Sphere

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Projection in two dimensions of a sphere defined by parallels and meridians.

3D model of a sphere

In geometry, a spherical surface is a surface of revolution formed by the set of all points in space that are equidistant from a point called the center.

For points whose distance is less than the length of the radius, it is said that they form the interior of the spherical surface. The union of the interior and the spherical surface is called a closed ball in topology, or sphere, as in elementary geometry of space. The sphere is a geometric solid.

The sphere, as a solid of revolution, is generated by rotating a semicircular surface around its diameter (Euclid, L. XI, def. 14).

Sphere comes from the Greek term σφαῖρα, sphaîra, which means ball (to play). Colloquially speaking, the word ball is used to describe the body delimited by a sphere.

Spherical geometry

As Surface

The sphere (spherical surface) is the set of points in three-dimensional space that have the same distance from a fixed point called the center; both the segment that joins a point with the center, and the length of the segment, is called the radius. In this case it is generated by rotating a semicircle using its diameter as its axis of rotation. This concept is used when defining the sphere in analytic geometry of space.

As Solid

A sphere (spherical solid) is the set of points in three-dimensional space that are, with respect to the center, at a distance equal to or less than the length of its radius. This concept coincides with the definition of closed ball in the actual analysis of ℝ³. It is generated by rotating a semicircle, having its diameter as its axis of rotation.

In this situation, topologically, we can speak of frontier(Fr) the set of points on the sphere with a distance equal to the radius; interior (Int), the set of points with a distance less than the radius; outside (Ext), the set of points with a distance greater than the radius. Being fulfilled that these three sets form a partition of the space, so that they are disjoint two by two and the union of the three is the same space.

Properties

Any segment that contains the center of the sphere and its ends are on the spherical surface, is a diameter.
Any flat section of a sphere is a circle.
Any section that passes through the center of a sphere is a circle, and if the section doesn't pass through the center is a lower circle.
If a circle of a sphere is given, the ends of the diameter perpendicular to that are called poles from that circle.

Volume

Data to find the area and volume of the sphere regarding the circumscribed cylinder.

The volume, $V,$ of a sphere is expressed according to its radio $r,$ like:

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

The volume of a sphere can be considered as 2/3 of the volume of the cylinder circumscribed to the sphere. Its base is a circle with the same diameter as the sphere. Its height has the same measurement as said diameter:

V = frac{2}{3} (pi r^2 cdot 2r)

This relationship of volumes is attributed to Archimedes.

It is possible to calculate the volume of a sphere with a margin of error of approximately 0.03% without using the value of π:

V = frac{67}{16} r^3

Area

The area is 4 times $pi ,$ by his radius square.

A = 4pi r^2

Demonstration
Archimedes showed that the area of the sphere is two thirds compared to the cylinder, using this definition: $A = frac{2}{3} (2r cdot 2pi r + 2 cdot pi r^2)$ $A = frac{2}{3} (4pi r^2 + 2pi r^2)$ $A = frac{2}{3} (6pi r^2)$ $A = 4pi r^2$

Demonstration
The area of the sphere is also equal to the derivative of its volume with respect to $r,$ . $V = frac{4}{3}pi r^3 = int_{0}^{r}A(r) dr$ $frac{dV}{dr}= A(r) = 4pi r^2$

Equations of the sphere

Cartesian Equation

In a Cartesian coordinate system in a three-dimensional Euclidean space, the equation of the unit sphere (of radius 1), centered at the origin, is:

$x^2 + y^2 + z^2 = 1,$

This equation is obtained considering that at the point M (x, y, z) of the sphere, the normal vector OM is equal to 1.

