Sphere
Sphere | |
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Smooth surface Algebraic surface | |
Euler char. | 2 |
Symmetry group | O(3) |
Surface area | 4πr2 |
Volume | 4/3πr3 |
A sphere (from
The sphere is a fundamental object in many fields of
Basic terminology
As mentioned earlier r is the sphere's radius; any line from the center to a point on the sphere is also called a radius.[3]
If a radius is extended through the center to the opposite side of the sphere, it creates a diameter. Like the radius, the length of a diameter is also called the diameter, and denoted d. Diameters are the longest line segments that can be drawn between two points on the sphere: their length is twice the radius, d = 2r. Two points on the sphere connected by a diameter are antipodal points of each other.[3]
A unit sphere is a sphere with unit radius (r = 1). For convenience, spheres are often taken to have their center at the origin of the coordinate system, and spheres in this article have their center at the origin unless a center is mentioned.
A great circle on the sphere has the same center and radius as the sphere, and divides it into two equal hemispheres.
Although the
Mathematicians consider a sphere to be a two-dimensional
Small spheres or balls are sometimes called spherules, e.g. in Martian spherules.
Equations
In analytic geometry, a sphere with center (x0, y0, z0) and radius r is the locus of all points (x, y, z) such that
Since it can be expressed as a quadratic polynomial, a sphere is a
Let a, b, c, d, e be real numbers with a ≠ 0 and put
Then the equation
has no real points as solutions if and is called the equation of an imaginary sphere. If , the only solution of is the point and the equation is said to be the equation of a point sphere. Finally, in the case , is an equation of a sphere whose center is and whose radius is .[2]
If a in the above equation is zero then f(x, y, z) = 0 is the equation of a plane. Thus, a plane may be thought of as a sphere of infinite radius whose center is a point at infinity.[4]
Parametric
A parametric equation for the sphere with radius and center can be parameterized using
The symbols used here are the same as those used in
Properties
Enclosed volume
In three dimensions, the volume inside a sphere (that is, the volume of a ball, but classically referred to as the volume of a sphere) is
where r is the radius and d is the diameter of the sphere.
Proof of sphere volume, using calculus
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At any given x, the incremental volume (δV) equals the product of the cross-sectional area of the disk at x and its thickness (δx):
The total volume is the summation of all incremental volumes: In the limit as δx approaches zero,[8] this equation becomes: At any given x, a right-angled triangle connects x, y and r to the origin; hence, applying the Pythagorean theorem yields: Using this substitution gives which can be evaluated to give the result An alternative formula is found using spherical coordinates, with volume element
so |
For most practical purposes, the volume inside a sphere inscribed in a cube can be approximated as 52.4% of the volume of the cube, since V = π/6 d3, where d is the diameter of the sphere and also the length of a side of the cube and π/6 ≈ 0.5236. For example, a sphere with diameter 1 m has 52.4% the volume of a cube with edge length 1 m, or about 0.524 m3.
Surface area
The surface area of a sphere of radius r is:
Proof of surface area, using calculus
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At any given radius r,[note 1] the incremental volume (δV) equals the product of the surface area at radius r (A(r)) and the thickness of a shell (δr): The total volume is the summation of all shell volumes: In the limit as δr approaches zero[8] this equation becomes: Substitute V: Differentiating both sides of this equation with respect to r yields A as a function of r: This is generally abbreviated as: where r is now considered to be the fixed radius of the sphere. Alternatively, the spherical coordinates by dA = r2 sin θ dθ dφ. The total area can thus be obtained by integration :
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The sphere has the smallest surface area of all surfaces that enclose a given volume, and it encloses the largest volume among all closed surfaces with a given surface area.[11] The sphere therefore appears in nature: for example, bubbles and small water drops are roughly spherical because the surface tension locally minimizes surface area.
The surface area relative to the mass of a ball is called the specific surface area and can be expressed from the above stated equations as
where ρ is the density (the ratio of mass to volume).
Other geometric properties
A sphere can be constructed as the surface formed by rotating a
A sphere is uniquely determined by four points that are not
Consequently, a sphere is uniquely determined by (that is, passes through) a circle and a point not in the plane of that circle.
By examining the
The angle between two spheres at a real point of intersection is the dihedral angle determined by the tangent planes to the spheres at that point. Two spheres intersect at the same angle at all points of their circle of intersection.[16] They intersect at right angles (are orthogonal) if and only if the square of the distance between their centers is equal to the sum of the squares of their radii.[4]
Pencil of spheres
If f(x, y, z) = 0 and g(x, y, z) = 0 are the equations of two distinct spheres then
is also the equation of a sphere for arbitrary values of the parameters s and t. The set of all spheres satisfying this equation is called a pencil of spheres determined by the original two spheres. In this definition a sphere is allowed to be a plane (infinite radius, center at infinity) and if both the original spheres are planes then all the spheres of the pencil are planes, otherwise there is only one plane (the radical plane) in the pencil.[4]
Properties of the sphere
In their book Geometry and the Imagination,
- The points on the sphere are all the same distance from a fixed point. Also, the ratio of the distance of its points from two fixed points is constant.
