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2013년 11월 14일 목요일

Sphere구

sphere (from Greek σφαῖρα — sphaira, "globe, ball"^[1]) is a perfectly round geometrical and circular object in three-dimensional space that resembles the shape of a completely round ball. Like a circle, which, in geometrical contexts, is in two dimensions, a sphere is the set of points that are all the same distance r from a given point in space. This distance r is the radius of the sphere, and the given point is the center of the sphere. The maximum straight distance through the sphere passes through the center and is thus twice the radius; it is the diameter.
In mathematics, a distinction is made between the sphere (a two-dimensional closed surface embedded in three-dimensional Euclidean space) and the ball (a three-dimensional shape that includes the interior of a sphere).

A two-dimensional perspective projection of a sphere

Area[edit]

The surface area of a sphere is:

$A = 4\pi r^2.$

Archimedes first derived this formula^{[citation
needed]} from the fact that the projection to the lateral surface of a circumscribed cylinder (i.e. the Lambert cylindrical equal-area projection) is area-preserving; it equals the derivative of the formula for the volume with respect to r because the total volume inside a sphere of radius r can be thought of as the summation of the surface area of an infinite number of spherical shells of infinitesimal thickness concentrically stacked inside one another from radius 0 to radius r. At infinitesimal thickness the discrepancy between the inner and outer surface area of any given shell is infinitesimal, and the elemental volume at radius r is simply the product of the surface area at radius r and the infinitesimal thickness.

At any given radius r, the incremental volume (δV) equals the product of the surface area at radius r (A(r)) and the thickness of a shell (δr):

$\delta V \approx A(r) \cdot \delta r. \,$

The total volume is the summation of all shell volumes:

$V \approx \sum A(r) \cdot \delta r.$

In the limit as δr approaches zero^[2] this equation becomes:

$V = \int_0^r A(r) \, dr.$

Substitute V:

$\frac{4}{3}\pi r^3 = \int_0^r A(r) \, dr.$

Differentiating both sides of this equation with respect to r yields A as a function of r:

$\!4\pi r^2 = A(r).$

Which is generally abbreviated as:

$\!A = 4\pi r^2$

Alternatively, the area element on the sphere is given in spherical coordinates by $dA = r^2 \sin\theta\, d\theta\, d\phi.$ . With Cartesian coordinates, the area element $dS=\frac{r}{\sqrt{r^{2}-\sum_{i\ne k}x_{i}^{2}}}\Pi_{i\ne k}dx_{i},\;\forall k$ . More generally, see area element.

The total area can thus be obtained by integration:

$A = \int_0^{2\pi} \int_0^\pi r^2 \sin\theta \, d\theta \, d\phi = 4\pi r^2.$

Enclosed volume[edit]

Circumscribed cylinder to a sphere

In 3 dimensions, the volume inside a sphere (that is, the volume of a ball) is derived to be

$\!V = \frac{4}{3}\pi r^3$

where r is the radius of the sphere and π is the constant pi. Archimedes first derived this formula, which shows that the volume inside a sphere is 2/3 that of a circumscribed cylinder. (This assertion follows from Cavalieri's principle.) In modern mathematics, this formula can be derived using integral calculus, i.e. disk integration to sum the volumes of an infinite number of circular disks of infinitesimally small thickness stacked centered side by side along the x axis from x = 0 where the disk has radius r (i.e. y = r) to x = r where the disk has radius 0 (i.e. y = 0).

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):

$\!\delta V \approx \pi y^2 \cdot \delta x.$

The total volume is the summation of all incremental volumes:

$\!V \approx \sum \pi y^2 \cdot \delta x.$

In the limit as δx approaches zero^[2] this equation becomes:

$\!V = \int_{-r}^{r} \pi y^2 dx.$

At any given x, a right-angled triangle connects x, y and r to the origin; hence, applying the Pythagorean theorem yields:

$\!y^2 = r^2 - x^2.$

Thus, substituting y with a function of x gives:

$\!V = \int_{-r}^{r} \pi (r^2 - x^2)dx.$

Which can now be evaluated as follows:

$\!V = \pi \left[r^2x - \frac{x^3}{3} \right]_{-r}^{r} = \pi \left(r^3 - \frac{r^3}{3} \right) - \pi \left(-r^3 + \frac{r^3}{3} \right) = \frac{4}{3}\pi r^3.$

Therefore the volume of a sphere is:

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

Alternatively this formula is found using spherical coordinates, with volume element

$\mathrm{d}V=r^2\sin\theta\,\mathrm{d}r\,\mathrm{d}\theta\,\mathrm{d}\varphi$

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 $\pi/6 \approx 0.5236$ . For example, a sphere with diameter 1m has 52.4% the volume of a cube with edge length 1m, or about 0.524m³.

