Sphere

A sphere (from Ancient Greek σφαῖρα (sphaîra) 'ball') is a surface analogous to the circle, a curve. In solid geometry, a sphere is the set of points that are all at the same distance r from a given point in three-dimensional space. That given point is the center of the sphere, and the distance r is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians.

The sphere is a fundamental surface in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings.

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. 'Radius' is used in two senses: as a line segment and also as its length.

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.

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 figure of Earth is not perfectly spherical, terms borrowed from geography are convenient to apply to the sphere. A particular line passing through its center defines an axis (as in Earth's axis of rotation). The sphere-axis intersection defines two antipodal poles (north pole and south pole). The great circle equidistant to the poles is called the equator. Great circles through the poles are called lines of longitude or meridians. Small circles on the sphere that are parallel to the equator are circles of latitude (or parallels). In geometry unrelated to astronomical bodies, geocentric terminology should be used only for illustration and noted as such, unless there is no chance of misunderstanding.

Mathematicians consider a sphere to be a two-dimensional closed surface embedded in three-dimensional Euclidean space. They draw a distinction between a sphere and a ball, which is a solid figure, a three-dimensional manifold with boundary that includes the volume contained by the sphere. An open ball excludes the sphere itself, while a closed ball includes the sphere: a closed ball is the union of the open ball and the sphere, and a sphere is the boundary of a (closed or open) ball. The distinction between ball and sphere has not always been maintained and especially older mathematical references talk about a sphere as a solid. The distinction between "circle" and "disk" in the plane is similar.

Small spheres or balls are sometimes called spherules (e.g., in Martian spherules).

In analytic geometry, a sphere with center (x₀, y₀, z₀) and radius r is the locus of all points (x, y, z) such that

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

Since it can be expressed as a quadratic polynomial, a sphere is a quadric surface, a type of algebraic surface.

Let a, b, c, d, e be real numbers with a ≠ 0 and put

\(x_0 = \frac{-b}{a}, \quad y_0 = \frac{-c}{a}, \quad z_0 = \frac{-d}{a}, \quad \rho = \frac{b^2 +c^2+d^2 - ae}{a^2}.\)

Then the equation

\(f(x,y,z) = a(x^2 + y^2 +z^2) + 2(bx + cy + dz) + e = 0\)

has no real points as solutions if \(\rho < 0\) and is called the equation of an imaginary sphere. If \(\rho = 0\), the only solution of \(f(x,y,z) = 0\) is the point \(P_0 = (x_0,y_0,z_0)\) and the equation is said to be the equation of a point sphere. Finally, in the case \(\rho > 0\), \(f(x,y,z) = 0\) is an equation of a sphere whose center is \(P_0\) and whose radius is \(\sqrt \rho\).

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.

A parametric equation for the sphere with radius \(r > 0\) and center \((x_0,y_0,z_0)\) can be parameterized using trigonometric functions. \[\begin{align} x &= x_0 + r \sin \theta \; \cos\varphi \\ y &= y_0 + r \sin \theta \; \sin\varphi \\ z &= z_0 + r \cos \theta \,\end{align}\]

The symbols used here are the same as those used in spherical coordinates. r is constant, while θ varies from 0 to π and \(\varphi\) varies from 0 to 2π.

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

\(V = \frac{4}{3}\pi r^3 = \frac{\pi}{6}\ d^3 \approx 0.5236 \cdot d^3\)

where r is the radius and d is the diameter of the sphere. Archimedes first derived this formula (On the Sphere and Cylinder c. 225 BCE) by showing that the volume inside a sphere is twice the volume between the sphere and the circumscribed cylinder of that sphere (having the height and diameter equal to the diameter of the sphere). This may be proved by inscribing a cone upside down into semi-sphere, noting that the area of a cross section of the cone plus the area of a cross section of the sphere is the same as the area of the cross section of the circumscribing cylinder, and applying Cavalieri's principle. This formula can also 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 side by side and centered along the x-axis from x = −r to x = r, assuming the sphere of radius r is centered at the origin.