- The first part is the usual definition of the sphere and determines it uniquely. The second part can be easily deduced and follows a similar result of Apollonius of Perga for the circle. This second part also holds for the plane.
- The contours and plane sections of the sphere are circles.
- This property defines the sphere uniquely.
- The sphere has constant width and constant girth.
- The width of a surface is the distance between pairs of parallel tangent planes. Numerous other closed convex surfaces have constant width, for example the Meissner body. The girth of a surface is the circumferenceof the boundary of its orthogonal projection on to a plane. Each of these properties implies the other.
- The width of a surface is the distance between pairs of parallel tangent planes. Numerous other closed convex surfaces have constant width, for example the
- All points of a sphere are umbilics.
- At any point on a surface a normal direction is at right angles to the surface because on the sphere these are the lines radiating out from the center of the sphere. The intersection of a plane that contains the normal with the surface will form a curve that is called a normal section, and the curvature of this curve is the normal curvature. For most points on most surfaces, different sections will have different curvatures; the maximum and minimum values of these are called the principal curvatures. Any closed surface will have at least four points called umbilical points. At an umbilic all the sectional curvatures are equal; in particular the principal curvatures are equal. Umbilical points can be thought of as the points where the surface is closely approximated by a sphere.
- For the sphere the curvatures of all normal sections are equal, so every point is an umbilic. The sphere and plane are the only surfaces with this property.
- The sphere does not have a surface of centers.
- For a given normal section exists a circle of curvature that equals the sectional curvature, is tangent to the surface, and the center lines of which lie along on the normal line. For example, the two centers corresponding to the maximum and minimum sectional curvatures are called the focal points, and the set of all such centers forms the focal surface.
- For most surfaces the focal surface forms two sheets that are each a surface and meet at umbilical points. Several cases are special:
- * For channel surfaces one sheet forms a curve and the other sheet is a surface
- * For both sheets form curves.
- * For the sphere the center of every osculating circle is at the center of the sphere and the focal surface forms a single point. This property is unique to the sphere.
- All geodesics of the sphere are closed curves.
- Geodesicsare curves on a surface that give the shortest distance between two points. They are a generalization of the concept of a straight line in the plane. For the sphere the geodesics are great circles. Many other surfaces share this property.
- Of all the solids having a given volume, the sphere is the one with the smallest surface area; of all solids having a given surface area, the sphere is the one having the greatest volume.
- It follows from isoperimetric inequality. These properties define the sphere uniquely and can be seen in soap bubbles: a soap bubble will enclose a fixed volume, and surface tension minimizes its surface area for that volume. A freely floating soap bubble therefore approximates a sphere (though such external forces as gravity will slightly distort the bubble's shape). It can also be seen in planets and stars where gravity minimizes surface area for large celestial bodies.
- The sphere has the smallest total mean curvature among all convex solids with a given surface area.
- The mean curvature is the average of the two principal curvatures, which is constant because the two principal curvatures are constant at all points of the sphere.
- The sphere has constant mean curvature.
- The sphere is the only imbedded surface that lacks boundary or singularities with constant positive mean curvature. Other such immersed surfaces as minimal surfaces have constant mean curvature.
- The sphere has constant positive Gaussian curvature.
- Gaussian curvature is the product of the two principal curvatures. It is an intrinsic property that can be determined by measuring length and angles and is independent of how the surface is embedded in space. Hence, bending a surface will not alter the Gaussian curvature, and other surfaces with constant positive Gaussian curvature can be obtained by cutting a small slit in the sphere and bending it. All these other surfaces would have boundaries, and the sphere is the only surface that lacks a boundary with constant, positive Gaussian curvature. The pseudosphere is an example of a surface with constant negative Gaussian curvature.
- The sphere is transformed into itself by a three-parameter family of rigid motions.
- Rotating around any axis a unit sphere at the origin will map the sphere onto itself. Any rotation about a line through the origin can be expressed as a combination of rotations around the three-coordinate axis (see rotation group SO(3). The plane is the only other surface with a three-parameter family of transformations (translations along the x- and y-axes and rotations around the origin). Circular cylinders are the only surfaces with two-parameter families of rigid motions and the surfaces of revolution and helicoidsare the only surfaces with a one-parameter family.
