare(16)(17)(18)and the surface area element is(19)The surface area is(20)which is exactly that of the usual sphere.Even though the pseudosphere has infinite extent, it has finite volume. The volume can be found by making the change of variables , giving , and substituting into the equation for a..

The Kuen surface is a special case of Enneper'snegative curvature surfaces which can be given parametrically by(1)(2)(3)(4)(5)for , (Reckziegel et al. 1986; Gray et al. 2006, p. 484).The Kuen surface appears on the cover of volume 2, number 1 of La Gaceta de laReal Sociedad Matemática Española (1999).The coefficients of the first fundamental formare(6)(7)(8)the second fundamental form coefficientsare(9)(10)(11)and the surface area element is(12)The Gaussian and meancurvatures are(13)(14)so the Kuen surface has constant negative Gaussian curvature, and the principal curvatures are(15)(16)(Gray 1997, p. 496).

A surface of constant negative curvature obtained by twisting a pseudosphere and given by the parametric equations(1)(2)(3)The above figure corresponds to , , , and .Dini's surface is pictured in the upper right-hand corner of Gray (1997; left figure), as well as on the cover of volume 2, number 3 of La Gaceta de la Real Sociedad Matemática Española (1999; right figure).The coefficients of the first fundamental formare(4)(5)(6)the coefficients of the second fundamentalform are(7)(8)(9)and the area element is(10)The Gaussian and meancurvatures are given by(11)(12)

A surface which can be interpreted as a self-intersecting rectangle in three dimensions. The Whitney umbrella is the only stable singularity of mappings from to . It is given by the parametric equations(1)(2)(3)for . The center of the "plus" shape which is the end of the line of self-intersection is a pinch point. The coefficients of the first fundamental form are(4)(5)(6)and the second fundamental form are(7)(8)(9)giving area element(10)and Gaussian curvature and meancurvature(11)(12)Note that the ruled cubicsurface given by the equation:(13)is the union of Whitney umbrella and the ray , , called the handle of the Whitney umbrella.

"Nordstrand's weird surface" is an attractive quarticsurface given by the implicit equationIt has 11 ordinary double points located at , , , , , , , , , , and .

Let , , and be tetrahedra in projective three-space . Then the tetrahedra are said to be desmically related if there exist constants , , and such thatA desmic surface is then defined as a quartic surfacewhich can be written asfor desmically related tetrahedra , , and . Desmic surfaces have 12 ordinary double points, which are the vertices of three tetrahedra in three-space (Hunt 1996).

A surface which a monkey can straddle with both legsand his tail. A simple Cartesian equation for such a surface is(1)which can also be given by the parametric equations(2)(3)(4)The monkey saddle has a single stationary point as summarized in the table below. While the second derivative test is not sufficient to classify this stationary point, it turns out to be a saddle point.point20saddle pointThe coefficients of the first fundamental formof the monkey saddle are(5)(6)(7)and the second fundamental form coefficientsare(8)(9)(10)giving Riemannian metric(11)area element(12)and Gaussian and meancurvatures(13)(14)(Gray 1997). The Gaussian curvature can be written implicitly as(15)so every point of the monkey saddle except the origin has negative Gaussian curvature.

Togliatti surfaces are quintic surfaces having the maximum possible number of ordinary double points (31).A related surface sometimes known as the dervish can be defined by(1)where(2)(3)(4)(5)(6)(7)(8)and(9)(10)(11)

Togliatti (1940, 1949) showed that quintic surfaces having 31 ordinary double points exist, although he did not explicitly derive equations for such surfaces. Beauville (1978) subsequently proved that 31 double points are the maximum possible, and quintic surfaces having 31 ordinary double points are therefore sometimes called Togliatti surfaces. van Straten (1993) subsequently constructed a three-dimensional family of solutions and in 1994, Barth derived the examplewhere is a parameter (Endraß 2003), illustrated above for .This surface is invariant under the group and contains exactly 15 lines. Five of these are the intersection of the surface with a -invariant cone containing 16 nodes, five are the intersection of the surface with a -invariant plane containing 10 nodes, and the last five are the intersection of the surface with a second -invariant plane containing no nodes (Endraß 2003)...

