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Sphere

2007 Schools Wikipedia Selection. Related subjects: Mathematics

   A sphere is a perfectly symmetrical geometrical object. In
   non-mathematical usage, the term is used to refer either to a round
   ball or to its two-dimensional surface. In mathematics, a sphere is the
   set of all points in three-dimensional space (R^3) which are at
   distance r from a fixed point of that space, where r is a positive real
   number called the radius of the sphere. The fixed point is called the
   centre or centre, and is not part of the sphere itself. The special
   case of r = 1 is called a unit sphere.

   This article deals with the mathematical concept of a sphere. In
   physics, a sphere is an object (usually idealized for the sake of
   simplicity) capable of colliding or stacking with other objects which
   occupy space.

Equations

   In analytic geometry, a sphere with centre (x[0], y[0], z[0]) and
   radius r is the locus of all points (x, y, z) such that

          (x - x_0 )^2 + (y - y_0 )^2 + ( z - z_0 )^2 = r^2. \,

   The points on the sphere with radius r can be parametrized via

          x = x_0 + r \sin \theta \; \sin \phi
          y = y_0 + r \cos \theta \; \sin \phi \qquad (0 \leq \theta \leq
          2\pi \mbox{ and } -\tfrac{\pi}{2} < \phi \leq \tfrac{\pi}{2}) \,
          z = z_0 + r \cos \phi \,

   (see also trigonometric functions and spherical coordinates).

   A sphere of any radius centered at the origin is described by the
   following differential equation:

          x \, dx + y \, dy + z \, dz = 0.

   This equation reflects the fact that the position and velocity vectors
   of a point travelling on the sphere are always orthogonal to each
   other.

   The surface area of a sphere of radius r is

          A = 4 \pi r^2 \,

   and its enclosed volume is

          V = \frac{4}{3}\pi r^3.

   The sphere has the smallest surface area among all surfaces enclosing a
   given volume and it encloses the largest volume among all closed
   surfaces with a given surface area. For this reason, the sphere appears
   in nature: for instance bubbles and small water drops are roughly
   spherical, because the surface tension locally minimizes surface area.
   An image of one of the most accurate spheres ever created by humans, as
   it refracts the image of Einstein in the background. This sphere was a
   fused quartz gyroscope for the Gravity Probe B experiment which differs
   in shape from a perfect sphere by no more than 40 atoms of thickness.
   It is thought that only neutron stars are smoother.
   An image of one of the most accurate spheres ever created by humans, as
   it refracts the image of Einstein in the background. This sphere was a
   fused quartz gyroscope for the Gravity Probe B experiment which differs
   in shape from a perfect sphere by no more than 40 atoms of thickness.
   It is thought that only neutron stars are smoother.

   The circumscribed cylinder for a given sphere has a volume which is 3/2
   times the volume of the sphere, and also the curved portion has a
   surface area which is 3/2 times the surface area of the sphere. This
   fact, along with the volume and surface formulas given above, was
   already known to Archimedes.

   A sphere can also be defined as the surface formed by rotating a circle
   about any diameter. If the circle is replaced by an ellipse, and
   rotated about the major axis, the shape becomes a prolate spheroid,
   rotated about the minor axis, an oblate spheroid.

Terminology

   Pairs of points on a sphere that lie on a straight line through its
   centre are called antipodal points. A great circle is a circle on the
   sphere that has the same centre and radius as the sphere, and
   consequently divides it into two equal parts. The shortest distance
   between two distinct non-polar points on the surface and measured along
   the surface, is on the unique great circle passing through the two
   points.

   If a particular point on a sphere is designated as its north pole, then
   the corresponding antipodal point is called the south pole and the
   equator is the great circle that is equidistant to them. Great circles
   through the two poles are called lines (or meridians) of longitude, and
   the line connecting the two poles is called the axis of rotation.
   Circles on the sphere that are parallel to the equator are lines of
   latitude. This terminology is also used for astronomical bodies such as
   the planet Earth, even though it is neither spherical nor even
   spheroidal (see geoid).

   A sphere is divided into two equal hemispheres by any plane that passes
   through its center. If two intersecting planes pass through its centre,
   then they will subdivide the sphere into four lunes or biangles, the
   vertices of which all coincide with the antipodal points lying on the
   line of intersection of the planes.