- Rotating around any axis a unit sphere at the origin will map the sphere onto itself. Any rotation about a line through the origin can be expressed as a combination of rotations around the three-coordinate axis (see
Treatment by area of mathematics
Spherical geometry
The basic elements of
Many theorems from
Any pair of points on a sphere that lie on a straight line through the sphere's center (i.e. the diameter) are called antipodal points—on the sphere, the distance between them is exactly half the length of the circumference.[note 2] Any other (i.e. not antipodal) pair of distinct points on a sphere
- lie on a unique great circle,
- segment it into one minor (i.e. shorter) and one major (i.e. longer) arc, and
- have the minor arc's length be the shortest distance between them on the sphere.[note 3]
Spherical geometry is a form of elliptic geometry, which together with hyperbolic geometry makes up non-Euclidean geometry.
Differential geometry
The sphere is a
A sphere of radius r has
A sphere of any radius centered at zero is an
This equation reflects that the position vector and
In Riemannian geometry, the filling area conjecture states that the hemisphere is the optimal (least area) isometric filling of the Riemannian circle.
Topology
Remarkably, it is possible to turn an ordinary sphere inside out in a three-dimensional space with possible self-intersections but without creating any creases, in a process called sphere eversion.
The antipodal quotient of the sphere is the surface called the real projective plane, which can also be thought of as the Northern Hemisphere with antipodal points of the equator identified.
Curves on a sphere
Circles
Circles on the sphere are, like circles in the plane, made up of all points a certain distance from a fixed point on the sphere. The intersection of a sphere and a plane is a circle, a point, or empty.[18] Great circles are the intersection of the sphere with a plane passing through the center of a sphere: others are called small circles.
More complicated surfaces may intersect a sphere in circles, too: the intersection of a sphere with a surface of revolution whose axis contains the center of the sphere (are coaxial) consists of circles and/or points if not empty. For example, the diagram to the right shows the intersection of a sphere and a cylinder, which consists of two circles. If the cylinder radius were that of the sphere, the intersection would be a single circle. If the cylinder radius were larger than that of the sphere, the intersection would be empty.
Loxodrome
In
Clelia curves
A Clelia curve is a curve on a sphere for which the longitude and the colatitude satisfy the equation
- .
Special cases are: Viviani's curve () and
Spherical conics
The analog of a conic section on the sphere is a spherical conic, a quartic curve which can be defined in several equivalent ways, including:
- as the intersection of a sphere with a quadratic cone whose vertex is the sphere center;
- as the intersection of a sphere with an elliptic or hyperbolic cylinder whose axis passes through the sphere center;
- as the locus of points whose sum or difference of great-circle distances from a pair of foci is a constant.
Many theorems relating to planar conic sections also extend to spherical conics.
Intersection of a sphere with a more general surface
If a sphere is intersected by another surface, there may be more complicated spherical curves.
- Example
- sphere – cylinder
The intersection of the sphere with equation and the cylinder with equation is not just one or two circles. It is the solution of the non-linear system of equations
(see implicit curve and the diagram)
Generalizations
Ellipsoids
An ellipsoid is a sphere that has been stretched or compressed in one or more directions. More exactly, it is the image of a sphere under an affine transformation. An ellipsoid bears the same relationship to the sphere that an ellipse does to a circle.
Dimensionality
Spheres can be generalized to spaces of any number of dimensions. For any natural number n, an n-sphere, often denoted S, is the set of points in ( nn + 1)-dimensional Euclidean space that are at a fixed distance r from a central point of that space, where r is, as before, a positive real number. In particular:
- S: a 0-sphere consists of two discrete points, 0−r and r
- S: a 1-sphere is a 1circle of radius r
- S: a 2-sphere is an ordinary sphere 2
- S: a 33-sphere is a sphere in 4-dimensional Euclidean space.
Spheres for n > 2 are sometimes called
The n-sphere of unit radius centered at the origin is denoted S and is often referred to as "the" nn-sphere. The ordinary sphere is a 2-sphere, because it is a 2-dimensional surface which is embedded in 3-dimensional space.
In
The sphere is the inverse image of a one-point set under the continuous function ‖x‖, so it is closed; Sn is also bounded, so it is compact by the Heine–Borel theorem.
Metric spaces
More generally, in a metric space (E,d), the sphere of center x and radius r > 0 is the set of points y such that d(x,y) = r.
If the center is a distinguished point that is considered to be the origin of E, as in a normed space, it is not mentioned in the definition and notation. The same applies for the radius if it is taken to equal one, as in the case of a unit sphere.
Unlike a
An octahedron is a sphere in taxicab geometry, and a cube is a sphere in geometry using the Chebyshev distance.