A surface given by the parametric equations(1)(2)(3)

A cyclide is a pair of focal conics which are the envelopes of two one-parameter families of spheres, sometimes also called a cyclid. The cyclide is a quartic surface, and the lines of curvature on a cyclide are all straight lines or circular arcs (Pinkall 1986). The standard tori and their inversions in an inversion sphere centered at a point and of radius , given byare both cyclides (Pinkall 1986). Illustrated above are ring cyclides, horn cyclides, and spindle cyclides. The figures on the right correspond to lying on the torus itself, and are called the parabolic ring cyclide, parabolic horn cyclide, and parabolic spindle cyclide, respectively.

A surface given by the parametricequations(1)(2)(3)

A surface of revolution defined by Kepler. It consists of less than half of a circular arc rotated about an axis passing through the endpoints of the arc. The equations of the upper and lower boundaries in the plane are(1)for and . The cross section of a lemon is a lens. The lemon is the inside surface of a spindle torus. The American football is shaped like a lemon.Two other lemon-shaped surfaces are given by the sexticsurface(2)called the "citrus" (or zitrus) surface by Hauser (left figure), and thesextic surface(3)whose upper and lower portions resemble two halves of a lemon, called the limão surface by Hauser (right figure).The citrus surface had bounding box , centroid at , volume(4)and a moment of inertia tensor(5)for a uniform density solid citrus with mass .

An algebraic surface of order 3. Schläfli and Cayley classified the singular cubic surfaces. On the general cubic, there exists a curious geometrical structure called double sixes, and also a particular arrangement of 27 (possibly complex) lines, as discovered by Schläfli (Salmon 1965, Fischer 1986) and sometimes called Solomon's seal lines. A nonregular cubic surface can contain 3, 7, 15, or 27 real lines (Segre 1942, Le Lionnais 1983). The Clebsch diagonal cubic contains all possible 27. The maximum number of ordinary double points on a cubic surface is four, and the unique cubic surface having four ordinary double points is the Cayley cubic.Examples of cubic surfaces include the Cayley cubic, Clebsch diagonal cubic, ding-dong surface, handkerchief surface, Möbius strip, monkey saddle, shoe surface, Wallis's conical edge, and Whitney umbrella.Schoute (1910) showed that the 27 lines can be put into a one-to-one correspondence..

The crossed trough is the surface(1)The coefficients of its first fundamental formare(2)(3)(4)and of the second fundamental form are(5)(6)(7)The Gaussian and meancurvatures are(8)(9)The Gaussian curvature can be written implicitly as(10)

The Kummer surfaces are a family of quartic surfacesgiven by the algebraic equation(1)where(2), , , and are the tetrahedral coordinates(3)(4)(5)(6)and is a parameter which, in the above plots, is set to .The above plots correspond to (7)(double sphere), 2/3, 1(8)(Roman surface), 2, 3(9)(four planes), and 5. The case corresponds to four real points.The following table gives the number of ordinary double points for various ranges of , corresponding to the preceding illustrations.parameterreal nodescomplex nodes412412160160The Kummer surfaces can be represented parametrically by hyperelliptic theta functions. Most of the Kummer surfaces admit 16 ordinary double points, the maximum possible for a quartic surface. A special case of a Kummer surface is the tetrahedroid.Nordstrand gives the implicit equations as(10)or(11)..

The order of an algebraic surface is the order of the polynomial defining a surface, which can be geometrically interpreted as the maximum number of points in which a line meets the surface.ordersurface3cubic surface4quartic surface5quintic surface6sextic surface7septic surface8octic surface9nonic surface10decic surface12dodecic surface

The kiss surface is the quintic surfaceof revolution given by the equation(1)that is closely related to the ding-dong surface. It is so named because the shape of the lower portion resembles that of a Hershey's Chocolate Kiss.It can be represented parametrically as(2)(3)(4)The coefficients of the first fundamental formare(5)(6)(7)and of the second fundamental form are(8)(9)(10)The Gaussian and meancurvatures are given by(11)(12)The Gaussian curvature can be given implicitlyby(13)The surface area and volumeenclosed of the top teardrop are given by(14)(15)Its centroid is at and the moment of inertia tensor is(16)for a solid kiss with uniform density and mass .

