Circle

A circle is a plane geometric figure. It is defined as the set of all points on a plane having a constant distance from a predetermined point in said plane ( the center ). The distance between the points on the circle to the center of the radius or radius of the circle, it is a positive real number. The circle is one of the classic and basic objects of Euclidean geometry.

Even the ancient Egyptians and Babylonians were trying to determine the area of ​​the circle approximation. Especially in Ancient Greece was the circle because of its perfection of great interest. For example, Archimedes tried unsuccessfully with the tools compass and ruler to transform the circle into a square with the same area in order to determine the area of ​​a circle can. Such a method of calculation of the surface area is called the quadrature of the circle. Only in 1882 was Ferdinand von Lindemann demonstrated by detection of a specific property of the circle number that this problem is unsolvable.

• 3.1 Time of the Egyptians and Babylonians
• 3.2 antiquity
• 3.3 Renaissance
• 3.4 19th century

• 4.1 Cartesian equation
• 4.2 Functional equation
• 4.3 Parameter representation
• 4.4 Complex representation

• 5.1 Kreiszahl
• 5.2 scope
• 5.3 circular area
• 5.4 diameter
• 5.5 curvature
• 5.6 Other Formulas

• 6.1 Approach by squares
• 6.2 counting in a grid
• 6.3 approximation by polygons

• 7.1 Symmetry and imaging characteristics
• 7.2 circle angles and angular rates
• 7.3 Theorems on tendons, secants and tangents
• 7.4 perimeters and incircles
• 7.5 Circle reflections and Möbius transformations

• 8.1 Thales circle
• 8.2 Construction of tangents
• 8.3 surface doubling
• 8.4 cyclotomic

• 9.1 The circle as curve
• 9.2 circumference
• 9.3 acreage
• 9.4 curvature
• 9.5 Isoperimetrisches problem

Word Definitions

Circular areas

According to this definition, a circle is a curve that is a one-dimensional structure, and any two-dimensional surface. Since the word "circle" but is often inaccurately used also for the enclosed area, often used to illustrate the concepts circle line, circle or circular peripheral edge instead of circle - in contrast to the circular area or circular disk. Mathematicians still distinguish between the closed circle or disc and open ( or circle inside ), depending on whether the circular line that includes or not.

Arc, chord, sector and segment

A contiguous portion of the circuit ( that is, the circle ) is a circular arc. A connecting two points of the circle is called a chord. For each string comprises two circular arcs (generally a shorter and a longer ). The longest chordal are those which pass through the center, which is the diameter. The corresponding arcs are called semicircles.

A sector of a circle ( circular section ) is an area that is bounded by two radii and an intermediate arc. Forming the two radii have a diameter, including the sectors are often referred to as half-circles.

Segments of a circle ( circular sections ) are enclosed by a circular arc and a chord.

Tangent, and secant Passante

For the location of a straight line with respect to a given circle, there are three options:

• If the distance between the center and line smaller than the radius of the circle, so you have a circle and straight line two (different ) points of intersection and the straight line is called a secant (Latin secare = cut ). Sometimes it refers to the special case of a secant line passing through the center of a circle as the center.
• Is the distance of the center point to the straight line with the radius coincide, then there is exactly one common point. It is said that the line touches the circle and the straight line is called a tangent (Latin tangere = touch ). A tangent at the contact point is perpendicular (orthogonal, normal) to the corresponding radius.
• If the distance of the circle center from the straight line is larger than the radius of the circle, then have in common circle and straight no point. In this case, is called the line as Passante. This term has no direct Latin origin, but was probably formed according to French or Italian passante = passers. The Latin root is passus = step.

Formal definition

In a plane is a circle of center and radius of the set of points

The radius is a positive real number and denotes the length of the track.

Is twice the radius of the diameter and is often referred to. Radius and diameter are related by the relationship or each other.

Sometimes, any line that connects the center to a point on the circle, referred to as the radius, and each track which passes through the center, and whose two end points are in the circle, as the diameter. In this way, the opening number is the length of each radius, and the number, the length of each diameter.

