Sphere

A ball is in the geometry of the abbreviation for spherical and spherical body.

• 3.2.1 Justification
• 3.2.2 Alternative derivation using the sphere volume
• 3.2.3 Alternative derivation using the sphere volume and the differential calculus
• 3.2.4 Derivation with the help of integral calculus
• 3.2.5 Derivation with the help of integral calculus in spherical coordinates

• 5.1 Higher-Dimensional Euclidean spaces
• 5.2 metric spaces

Spherical and spherical body

The spherical surface is formed to a diameter in the rotation of a circumference surface. It is a surface of revolution, and a specific surface of the second order and is described as the amount ( the locus ) of all the points in the three-dimensional Euclidean space, the distance from a fixed point in space is equal to a given positive real number. The fixed point is referred to as the center or the center of the ball, the number and radius of the sphere.

The spherical surface divides the space into two separate open subsets, one of which is exactly convex. This quantity is called the interior of the ball. The union of a sphere and its interior is called spherical body or a solid sphere. The spherical surface is also called sphere or sphere.

Both spherical and spherical bodies are often referred to briefly as a sphere, where it must be clear from the context which is meant the two meanings.

Is satisfied.

In vector notation,

Or

The points on the spherical surface having the radius and centered at the origin can be parameterized by spherical coordinates as follows:

Ball cuts

• Putting a level with a ball on the average, is always a circle. When the plane contains the center of the sphere is called the great circle cut line, otherwise small circle.
• The two resulting partial body hot spherical section or spherical segment, in the case of the great circle hemisphere ( hemisphere).
• The curved part of the surface of a spherical segment is called a ball cap, ball cap or calotte.
• A spherical segment and the cone with the circle of intersection as the base and the center of the sphere as a top result in a spherical section or ball sector.
• Two parallel, the ball cutting (not touching ) planes intersect the ball out from a spherical layer. The curved part of the surface of a spherical segment is called a spherical zone.
• Two intersecting planes whose line of intersection lies partly within the ball, cut the ball from an object whose curved surface is the lune.
• A spherical shell ( hollow sphere ) is the difference of two concentric spheres with different radius.

Formulas

Not identical with the h in the sketch below)

Volume

The spherical volume is the volume of a sphere, delimited by the spherical surface.

Kegelherleitung ( Archimedes' derivation )

After a consideration of the Greek mathematician Archimedes there is a hemisphere of radius a comparison body, the volume of which coincides with that of the hemisphere, but it is easy to calculate. This comparison body arises from the fact that one of a cylinder (more precisely, a right circular cylinder ) with base surface radius and height of a cone (more precisely, a right circular cone) away with basic surface radius and height.

To demonstrate that the hemisphere and the comparison body have equal volume, one can use the principle of Cavalieri. This principle is based on the idea to decompose the observed body in an infinite number of slices of infinitesimal ( infinitely small ) thickness. ( An alternative to this procedure would be the application of the integral calculus. ) According to the above principle is examined for both the cut surfaces of the body with the planes that are parallel to the respective base and have a predetermined distance from this.

In the case of the hemisphere, the sectional area is a circular area. The radius of this circle follows from the Pythagorean theorem:

This yields for the content of the cut surface

In the case of the reference block, the cut surface, however, is a circular ring with an outer radius and inner radius. The area of ​​this interface is therefore

For an arbitrary distance to the base of the two cut surfaces are therefore right in line in the area. This follows the principle of Cavalieri that the hemisphere and the reference block have the same volume.

The volume of the reference block and thus the hemisphere can now be easily calculated:

Subtract the cylinder volume of the cone volume.

Hence, in the volume of the (full) sphere:

Alternative derivation

The ball can be divided into an infinite number of pyramids of height ( peaks in the center of the sphere ), the entire surface area of ​​the surface of the sphere ( see below) corresponds. This is the total volume of all pyramids.

Derivation with the help of integral calculus

Radius at a distance:

Circular area at a distance:

Volume of the sphere:

In the same way, one can calculate the volume of a spherical segment of height:

Other derivations

The ball can be represented by the equation

Describe, wherein the spatial coordinates and represents the radius.

About the integral calculus can solve this problem in two ways:

We parameterize the ball through

With the Jacobian

One obtains the volume element required as

The volume of the sphere gives this as

A further possibility consists of the polar coordinates:

Now the Cartesian coordinate system is transformed into the polar coordinate system, which means that the integration after the " change" of the coordinate system by means of the variables and is maintained continuously, rather than as before and. Motivation for this transformation is the considerable simplification of the calculation in the further course. For the differential, this means: ( Keyword: surface element )

Another way using the formula for rotating bodies

Lets you rotate a patch about a fixed axis in space, you get a body with a certain volume. In a circular area the result is a ball. Clearly, one can think of this as a rotating coin.