The surface given by the parametricequations(1)(2)(3)For , the coefficients of the first fundamental form are(4)(5)(6)and of the second fundamental form are(7)(8)(9)The Gaussian and meancurvatures are given by(10)(11)and the principal curvatures are(12)(13)

A projection of the Veronese surface into three dimensions (which must contain singularities) is called a Steiner surface. A classification of Steiner surfaces allowing complex parameters and projective transformations was accomplished in the 19th century. The surfaces obtained by restricting to real parameters and transformations were classified into 10 types by Coffman et al. (1996). Examples of Steiner surfaces include the Roman surface (sometimes know as "the" Steiner surface; Coffman type 1) and cross-cap (type 3).The Steiner surface of type 2 is given by the implicit equationand can be transformed into the Roman surface or cross-cap by a complex projective change of coordinates (but not by a real transformation). It has two pinch points and three double lines and, unlike the Roman surface or cross-cap, is not compact in any affine neighborhood.The Steiner surface of type 4 has the implicit equationand two of the three..

A hyperbolic paraboloid is the quadratic and doubly ruled surface given by the Cartesian equation(1)(left figure). An alternative form is(2)(right figure; Fischer 1986), which has parametricequations(3)(4)(5)(Gray 1997, pp. 297-298).The coefficients of the first fundamental formare(6)(7)(8)and the second fundamental form coefficientsare(9)(10)(11)giving surface area element(12)The Gaussian curvature is(13)and the mean curvature is(14)The Gaussian curvature can be given implicitly as(15)Three skew lines always define a one-sheeted hyperboloid, except in the case where they are all parallel to a single plane but not to each other. In this case, they determine a hyperbolic paraboloid (Hilbert and Cohn-Vossen 1999, p. 15).

A set of quadratic surfaces which share foci. Ellipsoids and one- and two-sheeted hyperboloids can be confocal. These three types of surfaces can be combined to form an orthogonal coordinate system known as confocal ellipsoidal coordinates (Hilbert and Cohn-Vossen 1999, pp. 22-23).The planes of symmetry of the tangent cone from any point in space to any surface of the confocal system which does not enclose are the tangent planes at to the three surfaces of the system that pass through . As a limiting case, this result means that every surface of the confocal system when viewed from a point lying on a focal curve and not enclosed by the surface looks like a circle with its center on the line of sight, provided that the line of sight is tangent to the focal curve (Hilbert and Cohn-Vossen 1999, p. 24)...

A cubic algebraicsurface given by the equation(1)with the added constraint(2)The implicit equation obtained by taking the plane at infinity as is(3)(Hunt 1996), illustrated above.On Clebsch's diagonal surface, all 27 of the complex lines (Solomon's seal lines) present on a general smooth cubic surface are real. In addition, there are 10 points on the surface where 3 of the 27 lines meet. These points are called Eckardt points (Fischer 1986ab, Hunt 1996), and the Clebsch diagonal surface is the unique cubic surface containing 10 such points (Hunt 1996).If one of the variables describing Clebsch's diagonal surface is dropped, leaving the equations(4)(5)the equations degenerate into two intersecting planes given by the equation(6)

The surface given by the parametric equations(1)(2)(3)It is a sextic surface with algebraic equation(4)The coefficients of the first fundamental formare(5)(6)(7)the second fundamental form coefficientsare(8)(9)(10)the area element is(11)the Gaussian curvature is(12)and the mean curvature is a complicated expression.It has volume(13)

The hyperbolic cylinder is a quadratic surfacegiven by the equation(1)It is a ruled surface.It can be given parametrically by(2)(3)(4)The coefficients of the first fundamental formare(5)(6)(7)and of the second fundamental form are(8)(9)(10)The Gaussian and mean curvatures are(11)(12)

An algebraic surface which can be represented implicitly by a polynomial of degree six in , , and .Examples of quartic surfaces include the Barth sextic, Boy surface, heart surface, Hunt's surface, and sine surface.

An algebraic surface with affine equation(1)where is a Chebyshev polynomial of the first kind and is a polynomial defined by(2)where the matrices have dimensions . These represent surfaces in with only ordinary double points as singularities. The first few surfaces are given by (3)(4)(5)The th order such surface has(6)singular points (Chmutov 1992), giving the sequence 0, 1, 3, 14, 28, 57, 93, 154, 216, 321, 425, 576, 732, 949, 1155, ... (OEIS A057870) for , 2, .... For a number of orders , Chmutov surfaces have more ordinary double points than any other known equations of the same degree.Based on Chmutov's equations, Banchoff (1991) defined the simpler set of surfaces(7)where is even and is again a Chebyshev polynomial of the first kind. For example, the surfaces illustrated above have orders 2, 4, and 6 and are given by the equations (8)(9)(10)..