Generalization to other dimensions

   Spheres can be generalized to other dimensions. For any natural number
   n, an n-sphere, often written as S^n, is the set of points in
   (n+1)-dimensional Euclidean space which are at distance r from a fixed
   point of that space, where R is, as before, a positive real number. For
   n> 0, the n-sphere is the simply connected n-dimensional manifold of
   constant, positive curvature, and can also be thought of embedded in an
   n+1-dimensional manifold, as the surface or boundary of a ball in the
   n+1-dimensional manifold.
     * a 0-sphere is a pair of points on the line at ( − r,r)
     * a 1-sphere is a circle of radius r
     * a 2-sphere is an ordinary sphere
     * a 3-sphere is a sphere in 4-dimensional Euclidean space.

   Spheres for n > 2 are sometimes called hyperspheres.

   The n-sphere of unit radius centred at the origin is denoted S^n and is
   often referred to as "the" n-sphere. Note that the ordinary sphere is a
   2-sphere, because it is a 2-dimensional surface, though it is also a
   3-dimensional object because it can be embedded in ordinary 3-space.

   The surface area of the (n − 1)-sphere of radius 1 is

          2 \frac{\pi^{n/2}}{\Gamma(n/2)}

   where Γ(z) is Euler's Gamma function.

   Another formula for surface area is

          \begin{cases} {(2\pi)^{n/2}r^{n-1} \over 2 \cdot 4 \cdots n-2} &
          \mbox{if } n \mbox{ is even}; \\ \\ {2(2\pi)^{(n-1)/2}r^{n-1}
          \over 1 \cdot 3 \cdots n-2} & \mbox{if } n \mbox{ is odd}.
          \end{cases}

   and the volume within is the surface area times {r \over n} or

          \begin{cases} {(2\pi)^{n/2}r^n \over 2 \cdot 4 \cdots n} &
          \mbox{if } n \mbox{ is even}; \\ \\ {2(2\pi)^{(n-1)/2}r^n \over
          1 \cdot 3 \cdots n} & \mbox{if } n \mbox{ is odd}. \end{cases}

Generalization to metric spaces

   More generally, in a metric space (E,d), the sphere of centre x and
   radius r > 0 is the set

          S(x;r) = { y ∈ E | d(x,y) = r }.

   If the centre is a distinguished point considered as origin of E, e.g.
   in a normed space, it is not mentioned in the definition and notation.
   The same applies for the radius if it is taken equal to one, i.e. in
   the case of a unit sphere. In contrast to a ball, a sphere may be
   empty, even for a large radius. For example, in Z^n with Euclidean
   metric, a sphere of radius r is nonempty only if r ^2 can be written as
   sum of n squares of integers.

Topology

   In topology, an n-sphere is defined as a space homeomorphic to the
   boundary of an (n+1)-ball; thus, it is homeomorphic to the Euclidean
   n-sphere, but perhaps lacking its metric.
     * a 0-sphere is a pair of points with the discrete topology
     * a 1-sphere is a circle ( up to homeomorphism); thus, for example,
       (the image of) any knot is a 1-sphere
     * a 2-sphere is an ordinary sphere ( up to homeomorphism); thus, for
       example, any spheroid is a 2-sphere

   The n-sphere is denoted S^n. It is an example of a compact topological
   manifold without boundary. A sphere need not be smooth; if it is
   smooth, it need not be diffeomorphic to the Euclidean sphere.

   The Heine-Borel theorem is used in a short proof that a Euclidean
   n-sphere is compact. The sphere is the inverse image of a one-point set
   under the continuous function ||x||. Therefore the sphere is a closed.
   S^n is also bounded. Therefore it is compact.

Spherical geometry

   Great circle on a sphere
   Great circle on a sphere

   The basic elements of plane geometry are points and lines. On the
   sphere, points are defined in the usual sense, but the analogue of
   "line" may not be immediately apparent. If one measures by arc length
   one finds that the shortest path connecting two points lying entirely
   in the sphere is a segment of the great circle containing the points;
   see geodesic. Many theorems from classical geometry hold true for this
   spherical geometry as well, but many do not (see parallel postulate).
   In spherical trigonometry, angles are defined between great circles.
   Thus spherical trigonometry is different from ordinary trigonometry in
   many respects. For example, the sum of the interior angles of a
   spherical triangle exceeds 180 degrees. Also, any two similar spherical
   triangles are congruent.