Generalizing, the sphere of radius r, center Ω (a, b, c) has the equation:

$(x - a)^2 + (y - b)^2 + (z - c)^2 = r^2,$

The equation of the tangent plane at the point M (x', y', z') is obtained by doubling the variables: in the case of the unit sphere:

$x cdot x' + y cdot y' + z cdot z' = 0,$

and in the second example:

$(x - a) cdot x' + (y - b) cdot y' + (z - c) cdot z' = 0,$

In a three-dimensional Euclidean space, the points on the spherical surface can be parameterized as follows:

, x = x_0 + r cos theta ; sin varphi

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

, z = z_0 + r cos varphi ,

where r is the radius, (x0, y0, z0) are the coordinates of the center and (θ, φ) are the angular parameters of the equation.

Sections

A maximum circle divides the sphere into two equal hemispheres.

Section of one area by one level.

The intersection of a plane and a sphere is always a circle. The sphere is the only body that has this property. Logically, if the plane is tangent, the contact area is reduced to a point (it can be considered the limit case of the intersection).

If the plane passes through the center of the sphere, the radius of the circle is the same as the radius of the sphere, r. In this case, the circumference can be called the equator or great circle.

If the distance d, between the plane and the center, is less than the radius r of the sphere, applying the Pythagorean theorem, the radius of the section is:

r' = sqrt{r^2 - d^2}

Intersection of areas.

On the other hand, two spheres intersect if:

d le r + r'

and

r - r' le d

(these are triangular inequalities, and they are equivalent to the fact that no side is greater than the sum of the other two), that is, if there is a triangle with sides that measure r, r& #39; and d, where d is the distance between the centers of the spheres, r and r' their spokes.

In such a case, the intersection is also a circle. When one of the previous inequalities is an equality, the intersection will be a point, which is equivalent to a circle of zero radius.

In general, the radius is:

frac 2 d sqrt{m(m-r)(m-r')(m-d)} quad mbox{ con } quad m = frac {r+r'+d} 2

half perimeter.

Planes at a point on the spherical surface

Tangent plane

The plane that touches the area in one point is called so.. Each point in the sphere has associated a tangent plane. For the sphere the antipodale points have parallel tangent planes.

It is the plane whose distance to the center of the sphere is equal to the length of the radius. Or the limiting position of the dry planes of the sphere when their distance to the center tends to the length of the radius. Such a plane is unique and always exists, given a point $P=(x_0,y_0,z_0)$ radio R the tangent plane is given by:

${displaystyle x_{0}(x-x_{0})+y_{0}(y-y_{0})+z_{0}(z-z_{0})=0,}$

Normal shot

No information.

Binormal plane

It is the plane perpendicular to both the tangent plane and the normal plane.

Coordinates on the sphere

To locate a point on the spherical surface, Cartesian coordinates are not the most appropriate, for several reasons: firstly, because there are three Cartesian coordinates, while the spherical surface is a two-dimensional space. Secondly, in the case of a sphere, the angle is a more appropriate concept than the orthogonal coordinates.

The two orthogonal origins of the spherical coordinates

Choose an equator and a point on it as the origin of the horizontal angles; an orientation of the equator is chosen to define the sign of the angle φ; one of the two points on the sphere most distant from the equator –called poles– is chosen to define the sign of the angle θ

Determining points by angles

Every point on the sphere is uniquely located by the two angles θ and φ. With the value of an angle on the horizontal plane (plane of the equator) and another vertical (from a pole), any point on the sphere can be located.

In geometry, normally, these angles are expressed in radians (since it allows calculating the lengths of circumference arcs), while in geography the sexagesimal or centesimal degrees are used: in this case, θ is the latitude of the point and φ its longitude if one origin is taken at the point of the equator of the Greenwich meridian and the other origin at the north pole. Positive latitudes correspond to the Northern Hemisphere, and positive longitudes to the Eastern Hemisphere.

Introducing a third parameter r allows you to locate any point in space with spherical coordinates (r, φ, θ). If it is imposed to take φ in a semi-open interval of length 2π and θ in one of length π, then, any point in space has unique spherical coordinates, except those of the vertical axis, where any value of φ serves.