The open circle is formally defined as the set of points

The closed disk as

History

Time of the Egyptians and Babylonians

The circuit includes not only the point of the straight line and the oldest elements of pre-Greek geometry. Already two thousand years before Christ, the Egyptians dealt with it in their studies to the geometry. They could determine the area of ​​a circle approximately by one-ninth of its length withdrew from the diameter d and multiplied the result by yourself. So they counted

And given as approximately ( with a deviation of only about 0.6% ) the area of ​​a circle. This approximation has been found in ancient Egyptian Rhind papyrus paper, it can be obtained if one approaches the circle by an irregular octagon.

The Babylonians ( 1900-1600 BC ) used a very different method to calculate the area of the circular disk. Unlike the Egyptians, they went out from the circumference, which they estimated as three times the diameter. The area was then estimated at one-twelfth the square of the circumference, ie

But the Babylonians dealt already with sectors. You could calculate the length of the tendon or the height of the circular segment (which is perpendicular to the mid-chord line between tendon and scope ). Thus they founded the tendon geometry, which was later further developed by Hipparchus and Claudius Ptolemy presented at the beginning of his astronomical textbook Almagest.

Antiquity

The Greeks are usually regarded as the founder of the science of nature. As the first significant philosopher of this time, who occupied himself with mathematics, Thales of Miletus is considered ( 624-546 BC). He brought knowledge of the geometry from Egypt to Greece, such as the statement that the diameter bisects the circle. Other statements on the geometry established by Thales himself. The named today after Thales theorem states that peripheral angles are right angles in a semicircle. In particular, Thales was the first in which the concept of the angle occurred.

The first known definition of the circle goes back to the Greek philosopher Plato ( 428/427-348/347 BC), which he formulated in his dialogue Parmenides:

" Round but is probably the whose outermost parts are everywhere equally distant from the center far. "

Lived the Greek mathematician Euclid of Alexandria about 300 years before Christ. About himself little is known, but his work in the field of geometry was remarkable. His name is still in contexts such as Euclidean space, Euclidean geometry, or Euclidean metric in use. His most important work was The elements, a thirteen- volume treatise, in which he summarized the arithmetic and geometry of his time and systematized. He concluded the mathematical statements of postulates, arguing that Euclidean geometry. The third volume of the elements dealt with the doctrine of the circle.

From Archimedes, who probably lived between 287 BC and 212 BC in Sicily, a detailed treatise entitled circle measurement is handed down. He proved in this work that the area of ​​a circle is equal to the area of a right triangle with the circle radius as the one and circumference than the other cathetus. The area of ​​the circle can thus be specified as ½ · radius · scope. With this realization, he brought back the problem of squaring the circle to the issue of constructability of the circumference of the given radius. , In his treatise Archimedes measurement circuit could also show that the circumference of a circle is greater than 310 /71 and less than 31/7 of the diameter. For practical purposes, this approximation is still used today 22/7 ( 3143 ~ ). From these two statements we deduce that the area of ​​a circle to the square of its diameter behaves almost like 11/14. Euclid was already known that the area of ​​a circle is proportional to the square of its diameter. Archimedes are here at a good approximation of the constant of proportionality.

In another work On Spirals, Archimedes describes the construction of the later named after him Archimedean spiral. With this construction, it was Archimedes possible to remove the circumference of a circle on a straight line. In this way, the area of ​​a circle could now be determined exactly. However, this spiral can not be constructed with compass and straightedge.

Apollonius of Perga lived about 200 years before Christ. In its conic teaching Konika he took on, among other things, the ellipse and the circle as sections of a right circular cone - just as it is still defined in algebraic geometry. His findings back to his predecessors Euclid and Aristaeus ( about 330 BC ), their written essays, however, are not handed over conic sections.

According to Apollonius the Apollonian problem may still be named to three given circles with the Euclidean tools ruler and compasses to construct the circles which touch the given. However, compared to Euclid's Elements, which formed the basis of the geometry in the Middle Ages, the works of Apollonius found initially only in the Islamic world attention. In Western Europe, his books gained greater importance in the 17th century, when Johannes Kepler recognized the ellipse as the true orbit of a planet around the sun.

Renaissance

In the history of science are usually called the period between 1400 AD and 1630 AD Renaissance, even if the timing section does not coincide with the periodization about the history of art. During this time, Euclid's Elements were again more attention. They were among the first printed books and have been published in the following centuries in many different editions. Erhard Ratdolt presented in Venice in 1482, the first printed edition of the Elements ago. One of the most important editions of Euclid's Elements, published by the Jesuit Christoph Clavius ​​. He added the actual texts of Euclid addition to the late antique books XIV and XV still a sixteenth book and further extensive additions added. For example, he added a construction of common tangents of two circles.