In higher dimensions, the sphere (or hypersphere) is usually called an n-ball. General recursive formulas exist for the volume of an n-ball.

Equations in $\mathbb{R}^3$ [edit]

In analytic geometry, a sphere with centre (x₀, y₀, z₀) 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 parameterized via

$\, x = x_0 + r \cos \theta \; \sin \varphi$

$\, y = y_0 + r \sin \theta \; \sin \varphi \qquad (0 \leq \theta \leq 2\pi \mbox{ and } 0 \leq \varphi \leq \pi ) \,$

$\, z = z_0 + r \cos \varphi \,$

(see also trigonometric functions and spherical coordinates).

A sphere of any radius centred at zero is an integral surface of the following differential form:

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

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

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 nanometers) of thickness. It was announced on 1 July 2008 that Australian scientists had created even more perfect spheres, accurate to 0.3 nanometers, as part of an international hunt to find a new global standard kilogram.^[3]

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. 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 sphere is called the specific surface area and can be expressed from the above stated equations as

$SSA = \frac{A}{V\rho} = \frac{3}{r\rho},$

where $\rho$ is the ratio of mass to volume.

A sphere can also be defined as the surface formed by rotating a circle about any diameter. Replacing the circle with an ellipse rotated about its major axis, the shape becomes a prolate spheroid; rotated about the minor axis, an oblate spheroid.

Terminology[edit]

Pairs of points on a sphere that lie on a straight line through the sphere's center are called antipodal points. A great circle is a circle on the sphere that has the same center and radius as the sphere and consequently divides it into two equal parts. The shortest distance along the surface between two distinct non-antipodal points on the surface is on the unique great circle that includes the two points. Equipped with the great-circle distance, a great circle becomes the Riemannian circle.

If a particular point on a sphere is (arbitrarily) 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 such approximately spheroidal astronomical bodies as the planet Earth (see geoid).

Hemisphere[edit]

Any plane that includes the center of a sphere divides it into two equal "hemispheres". Any two intersecting planes that include the center of a sphere 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.

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.

The round hemisphere is conjectured to be the optimal (least area) filling of the Riemannian circle.

The circles of intersection of any plane not intersecting the sphere's center and the sphere's surface are called spheric sections.^[4]

Generalization to other dimensions[edit]

Main article: n-sphere

Spheres can be generalized to spaces of any dimension. For any natural number n, an "n-sphere," often written as Sⁿ, is the set of points in (n + 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:

a 0-sphere is a pair of endpoints of an interval (−r, r) of the real line
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 centered at the origin is denoted Sⁿ 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 (which is embedded in 3-dimensional 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 expression for the surface area is

$\begin{cases} \displaystyle \frac{(2\pi)^{n/2}\,r^{n-1}}{2 \cdot 4 \cdots (n-2)}, & \text{if } n \text{ is even}; \\ \\ \displaystyle \frac{2(2\pi)^{(n-1)/2}\,r^{n-1}}{1 \cdot 3 \cdots (n-2)}, & \text{if } n \text{ is odd}. \end{cases}$

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

$\begin{cases} \displaystyle \frac{(2\pi)^{n/2}\,r^n}{2 \cdot 4 \cdots n}, & \text{if } n \text{ is even}; \\ \\ \displaystyle \frac{2(2\pi)^{(n-1)/2}\,r^n}{1 \cdot 3 \cdots n}, & \text{if } n \text{ is odd}. \end{cases}$

Generalization to metric spaces[edit]

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 ball, even a large sphere may be an empty set. For example, in Zⁿ with Euclidean metric, a sphere of radius r is nonempty only if r² can be written as sum of n squares of integers.

Topology[edit]

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 Sⁿ. 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 implies 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 closed. Sⁿ is also bounded; therefore it is compact.

Smale's paradox shows that it is possible to turn an ordinary sphere inside out in a three-dimensional space with possible self-intersections but without creating any crease, a process more commonly and historically called sphere eversion.

Spherical geometry[edit]

Great circle on a sphere

Main article: Spherical geometry

The basic elements of Euclidean 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. Measuring by arc length yields that the shortest path between two points that entirely lie in the sphere is a segment of the great circle the includes the points; see geodesic. Many, but not all (see parallel postulate) theorems from classical geometry hold true for this spherical geometry as well. In spherical trigonometry, angles are defined between great circles. Thus spherical trigonometry differs 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.

Wikipedia

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