History
The geometry of the sphere was studied by the Greeks. Euclid's Elements defines the sphere in book XI, discusses various properties of the sphere in book XII, and shows how to inscribe the five regular polyhedra within a sphere in book XIII. Euclid does not include the area and volume of a sphere, only a theorem that the volume of a sphere varies as the third power of its diameter, probably due to Eudoxus of Cnidus. The volume and area formulas were first determined in Archimedes's On the Sphere and Cylinder by the method of exhaustion. Zenodorus was the first to state that, for a given surface area, the sphere is the solid of maximum volume.[3]
Archimedes wrote about the problem of dividing a sphere into segments whose volumes are in a given ratio, but did not solve it. A solution by means of the parabola and hyperbola was given by
Gallery
-
An image of one of the most accurate human-made spheres, as it refracts the image of Einstein in the background. This sphere was a fused quartz gyroscope for the Gravity Probe B experiment, and differs in shape from a perfect sphere by no more than 40 atoms (less than 10 nm) of thickness. It was announced on 1 July 2008 that Australian scientists had created even more nearly perfect spheres, accurate to 0.3 nm, as part of an international hunt to find a new global standard kilogram.[20]
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Deck of playing cards illustrating engineering instruments, England, 1702.King of spades: Spheres
Regions
- Hemisphere
- Spherical cap
- Spherical lune
- Spherical polygon
- Spherical sector
- Spherical segment
- Spherical wedge
- Spherical zone
See also
- 3-sphere
- Affine sphere
- Alexander horned sphere
- Celestial spheres
- Curvature
- Directional statistics
- Dyson sphere
- Gauss map
- M.C. Escherself-portrait drawing illustrating reflection and the optical properties of a mirror sphere
- Hoberman sphere
- Homology sphere
- Homotopy groups of spheres
- Homotopy sphere
- Lenart Sphere
- Napkin ring problem
- Orb (optics)
- Pseudosphere
- Riemann sphere
- Solid angle
- Sphere packing
- Spherical coordinates
- Spherical cow
- Spherical helix, tangent indicatrix of a curve of constant precession
- Spherical polyhedron
- Sphericity
- Tennis ball theorem
- Zoll sphere
Notes and references
Notes
References
- ^ σφαῖρα, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus.
- ^ a b Albert 2016, p. 54.
- ^ a b c d e f Chisholm, Hugh, ed. (1911). . Encyclopædia Britannica. Vol. 25 (11th ed.). Cambridge University Press. pp. 647–648.
- ^ a b c Woods 1961, p. 266.
- ^ Kreyszig (1972, p. 342).
- ^ Steinhaus 1969, p. 223.
- ^ "The volume of a sphere – Math Central". mathcentral.uregina.ca. Retrieved 10 June 2019.
- ^ ISBN 978-0-00-434347-1.
- ^ a b c Weisstein, Eric W. "Sphere". MathWorld.
- ^ Steinhaus 1969, p. 221.
- . Retrieved 14 December 2019.
- ^ Albert 2016, p. 60.
- ^ Albert 2016, p. 55.
- ^ Albert 2016, p. 57.
- ^ Woods 1961, p. 267.
- ^ Albert 2016, p. 58.
- ISBN 978-0-8284-1087-8.
- ^ Weisstein, Eric W. "Spheric section". MathWorld.
- ISBN 978-0-19-938113-5. Retrieved 4 November 2022.
More significantly, Vitruvius (On Architecture, Vitr. 9.8) associated conical sundials with Dionysodorus (early 2nd century bce), and Dionysodorus, according to Eutocius of Ascalon (c. 480–540 ce), used conic sections to complete a solution for Archimedes' problem of cutting a sphere by a plane so that the ratio of the resulting volumes would be the same as a given ratio.
- ^ New Scientist | Technology | Roundest objects in the world created.
Further reading
- Albert, Abraham Adrian (2016) [1949], Solid Analytic Geometry, Dover, ISBN 978-0-486-81026-3.
- Dunham, William (1997). The Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Problems and Personalities. New York: Wiley. pp. 28, 226. ISBN 978-0-471-17661-9.
- Kreyszig, Erwin (1972), Advanced Engineering Mathematics (3rd ed.), New York: ISBN 978-0-471-50728-4.
- Steinhaus, H. (1969), Mathematical Snapshots (Third American ed.), Oxford University Press.
- Woods, Frederick S. (1961) [1922], Higher Geometry / An Introduction to Advanced Methods in Analytic Geometry, Dover.
- John C. Polking (15 April 1999). "The Geometry of the Sphere". www.math.csi.cuny.edu. Retrieved 21 January 2022.