19th century

After inputs of Leonard Euler, who established the Euler's identity, Johann Heinrich Lambert and Charles Hermite Ferdinand von Lindemann was able to prove in 1882 that the number π is transcendental. That is, there is no polynomial with rational coefficients for which π is a root. As has been shown already in the 17th century that the circuit number must π is a root of such a polynomial be in order to square the circle was working with ruler and compass, was thus also proved that there can be no such procedure.

Equations

In analytical geometry geometric objects with the aid of equations will be described. Points in the plane are usually represented to by their Cartesian coordinates and a circle is the set of all points whose coordinates satisfy the respective equation.

Cartesian equation

The Euclidean distance of a point from the point calculated as

By squaring the equation defining the coordinates equation

For the points on the circle having its center and radius. An important special case is the Cartesian equation of the unit circle

Functional equation

Since the circle is not a function graph, it can also not be represented by a functional equation. Makeshift, a pair of functional equations

Be used. For the unit circle, this simplifies to

Parameter representation

Another way to describe a circle as coordinates, has the parameter representation (see also polar ):

Here the coordinates and the parameters are expressed, which can accept with all values.

Applying these equations specific to the unit circle, so we get:

It is also a parametric representation without recourse to trigonometric function possible (rational parameterization ), but here the entire set of real numbers is required as a parameter area and the point will be reached only as a limit for.

For the unit circle is then as follows:

Complex representation

In the complex plane can the circle with radius through the equation

Represent. With the help of the complex exponential function we obtain the parametric representation

Circuit calculation

Kreiszahl

Since all circles are similar, the ratio of circumference and diameter for all circles is constant. The numerical value of this ratio is used in elementary geometry as a definition of the circle number. It is a transcendental number, in which has also been shown that it has an outstanding importance in many areas of higher mathematics.

Scope

In the context of elementary geometry the ratio of the circumference to its diameter, and that for any circuits. Thus applies

The radius of the circle is meant.

Circular area

The area of ​​the circular area (lat. area: area) is proportional to the square of the radius or diameter of the circle. They described him as a circle content.

To obtain the formula for the circuit content limit are essential considerations. Law clearly gives a those from the adjacent drawing:

The circular area is equidecomposable with the surface of the right figure. These approaches - at finer sector classification - a rectangle with the length and the width. The area formula is thus

The area formula can be proved, for example, by integrating the equation of a circle or using the approach described below by regular polygons.

Diameter

The diameter of a circle with the area and the radius can be extended by

Calculate.

Curvature

In comparison with the sizes described as yet less basic property of the circuit is the curvature. For the precise definition of the curvature terms from the analysis are needed, they can, however, simply because of the symmetry properties of the circle calculated. Clearly indicates the curvature at each point how much the companies differ in the immediate neighborhood of the point of a straight line. The curvature of the circle at the point can be extended by

Calculate, where again the radius of the circle. In contrast to other mathematical curves of the circuit has the same curvature at any point. Apart from the circle only has the straight constant curvature. For all other curves, the curvature of the point depends.

Other formulas

In the following formulas, the sector angle designated in radians. Specifies the angle in degrees, the conversion is valid.

Approximations for the area

Since the wave number is a transcendental number, there is no method of construction by ruler and compass, with which one can determine the exact acreage. In addition transcendental numbers are irrational, and therefore has no finite decimal expansion, which is why the rotor disc area also has no finite decimal expansion at rational radius. For these reasons, up to now, different approximation method for the surface area and thus the circumference of a circle have been developed. Some of approximation methods, such as the explained by polygons in the section approximation methods that can provide an arbitrarily accurate result by multiple repetition. So Could such processes are repeated infinitely often, they would provide the exact result.

Approach by squares

A circle with radius is circumscribed by a square of side length. He is further inscribed with the diagonal of a square. The area of ​​the outer square is that of the inner triangle area on the formula, and the mean is consequently. With this approximation, the circular area with a relative error less than 5 % is determined exactly.

Counting in a grid

The circular area can be determined approximately by underlaid her many small squares (eg with graph paper ). If all squares that lie completely inside the circle, so you get a little too low for the area, you count all squares that merely cut the circle, then the value is too large. The average of the two results gives an approximation for the area of ​​a circle whose quality increases with the fineness of the grid square.