The general formula for the rotation body which rotates about the x- axis results,

The equation for the circle is

With center

Inserted into the equation for the circle, we get

By inserting into the formula for the rotary body about the x- axis is obtained

Surface

The spherical surface is the two-dimensional surface, which forms the edge of the ball. It is thus the set of all points whose distance from the center of the sphere has a fixed value. It is a closed two-dimensional manifold.

Its area is the same as that of the lateral surface of the circular cylinder, which encloses the ball. The sphere has the smallest surface area for a given volume of all possible body.

Grounds

If you divide a sphere into:

• Layers with a height of each and
• " Meridians " that also have the distance to each other at the equator

And allowed according to strive, as is

• The length of each field is inversely proportional to - so to its distance from the central axis
• Its width, however, is proportional to.

From the upper right drawing it is clear that is inversely proportional to ( the distance of the point of tangency to the central axis ). The tangent is perpendicular to the " spoke" and the two ( right-angled ) triangles are similar. Accordingly, the following applies:

The length multiplied by the width is therefore always the same.

Since all square fields have the same area and this is at the equator and total (number of fields in the horizontal direction multiplied by the number of fields in the vertical direction, that is ) are fields, the total area of all fields.

Alternative derivation using the sphere volume

A ball can be imagined as composed of an infinite number of infinitesimal ( infinitely small ) pyramids. The bases of these pyramids together form the spherical surface; the heights of the pyramids are each equal to the radius of the sphere. Because the pyramidal volume is represented by the formula, a corresponding relation for the total volume of all pyramids, that is, the sphere volume the following applies:

Because the result is:

Alternative derivation using the sphere volume and the differential calculus

Since the spherical volume with

Is defined, and on the other hand, the surface of which a change in the volume according to

Is, the surface formula gives immediately from the derivative of the volume formula.

Derivation with the help of integral calculus

From the first Guldin'schen rule

For the lateral surface of a rotating body gives:

Derivation with the help of integral calculus in spherical coordinates

For the surface element on surfaces = constant applies in spherical coordinates:

Thus, the surface can be easily calculated:

Properties

The ball has an infinite number of planes of symmetry, namely the planes through the ball center point. Further, the ball is rotationally symmetric with respect to any axis through the center and each rotation angle and point-symmetric with respect to its center.

The ball has neither edges nor corners. Their surface can not propagate without distortion in the plane, see also the article card network design.

The sphere has the smallest surface area of all bodies with a predetermined volume. Of all the bodies with a predetermined surface it encloses the largest volume. For this reason, the ball also occurs in nature bubbles ( see soap bubble ), and water drops are balls (not including gravity ), as the surface tension tries to minimize its surface. Planets are approximately spheres because they were liquid at its origin and the sphere is the shape with the largest gravitational binding energy. The mathematical sphere is an ideal shape. In nature, occurring balls always have only approximately spherical shape.

The a sphere circumscribed cylinder has the times the volume of the sphere. This, as well as the surface and volume formulas were already the Greek Archimedes known in antiquity.

A ball can also be interpreted as a rotating body: Lets you rotate a semicircular surface around its diameter, then this provides a ball. If the circle is replaced by an ellipse that rotates about one of its axes, a rotation ellipsoid gives (also called spheroid ).

Generalization

Higher dimensional Euclidean spaces

The term of the sphere can be transferred to other rooms dimension. Analogous to the three dimensional solid sphere is defined for a natural number of one- dimensional set of points of the ball as -dimensional Euclidean space, the distance to a given point ( the center ) is equal to a positive real number ( the radius ) is smaller. The edge of the ball -dimensional, in other words the set of points whose distance from the center is equal to is referred to as () - dimensional sphere or short () sphere. If you speak no further details of the - dimensional sphere, one usually refers to the -dimensional unit sphere; In this case, the center point lies at the origin of the coordinate system, and the radius is equal to 1

By this definition, a three-dimensional sphere is thus an ordinary ball; their surface corresponds to a 2-sphere. A two-dimensional sphere is a circle, the corresponding edge of the circle a 1- sphere. A one-dimensional ball eventually is a stretch, with the two line endpoints can be regarded as 0 - sphere.

Note: These terms are not used consistently. Spheres in the sense defined herein are sometimes called beads. In addition, some authors speak of spheres when they mean (-1 )-dimensional sphere in -dimensional space.

Is the -dimensional volume of an -dimensional sphere of radius

Here is the gamma function, a continuous expansion of the faculty. The ( )-dimensional content of the ( )-dimensional surface, ie, the () - sphere is obtained by deriving the volume according to the radius:

So for a unit sphere in dimensions we find the following volumes and surface areas:

A sphere is an example of a compact manifold.

Metric spaces

The term of the ball can be generalized to all rooms where you have a notion of distance, these are the metric spaces.

If a metric space, and, so called

The open ball with center and radius. The amount

Ie closed ball.

Some authors write for the open and the closed balls. Other spellings for the open balls are.