A surface with tetrahedral symmetry which looks likean inflatable chair from the 1970s. It is given by the implicit equationThe surface illustrated above has , , and .

The dodecic surface defined by(1)where(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16) and are both invariants of order 12. It was discovered by A. Sarti in 1999.The version with arbitrary and has exactly 600 ordinary points (Endraß), and taking gives the surface with 560 real ordinary points illustrated above.The Sarti surface is invariant under the bipolyhedralgroup.

The Roman surface, also called the Steiner surface (not to be confused with the class of Steiner surfaces of which the Roman surface is a particular case), is a quartic nonorientable surface The Roman surface is one of the three possible surfaces obtained by sewing a Möbius strip to the edge of a disk. The other two are the Boy surface and cross-cap, all of which are homeomorphic to the real projective plane (Pinkall 1986).The center point of the Roman surface is an ordinary triple point with , and the six endpoints of the three lines of self-intersection are singular pinch points, also known as pinch points. The Roman surface is essentially six cross-caps stuck together and contains a double infinity of conics.The Roman surface can given by the equation(1)Solving for gives the pair of equations(2)If the surface is rotated by about the z-axis via the rotation matrix(3)to give(4)then the simple equation(5)results.The Roman surface can also..

Cayley's cubic surface is the unique cubic surface having four ordinary double points (Hunt), the maximum possible for cubic surface (Endraß). The Cayley cubic is invariant under the tetrahedral group and contains exactly nine lines, six of which connect the four nodes pairwise and the other three of which are coplanar (Endraß).If the ordinary double points in projective three-space are taken as (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1), then the equation of the surface in projective coordinates is(1)(Hunt). Defining "affine" coordinates with plane at infinity and(2)(3)(4)then gives the equation(5)plotted in the left figure above (Hunt). The slightly different form(6)is given by Endraß (2003) which, when rewritten in tetrahedralcoordinates, becomes(7)plotted in the right figure above.The Hessian of the Cayley cubic is given by(8)in homogeneous coordinates , , , and . Taking the plane at infinity..

A quintic surface is an algebraic surface of degree 5. Togliatti (1940, 1949) showed that quintic surfaces having 31 ordinary double points exist, although he did not explicitly derive equations for such surfaces. Beauville (1978) subsequently proved that 31 double points was the maximum possible, and quintic surfaces having 31 ordinary double points are therefore sometimes called Togliatti surfaces. van Straten (1993) subsequently constructed a three-dimensional family of solutions and in 1994, Barth derived the example known as the dervish.Examples of quartic surfaces include the dervish, kiss surface, peninsula surface, swallowtail catastrophe surface, and Togliatti surface.

A heart-shaped surface given by the sextic equation(Taubin 1993, 1994). The figures above show a ray-traced rendering (left) and the cross section (right) of the surface.A slight variation of the same surface is given by(Nordstrand, Kuska 2004).

The quartic surface obtained by replacing the constant in the equation of the Cassini ovals with , obtaining(1)As can be seen by letting to obtain(2)(3)the intersection of the surface with the plane is a circle of radius .The Gaussian curvature of the surface is givenimplicitly by(4)Let a torus of tube radius be cut by a plane perpendicular to the plane of the torus's centroid. Call the distance of this plane from the center of the torus hole , let , and consider the intersection of this plane with the torus as is varied. The resulting curves are Cassini ovals, and the surface having these curves as cross sections is the Cassini surface(5)which has a scaled on the right side instead of .

An algebraic surface of surface order 4. Unlike cubic surfaces, quartic surfaces have not been fully classified.Examples of quartic surfaces include the apple surface, Bohemian dome, Cassini surface, Cayley cubic Hessian, crossed trough, cushion surface, double sphere, eight surface, elliptic torus, Goursat's surface, lemon surface, Menn's surface, miter surface, Nordstrand's weird surface, Peano surface, piriform surface, quartoid, Roman surface, Steiner surface types 2 and 4, tanglecube, tooth surface, and torus.

A surface given by the parametric equations(1)(2)(3)The handkerchief surface has stationary points summarized in the table below, where the type of point can be found using the second derivative test.point48local maximum4saddle pointIts first fundamental form has coefficients(4)(5)(6)and its second fundamental form has coefficients(7)(8)(9)The Gaussian curvature can be given implicitlyby(10)

A general quartic surface defined by(1)(Gray 1997, p. 314). The above two images correspond to , and , respectively.Additional cases are illustrated above.The "rounded cube" case corresponding to is a superellipsoid with volume(2)where is the gamma function.The volume of the case is given by(3)(4)where(5) is the real part of and is a Bessel function of the first kind (E. W. Weisstein and M. Trott, pers. comm., Nov. 9, 2008), which can probably be expressed in closed form in terms of bivariate hypergeometric functions.The related surface(6)for an even integer is also considered by Gray (1997, p. 292) and is a special case of the superellipsoid.