Approximation by polygons

In another possibility for determining circular area in the circle is inscribe a regular hexagon whose vertices lie on the circle. If now the Middles projected from the center of the circle and these new points connected with the old corners, then a regular dodecagon. If this process is repeated continuously arise one after the 24 -gon, a 48 -gon and such.

In each hexagon, the sides are the same length as the radius radius. The sides of the following polygons arise using the Pythagorean theorem each of the sides of the foregoing. From the sides, the faces of the polygons by triangle area calculation can be determined exactly. They are all slightly smaller than the circular area, which they approach with increasing number of corners, however.

Accordingly, one can proceed with a hexagon, which is drawn from the outside of the circle, so the middle of each side lie on it. To obtain a decreasing sequence of surface dimensions which limit in turn the circular area.

Geometric principles and concepts around the circle

Symmetry and imaging characteristics

The circle is a geometric figure of very high symmetry. Each straight line through its center is an axis of symmetry. In addition, the county is rotationally symmetric, that is, any rotation around the center is the circle onto itself from. In group theory, the above symmetry properties of the circle be characterized by its symmetry group. Formal results for the orthogonal group which is the group of orthogonal matrices.

All circles of the same radius are congruent to each other, so can be described by successive parallel shifts represent. Any two circles are similar to each other. You can always be mapped to each other by a central dilation and a parallel shift.

Circle angles and angular rates

A chord with endpoints A and B divides a given circle into two arcs. An angle with vertex C on one of the arcs of a circle is called the circumference angle or peripheral angle. The angle with vertex at the center of M is called a central angle or central angle.

In the special case that the string containing the center, that is a diameter of the circle, the central angle is a straight angle of 180 °. In this situation, a basic statement of the circular geometry, Thales' theorem is true: He states that circumferential angle always are right angles on a diameter, so be 90 °. The circle around the right triangle is in this situation also called Thales circle.

Also in the case of an arbitrary chord circumferential angle are all lying on the same circular arc, the same size. This statement is also known as peripheral angle theorem. The circular arc on which are the vertices of the circumferential angle, ie barrel arc. Lying circumferential angles and central angles on the same side of the chord, then the central angle is twice as large as the peripheral angle ( circle angle set). Two circumferential angles on opposite sides of the chord, complement each other to 180 °.

The circumferential angle is the same as the top chord tangent angle between the chord and the plane passing through one of its endpoints tangent ( tangent angle tendons rate).

Theorems about chords, secants and tangents

For circles of tendons sentence which applies: two tendons [AC ] and [ BD] cutting each other at a point S, the following applies

I.e., the products of each chordal portions are the same.

Two chords of a circle, which do not intersect, can be extended to secants that are either parallel or intersect at a point S outside the circle. If the latter is the case, applies analogously to the set of tendons Sekantensatz

In the case of a secant that intersects the circle at the points A and C, and a tangent that touches the circle at point B, the secant - tangent theorem: If S is the intersection of the secant and tangent, as follows

Perimeters and incircles

If A, B, C are three points that do not lie on a straight line, thus form a nondegenerate triangle, then there exists a uniquely determined circle through these points, namely the area of the triangle ABC. The center of the circumscribed circle is the intersection of the three mid-perpendiculars of the triangle. Likewise, every triangle a unique circle to inscribe certain that touches the three sides, that is, the sides of the triangle form tangents of the circle. This circle is called the inscribed circle of the triangle. Its center is the intersection of the three angle bisectors.

In elementary geometry still more circles be considered a Triangle: The excircles lie outside the triangle and touch one side and the extensions of the other two sides. Another interesting triangle is the circle at the Feuerbach circle, named after Karl Wilhelm Feuerbach. Are the three midpoints and the three feet of the altitudes on it. Moreover, since the lie to him three midpoints of the segments between the orthocenter and the vertices of the triangle, the Feuerbach circle is also called nine -point circle. Its center lies as the focus, the circumcenter and the orthocenter on the Eulerian straight.

Unlike triangles polygons have more than three corners in general no perimeter or inscribed circle. For regular polygons but both always exist. A quadrilateral that has a radius, inscribed quadrilateral is called. A convex quadrilateral is exactly then a cyclic quadrilateral when complete opposite angle to 180 °. A quadrilateral has an inscribed circle is called the tangent quadrilateral. A convex quadrilateral is a tangent rectangle if and when the sum of the side lengths of two opposite sides is equal to the sum of the other two side lengths.