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 d, 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 m.

The surface area of a sphere of radius r is:

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

Archimedes first derived this formula from the fact that the projection to the lateral surface of a circumscribed cylinder is area-preserving. Another approach to obtaining the formula comes from the fact that 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.

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

\(\mathrm{SSA} = \frac{A}{V\rho} = \frac{3}{r\rho}\)

where ρ is the density (the ratio of mass to volume).

A sphere can be constructed as the surface formed by rotating a circle one half revolution about any of its diameters; this is very similar to the traditional definition of a sphere as given in Euclid's Elements. Since a circle is a special type of ellipse, a sphere is a special type of ellipsoid of revolution. 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.

A sphere is uniquely determined by four points that are not coplanar. More generally, a sphere is uniquely determined by four conditions such as passing through a point, being tangent to a plane, etc. This property is analogous to the property that three non-collinear points determine a unique circle in a plane.

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 common solutions of the equations of two spheres, it can be seen that two spheres intersect in a circle and the plane containing that circle is called the radical plane of the intersecting spheres. Although the radical plane is a real plane, the circle may be imaginary (the spheres have no real point in common) or consist of a single point (the spheres are tangent at that point).

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. 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.

In their book Geometry and the Imagination, David Hilbert and Stephan Cohn-Vossen describe eleven properties of the sphere and discuss whether these properties uniquely determine the sphere. Several properties hold for the plane, which can be thought of as a sphere with infinite radius. These properties are:

Condensed: the full section is in Wikipedia.

The basic elements of Euclidean plane geometry are points and lines. On the sphere, points are defined in the usual sense. The analogue of the "line" is the geodesic, which is a great circle; the defining characteristic of a great circle is that the plane containing all its points also passes through the center of the sphere. Measuring by arc length shows that the shortest path between two points lying on the sphere is the shorter segment of the great circle that includes the points.

Many theorems from classical geometry hold true for spherical geometry as well, but not all do because the sphere fails to satisfy some of classical geometry's postulates, including the parallel postulate. In spherical trigonometry, angles are defined between great circles. Spherical trigonometry differs from ordinary trigonometry in many respects. For example, the sum of the interior angles of a spherical triangle always exceeds 180 degrees. Also, any two similar spherical triangles are congruent.

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. 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.

Spherical geometry is a form of elliptic geometry, which together with hyperbolic geometry makes up non-Euclidean geometry.

The sphere is a smooth surface with constant Gaussian curvature at each point equal to 1/r. As per Gauss's Theorema Egregium, this curvature is independent of the sphere's embedding in 3-dimensional space. Also following from Gauss, a sphere cannot be mapped to a plane while maintaining both areas and angles. Therefore, any map projection introduces some form of distortion.

A sphere of radius r has area element \(dA = r^2 \sin \theta\, d\theta\, d\varphi\). This can be found from the volume element in spherical coordinates with r held constant.

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

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

This equation reflects that the position vector and tangent plane at a point are always orthogonal to each other. Furthermore, the outward-facing normal vector is equal to the position vector scaled by 1/r.

In Riemannian geometry, the filling area conjecture states that the hemisphere is the optimal (least area) isometric filling of the Riemannian circle.

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.

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. 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.

In navigation, a loxodrome or rhumb line is a path whose bearing, the angle between its tangent and due North, is constant. Loxodromes project to straight lines under the Mercator projection. Two special cases are the meridians which are aligned directly North-South and parallels which are aligned directly East-West. For any other bearing, a loxodrome spirals infinitely around each pole. For the Earth modeled as a sphere, or for a general sphere given a spherical coordinate system, such a loxodrome is a kind of spherical spiral.

Another kind of spherical spiral is the Clelia curve, for which the longitude (or azimuth) \(\varphi\) and the colatitude (or polar angle) \(\theta\) are in a linear relationship, ⁠\(\varphi = c\theta\)⁠. Clelia curves project to straight lines under the equirectangular projection. Viviani's curve (⁠\(c=1\)⁠) is a special case. Clelia curves approximate the ground track of satellites in polar orbit.