An Enriques surface is a smooth compact complex surface having irregularity and nontrivial canonical sheaf such that (Endraß). Such surfaces cannot be embedded in projective three-space, but there nonetheless exist transformations onto singular surfaces in projective three-space. There exists a family of such transformed surfaces of degree six which passes through each edge of a tetrahedron twice. A subfamily with tetrahedral symmetry is given by the two-parameter () family of surfacesand the polynomial is a sphere with radius ,(Endraß).

A second-order algebraic surface given by thegeneral equation(1)Quadratic surfaces are also called quadrics, and there are 17 standard-form types. A quadratic surface intersects every plane in a (proper or degenerate) conic section. In addition, the cone consisting of all tangents from a fixed point to a quadratic surface cuts every plane in a conic section, and the points of contact of this cone with the surface form a conic section (Hilbert and Cohn-Vossen 1999, p. 12).Examples of quadratic surfaces include the cone, cylinder, ellipsoid, elliptic cone, elliptic cylinder, elliptic hyperboloid, elliptic paraboloid, hyperbolic cylinder, hyperbolic paraboloid, paraboloid, sphere, and spheroid.Define(2)(3)(4)(5)(6)and , , as are the roots of(7)Also define(8)Then the following table enumerates the 17 quadrics and their properties (Beyer 1987).surfaceequationcoincident planes11ellipsoid (imaginary)341ellipsoid..

A quartic surface which can be constructed as follows. Given a circle and plane perpendicular to the plane of , move a second circle of the same radius as through space so that its center always lies on and it remains parallel to . Then sweeps out the Bohemian dome. It can be given by the parametric equations(1)(2)(3)where . In the above plot, , , and .The Gaussian curvature and meancurvature of the surface are given by(4)(5)

Endraß surfaces are a pair of octic surfaces which have 168 ordinary double points. This is the maximum number known to exist for an octic surface, although the rigorous upper bound is 174. The equations of the surfaces arewhere is a parameter. All ordinary double points of are real, while 24 of those in are complex. The surfaces were discovered in a five-dimensional family of octics with 112 nodes, and are invariant under the group .The surfaces illustrated above take . The first of these has 144 real ordinary double points, and the second of which has 144 complex ordinary double points, 128 of which are real.

The Barth sextic is a sextic surface in complex three-dimensional projective space having the maximum possible number of ordinary double points, namely 65. The surface was discovered by W. Barth in 1994, and is given by the implicit equationwhere is the golden ratio.Taking gives the surface in 3-space illustrated above, which retains 50 ordinary double points.Of these, 20 nodes are at the vertices of a regular dodecahedron of side length and circumradius (left figure above), and 30 are at the vertices of a concentric icosidodecahedron and circumradius 1 (right figure).The Barth sextic is invariant under the icosahedralgroup. Under the mapthe surface is the eightfold cover of the Cayley cubic(Endraß 2003).The Barth sextic appeared on the cover of the March 1999 issue of Notices of theAmerican Mathematical Society (Dominici 1999)...

The Barth decic is a decic surface in complex three-dimensional projective space having the maximum possible number of ordinary double points, namely 345. It is given by the implicit equationwhere is the golden ratio and is a parameter.The case , illustrated in the above plot, has 300 ordinary double points. These 300 points lie symmetrically at the vertices of two 60-faced and two 90-faced solids, illustrated above, with the 60-faces solids corresponding to a truncation of the icosahedron (but not the symmetrically truncated truncated icosahedron).The concentric groups of points lie at distances , , 1, and from the origin, illustrated above.The Barth decic is invariant under the icosahedralgroup.