The Cartesian coordinates (x, y, z) in the spherical coordinate system (r, φ, θ) will be:

<math alttext="{displaystyle left{{begin{array}{lll}x&=&rsin theta ;cos varphi \y&=&rsin theta ;sin varphi \z&=&rcos theta end{array}}right.qquad {text{con}};-{cfrac {pi }{2}}<varphi leq {cfrac {pi }{2}};,;{text{y}}quad 0{x=rwithout θ θ # φ φ and=rwithout θ θ without φ φ z=r# θ θ with− − π π 2.φ φ ≤ ≤ π π 2,and0.θ θ ≤ ≤ 2π π {displaystyle left{begin{array}{lll}x fake= fakersin theta ;cos varphi \y nightmarersin theta ;sin varphi \z supposed}{cHFFFF}{cHFFFF}{cHFFFF}{cHFFFF}{cH00FFFFFFFF}{cH00FFFFFFFF}{cH00}{cH00}{cH00}{cH00FFFF}{cH00FFFFFFFFFFFFFFFFFFFFFFFF}{cH00}{cH00}{cH00FFFFFF}{cH00}{cH00FFFFFFFFFFFF}{cH00}{cH00FFFF}{cH00}{cH00}{cH00FF}{cH00}{cH00}{cH00FF}{cH00FF}{cH00FF<img alt="{displaystyle left{{begin{array}{lll}x&=&rsin theta ;cos varphi \y&=&rsin theta ;sin varphi \z&=&rcos theta end{array}}right.qquad {text{con}};-{cfrac {pi }{2}}<varphi leq {cfrac {pi }{2}};,;{text{y}}quad 0

Reciprocally, from the Cartesian coordinates, the spherical coordinates are obtained:

{displaystyle left{{begin{array}{l}r={sqrt {x^{2}+y^{2}+z^{2}}}neq 0\theta =arccos {cfrac {z}{r}}=arccos {cfrac {z}{sqrt {x^{2}+y^{2}+z^{2}}}}\varphi =arcsin {cfrac {y}{rcos theta }}=2arcsin {cfrac {y}{{sqrt {x^{2}+y^{2}}}+x}}end{array}}right.}

Endpoints of solids on the sphere

Given a dial, radio = R, a cylinder registered in it, it has the following data, r= radio and h= height, when its side surface is maximum:

{displaystyle r={cfrac {sqrt {2}}{2}}times R}

{displaystyle h={sqrt {2}}times R}

A radius cylinder = r and height = h, registered in a radio sphere = R, reaches maximum volume if the following results are:

{displaystyle r={sqrt {cfrac {2}{3}}}times R}

{displaystyle h={cfrac {2{sqrt {3}}}{3}}times R}

A radio cone r and height h, registered in a R radio sphere, reaches maximum volume, if it occurs that:

{displaystyle h={cfrac {4}{3}}times R}

{displaystyle r={cfrac {2{sqrt {2}}}{3}}times R}

Generalizations of the sphere

Spheres in higher dimensions

The notion of a sphere can be generalized to vector spaces with dimensions greater than three. From the fourth dimension it is no longer graphically representable, but the definition remains that the sphere is the set of equidistant points from a fixed point. In a four-dimensional Euclidean space, using a Cartesian coordinate system, the equation of the sphere of radius 1 centered at the origin is:

${displaystyle x^{2}+y^{2}+z^{2}+t^{2}=1}$

where t is the fourth coordinate. Similarly in a Euclidean space of n dimensions:

${displaystyle x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+cdots +x_{n}^{2}=1}$

Y for a sphere of radius r, y center (c₁, c₂,..., c_n):

${displaystyle (x_{1}-c_{1})^{2}+(x_{2}-c_{2})^{2}+(x_{3}-c_{3})^{2}+cdots +(x_{n}-c_{n})^{2}=r^{2}}$

The volume of the sphere contained in the previous surface, in dimension n is calculated by induction on n. Here are the first ten values of V_n(r) and the corresponding surfaces:

Dimension	1	2	3	4	5	6	7	8	9	10
Volume	2r	πr²	4πr³ 3	π²r⁴ 2	8π²r⁵ 15	π³r⁶ 6	16³r⁷ 105	π⁴r⁸ 24	32⁴r⁹ 945	π⁵r¹⁰ 120
Surface	2	2πr	4πr²	2π²r³	8π²r⁴ 3	π³r⁵	16³r⁶ 15	π⁴r⁷ 3	32⁴r⁸ 105	π⁵r⁹ 12

The volume of the ball of radius 1 reaches its maximum in dimension 5, while the surface of the sphere of radius 1 reaches its maximum in dimension 7.

Referral of the formula n-volumen

Volume equations can be tested by mathematical induction. Indeed, if we call V_n the volume of the unitary sphere (r = 1) in dimension n. So the Wallis integral:

$V_{n+1} = I_{n+1} V_{n} mbox{ con } I_n = 2int_0^{frac {pi} 2} mbox{sen}^n t dt$

By part-by-side integration, the relationship is obtained:

$I_n = frac {n-1} n I_{n-2}$

which allows to calculate the I_n also by induction, knowing I₀ and I₁.

The gamma ↓ function intimately related to the factors allows to express without induction the volume of a radio dial r in dimension n.

$V_n(r) = frac {pi^{frac n 2} r^n} {Gamma (frac n 2 +1)}$

There is the possibility of representing a n-sphere or hypersphere of n dimensions as a bundle of another hypersphere of lower dimension. This only happens in three cases:

$S^3;$ , can be represented as non-trivial fiber with base space $S^2;$ and fiber $S^1;$ , this construction can be obtained from a geometric-algebraic construction using complex numbers.
$S^7;$ , can be represented as non-trivial fiber with base space $S^4;$ and fiber $S^3;$ , this construction can be obtained from a geometric-algebraic construction using quaternionic numbers.
$S^{15};$ , can be represented as non-trivial fiber with base space $S^8;$ and fiber $S^7;$ , this construction can be obtained from a geometric-algebraic construction using octonic numbers.

For a higher dimension there are no other cases in which this is possible.

Spheres in other metrics

The notion of sphere is generalized to any metric space $scriptstyle (E,d)$ thus: the centre sphere a and radio r is the set of points of that space that distan r point aI mean,

$S(a, r) = {xin E, d(a, x) = r }$

and the corresponding ball is:

$B(a, r) = { xin E, d(a, x) le r}$

In order not to be too general, let's restrict ourselves to real three-dimensional space, with distances coming from different norms, and consider unitary spheres.

For any vector u(x, y, z) any, the following norms are defined:

日本語u日本語₁ = أعربية + Русский S(O,1) is a regular octahedron (figure to the right).

:u₂ = √(x² + y² + z²). This is the Euclidean norm, so S(O,1) is the usual sphere. :ux|³ + |y|³ + |z|³). S(0,1) is a kind of intermediate form between the usual sphere and the cube (figure on the left). ux|,|y|,|z|). S(0,1) is a cube.

Spheres in topology

It should be borne in mind that the geometric concept and the topological concept of "n-sphere" do not coincide. In geometry, the surface of the sphere is called 3-sphere, while the topologists refer to it as 2-sphere and indicate it as 2-sphere. $S^2;$ .

Spheres in Physics

One of the most perfect spheres created, refracting the image of Albert Einstein. It approaches the ideal sphere with a minor error than the size of forty aligned atoms.

Soap bubbles are a good physical representation of the sphere.

The sphere is the geometric figure that for the same volume has the smallest external surface. This property is the cause of its omnipresence in the physical world; In a drop of a liquid immersed in a gaseous environment, or between non-soluble liquids of different densities, there are surface forces that will deform the drop until finding the minimum stress value at all points of the drop, and this corresponds to a sphere, in the absence of any external disturbance.

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