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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

The Reuleaux tetrahedron, sometimes also called the spherical tetrahedron, is the three-dimensional solid common to four spheres of equal radius placed so that the center of each sphere lies on the surface of the other three. The centers of the spheres are therefore located at the vertices of a regular tetrahedron, and the solid consists of an "inflated" tetrahedron with four curved edges.Note that the name, coined here for the first time, is based on the fact that the geometric shape is the three-dimensional analog of the Reuleaux triangle, not the fact that it has constant width. In fact, the Reuleaux tetrahedron is not a solid of constant width. However, Meißner (1911) showed how to modify the Reuleaux tetrahedron to form a surface of constant width by replacing three of its edge arcs by curved patches formed as the surfaces of rotation of a circular arc. Depending on which three edge arcs are replaced (three that have a common..

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)

A spheroid is an ellipsoid having two axes of equal length, making it a surface of revolution. By convention, the two distinct axis lengths are denoted and , and the spheroid is oriented so that its axis of rotational symmetric is along the -axis, giving it the parametric representation(1)(2)(3)with , and .The Cartesian equation of the spheroid is(4)If , the spheroid is called oblate (left figure). If , the spheroid is prolate (right figure). If , the spheroid degenerates to a sphere.In the above parametrization, the coefficients of the firstfundamental form are(5)(6)(7)and of the second fundamental form are(8)(9)(10)The Gaussian curvature is given by(11)the implicit Gaussian curvature by(12)and the mean curvature by(13)The surface area of a spheroid can be variously writtenas(14)(15)(16)(17)where(18)(19)and is a hypergeometric function.The volume of a spheroid can be computed from the formula for a general ellipsoid with ,(20)(21)(Beyer..

A prolate spheroid is a spheroid that is "pointy" instead of "squashed," i.e., one for which the polar radius is greater than the equatorial radius , so (called "spindle-shaped ellipsoid" by Tietze 1965, p. 27). A symmetrical egg (i.e., with the same shape at both ends) would approximate a prolate spheroid. A prolate spheroid is a surface of revolution obtained by rotating an ellipse about its major axis (Hilbert and Cohn-Vossen 1999, p. 10), and has Cartesian equations(1)The ellipticity of the prolate spheroid is definedby(2)The surface area of a prolate spheroid can be computedas a surface of revolution about the z-axis,(3)with radius as a function of given by(4)The integrand is then(5)and the integral is given by(6)(7)Using the identity(8)gives(9)(Beyer 1987, p. 131). Note that this is the conventional form in which the surface area of a prolate spheroid is written, although it..

A "squashed" spheroid for which the equatorial radius is greater than the polar radius , so (called an oblate ellipsoid by Tietze 1965, p. 27). An oblate spheroid is a surface of revolution obtained by rotating an ellipse about its minor axis (Hilbert and Cohn-Vossen 1999, p. 10). To first approximation, the shape assumed by a rotating fluid (including the Earth, which is "fluid" over astronomical time scales) is an oblate spheroid.For a spheroid with z-axisas the symmetry axis, the Cartesian equation is(1)The ellipticity of an oblate spheroid is definedby(2)The surface area of an oblate spheroid can be computedas a surface of revolution about the z-axis,(3)with radius as a function of given by(4)Therefore(5)(6)(7)(8)where the last step makes use of the logarithm identity(9)valid for . Re-expressing in terms of the ellipticity then gives(10)yielding the particular simple form(11)(Beyer 1987, p. 131)...

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).

An (ordinary) torus is a surface having genus one, and therefore possessing a single "hole" (left figure). The single-holed "ring" torus is known in older literature as an "anchor ring." It can be constructed from a rectangle by gluing both pairs of opposite edges together with no twists (right figure; Gardner 1971, pp. 15-17; Gray 1997, pp. 323-324). The usual torus embedded in three-dimensional space is shaped like a donut, but the concept of the torus is extremely useful in higher dimensional space as well.In general, tori can also have multiple holes, with the term -torus used for a torus with holes. The special case of a 2-torus is sometimes called the double torus, the 3-torus is called the triple torus, and the usual single-holed torus is then simple called "the" or "a" torus.A second definition for -tori relates to dimensionality. In one dimension, a line bends into..

A surface of revolution which is generalization of the ring torus. It is produced by rotating an ellipse having horizontal semi-axis , vertical semi-axis , embedded in the -plane, and located a distance away from the -axis about the -axis. It is given by the parametric equations(1)(2)(3)for .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)By Pappus's centroid theorems, the surface area and volume are(12)(13)(14)(15)where is a complete elliptic integral of the first kind and(16)is the eccentricity of the ellipse cross section.

A surface of revolution obtained by rotating a closed plane curve about an axis parallel to the plane which does not intersect the curve. The simplest toroid is the torus. The word is also used to refer to a toroidal polyhedron (Gardner 1975).

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.

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 ,..

A sphere is defined as the set of all points in three-dimensional Euclidean space that are located at a distance (the "radius") from a given point (the "center"). Twice the radius is called the diameter, and pairs of points on the sphere on opposite sides of a diameter are called antipodes.Unfortunately, geometers and topologists adopt incompatible conventions for the meaning of "-sphere," with geometers referring to the number of coordinates in the underlying space ("thus a two-dimensional sphere is a circle," Coxeter 1973, p. 125) and topologists referring to the dimension of the surface itself ("the -dimensional sphere is defined to be the set of all points in satisfying ," Hocking and Young 1988, p. 17; "the -sphere is ," Maunder 1997, p. 21). As a result, geometers call the surface of the usual sphere the 3-sphere, while topologists refer to it as the..

A (finite, circular) conical surface is a ruled surface created by fixing one end of a line segment at a point (known as the vertex or apex of the cone) and sweeping the other around the circumference of a fixed circle (known as the base). When the vertex lies above the center of the base (i.e., the angle formed by the vertex, base center, and any base radius is a right angle), the cone is known as a right cone; otherwise, the cone is termed "oblique." When the base is taken as an ellipse instead of a circle, the cone is called an elliptic cone.In discussions of conic sections, the word "cone" is commonly taken to mean "double cone," i.e., two (possibly infinitely extending) cones placed apex to apex. The infinite double cone is a quadratic surface, and each single cone is called a "nappe." The hyperbola can then be defined as the intersection of a plane with both nappes of the double cone.As can be seen from the above,..

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..

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).

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.

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..

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 .

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 .

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 ...

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