Eleven properties of the sphere

   In their book Geometry and the imagination David Hilbert and Stephan
   Cohn-Vossen describe eleven properties of the sphere and discuss
   whether these properties are uniquely determine the sphere. Several
   properties hold for the plane which can be thought of as a sphere with
   infinite radius. These properties are:
    1. 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 be
                deduced and follows a similar result of Apollonius of
                Perga for the circle. This second part also holds for the
                plane.

    2. The contours and plane sections of the sphere are circles.

                This property defines the sphere uniquely.

    3. The sphere has constant width and constant girth.

                The width of a surface is the distance between pairs of
                parallel tangent planes. There are numerous other closed
                convex surfaces which have constant width, for example
                Meissner's tetrahedron. The girth of a surface is the
                circumference of the boundary of its orthogonal projection
                on to a plane. It can be proved that each of these
                properties implies the other.

                A normal vector to a sphere, a normal plane and its normal
                section. The curvature of the curve of intersection is the
                sectional curvature. For the sphere each normal section
                through a given point will be a circle of the same radius,
                the radius of the sphere. This means that every point on
                the sphere will be an umbilical point.

                A normal vector to a sphere, a normal plane and its normal
                section. The curvature of the curve of intersection is the
                sectional curvature. For the sphere each normal section
                through a given point will be a circle of the same radius,
                the radius of the sphere. This means that every point on
                the sphere will be an umbilical point.

    4. All points of a sphere are umbilics.

                At any point on a surface we can find a normal direction
                which is at right angles to the surface, for the sphere
                these on the lines radiating out from the centre of the
                sphere. The intersection of a plane containing the normal
                with the surface will form a curve called a normal section
                and the curvature of this curve is the sectional
                curvature. For most points on a surfaces different
                sections will have different curvatures, the maximum and
                minimum values of these are called the principal
                curvatures. It can be proved that 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 curvature's 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.

    5. The sphere does not have a surface of centers

                For a given normal sections there is a circle whose
                curvature is the same as the sectional curvature, is
                tangent to the surface and whose center lines along on the
                normal line. Take the two centre corresponding to the
                maximum and minimum sectional curvatures these 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 each
                of which is a surface and which come together at umbilical
                points. There are a number of special cases. For canal
                surfaces one sheet forms a curve and the other sheet is a
                surface; For cones, cylinders, toruses and cyclides both
                sheets form curves. For the sphere the center of every
                osculating circle is at the centre of the sphere and the
                focal surface forms a single point. This is a unique
                property of the sphere.

    6. All geodesics of the sphere are closed curves

                Geodesics are curves on a surface which give the shortest
                distance between two points. They are generalisation of
                the concept of a straight line in the plane. For the
                sphere the geodesics are great circles. There are many
                other surfaces with this property.

    7. 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.

                These properties define the sphere uniquely. These
                properties can be seen by observing soap bubbles. A soap
                bubble will enclose a fixed volume and due to surface
                tension it will try to minimise its surface area.
                Therefore a free floating soap bubble will be
                approximately a sphere, factors like gravity will cause a
                slight distortion.

    8. 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 and as these are constant at all points of the
                sphere then so is the mean curvature.

    9. The sphere has constant positive mean curvature

                The sphere is the only surface without boundary or
                singularities with constant positive mean curvature. There
                are other surfaces with constant mean curvature, the
                minimal surfaces have zero mean curvature.

   10. The sphere has constant positive Gaussian curvature

                Gaussian curvature is the product of the two principle
                curvatures. It is an intrinsic property which can be
                determined by measuring length and angles and does not
                depend on the way 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 without
                boundary with constant positive Gaussian curvature. The
                pseudosphere is an example of a surface with constant
                negative Gaussian curvature.

   11. The sphere is transformed into itself by a three parameter family
       of rigid motions

                Consider a unit sphere place at the origin, a rotation
                around the x, y or z axis will map the sphere onto itself,
                indeed any rotation about a line through the origin can be
                expressed as a combination of rotations around the three
                coordinate axis, see Euler angles. Thus there is a three
                parameter family of rotations which transform the sphere
                onto itself, this is the rotation group, SO(3). The plane
                is the only other surface with a three parameter family of
                transformations (translations along the x and y axis and
                rotations around the origin). Circular cylinders are the
                only surfaces with two parameter families of rigid motions
                and the surfaces of revolution and Helicoids are the only
                surfaces with a one parameter family.

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