A quadratic surface which has elliptical cross section. The elliptic paraboloid of height , semimajor axis , and semiminor axis can be specified parametrically by(1)(2)(3)for and .This gives first fundamental form coefficientsof(4)(5)(6)second fundamental form coefficients of(7)(8)(9)The Gaussian curvature and meancurvature are(10)(11)The Gaussian curvature can be expressed implicitly as(12)

The surface which is the inverse of the ellipsoid in the sense that it "goes in" where the ellipsoid "goes out." It is given by the parametric equations(1)(2)(3)for and .The special case corresponds to the hyperbolic octahedron. Like the hyperbolic octahedron, the astroidal ellipse is an algebraic surface of degree 18 with very complicated terms.The astroidal ellipsoid has first fundamentalform coefficients(4)(5)(6)while the coefficients of the second fundamentalform are more complicated.The Gaussian curvature is(7)while the mean curvature has a complicated expression.

The elliptic hyperboloid is the generalization of the hyperboloid to three distinct semimajor axes. The elliptic hyperboloid of one sheet is a ruled surface and has Cartesian equation(1)and parametric equations(2)(3)(4)for , or(5)(6)(7)or(8)(9)(10)Taking the second of these with upper signs gives firstfundamental form coefficients of(11)(12)(13)second fundamental form coefficients of(14)(15)(16)The Gaussian curvature and meancurvature are(17)(18)The Gaussian curvature can be giving implicitly by(19)(20)(21)The two-sheeted elliptic hyperboloid oriented along the z-axishas Cartesian equation(22)and parametric equations(23)(24)(25)The two-sheeted elliptic hyperboloid oriented along the x-axishas Cartesian equation(26)and parametric equations(27)(28)(29)The Gaussian curvature can be giving implicitly by(30)(31)(32)..

A surface of revolution defined by Kepler. It consists of more than half of a circular arc rotated about an axis passing through the endpoints of the arc. The equations of the upper and lower boundaries in the - plane are(1)for and . It is the outside surface of a spindle torus.It is also a quartic surface given by Cartesianequation(2)or(3)

The surface of revolution given by the parametric equations(1)(2)(3)for and .It is a quartic surface with equation(4)An essentially equivalent surface called by Hauser the octdong surface follows by making the transformation in the above, leading to(5)Setting , , and (i.e., scaling by half and relabeling the -axis as the -axis) gives the eight curve, of which the eight surface is therefore "almost" a surface of revolution.The coefficients of the first fundamental formare(6)(7)(8)and of the second fundamental form are(9)(10)(11)The Gaussian and meancurvatures are given by(12)(13)The Gaussian curvature can be given implicitly as(14)The eight surface has surface area and volumegiven by(15)(16)Its centroid is at and its moment of inertia tensor is(17)for a solid with uniform density and mass ...

The set of roots of a polynomial . An algebraic surface is said to be of degree , where is the maximum sum of powers of all terms . The following table lists the names of algebraic surfaces of a given degree.ordersurfaceexamples2quadratic surfacecone, cylinder, ellipsoid, elliptic cone, elliptic cylinder, elliptic hyperboloid, elliptic paraboloid, hyperbolic cylinder, hyperbolic paraboloid, paraboloid, sphere, spheroid3cubic surfaceCayley cubic, Clebsch diagonal cubic, ding-dong surface, handkerchief surface, Möbius strip, monkey saddle, shoe surface, Wallis's conical edge, Whitney umbrella4quartic surfaceapple surface, Bohemian dome, Cassini surface, Cayley cubic Hessian, cross-cap, crossed trough, cushion surface, double sphere, eight surface, elliptic torus, Goursat's surface, Klein quartic surface, lemon surface, Menn's surface, miter surface, Nordstrand's weird surface, Peano surface, piriform surface,..

The Boy surface is a nonorientable surface that is one possible parametrization of the surface obtained by sewing a Möbius strip to the edge of a disk. Two other topologically equivalent parametrizations are the cross-cap and Roman surface. The Boy surface is a model of the projective plane without singularities and is a sextic surface.A sculpture of the Boy surface as a special immersion of the real projective plane in Euclidean 3-space was installed in front of the library of the Mathematisches Forschungsinstitut Oberwolfach library building on January 28, 1991 (Mathematisches Forschungsinstitut Oberwolfach; Karcher and Pinkall 1997).The Boy surface can be described using the general method for nonorientable surfaces, but this was not known until the analytic equations were found by Apéry (1986). Based on the fact that it had been proven impossible to describe the surface using quadratic polynomials, Hopf had conjectured..

The maximal number of regions into which lines divide a plane arewhich, for , 2, ... gives 2, 4, 7, 11, 16, 22, ... (OEIS A000124), the same maximal number of regions into which a circle, square, etc. can be divided by lines.

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