Circular reflections and Möbius transformations

The mirroring circuit, also known as inversion, is a special picture plane geometry, which describes a " reflection " of the Euclidean plane at a given circle having its center and radius. Is a given point, then its image point is determined by the fact that it lies on the half-line and its distance from the equation

Met. The circular mirror forms the interior of the given circle, and vice versa on its exterior. All circuit points are mapped to themselves. Circular reflections are angle preserving, orientation- and circular faithful. The latter means that generalized circles - these are circles and lines - are again mapped to generalized circles.

The sequential execution of two circular reflections yields a Möbius transformation. Möbius transformations - another important class of mappings of the plane - are therefore also conformal and loyal circle, but orientation-preserving.

Circular reflections and Möbius transformations can be particularly clearly visualized with the aid of complex numbers: In a circular reflection of a point on the circle, the formula for the pixel

For the simple reflection in the unit circle is considered.

Möbius transformations of the complex plane are broken linear functions of the form

With and displayed.

Constructions with ruler and compass

A classic problem of geometry is the construction of geometric objects with ruler and compass in a finite number of construction steps from a given set of points. In each step of this straight line may be drawn through given or already constructed points and circles are drawn around those points with a given radius or the constructed already. The thus constructed points are obtained as intersections of two lines, two circles or a line and a circle. Naturally, therefore, play an important role in all constructions with ruler and compass circles.

In the following, an example will be discussed, some structures that are related to the geometry of circles of importance.

Thales circle

For the construction of the Thales circle over a given route initially the focus of this track is constructed, which is also the center of Thales circle. And as for and circles with the same radius are beaten, which must be chosen so large that the two circles intersect at two points and. This is for example the case. The line then crosses the center. The searched Thales circle is now the circle with center and radius.

Construction of tangents

Given a point outside a circle with center and there are the two tangents are constructed to the circle passing through the point. This elementary construction task can be easily solved with the help of the theorem of Thales: Thales We construct the circle with the line as a diameter. The points of intersection of this circle with the contact points are then the desired tangents.

Surface doubling

The area of ​​a circle can be doubled geometrically by a square is drawn, one corner of which is located in the center of the circle, with two other vertices lie on the arc. By the fourth corner a circle around the old center is pulled. This procedure was shown in the 13th century in Bauhütte book of Villard de Honnecourt. This method works because ( according to the Pythagorean theorem )

And thus the surface area of the large circle

Is exactly twice as large as that of the small circle.

Cyclotomic

Another already -studied in ancient construction problem is the circular pitch. Here it should be inscribed in a given natural number in a given circle a regular -gon. Located on the circle then divide these vertices into equal arcs. This construction is not possible for all: With the help of the algebraic theory of field extensions can be shown that it is precisely then feasible if a factorization of the form

And has pairwise distinct Fermat primes, ie primes of the form. Thus, the construction is so, for example, possible, but not for eg. The construction of the regular Siebzehnecks succeeded Carl Friedrich Gauss in 1796.

Circuit calculation in the Analysis

In modern analysis the trigonometric functions and the county number are usually defined initially without recourse to the elementary geometric intuition and on special properties of the circle. So can be defined as the sine and cosine over its representation as a power series. A common definition of the value of is then the double of the smallest positive root of the cosine.

The circle as curve

In differential geometry, a branch of analysis, the geometric shapes investigated by means of differential and integral calculus, circuits are considered special curves. These curves can be described using the above parameters View as way. If one of the coordinate origin at the center of a circle with radius, then by working with

Been such a parameterization. Using the trigonometric formula follows for the Euclidean norm of the parameterized points, that is, they are actually on a circle with radius. Since sine and cosine are periodic functions, corresponds to the definition interval of exactly one circle round.

Circumference

The circumference of the circle is calculated as the length of the path through integration

Analogously, for the length of the given part of the arc. This is obtained after the arc length parametrization of the circle as

With.

Area

The area of ​​the circular disc, that is the measure of the amount, can be used as ( two-dimensional ) Integral

Are shown. To avoid the somewhat tedious calculation of this integral in Cartesian coordinates, it is convenient to transform to use on polar coordinates. This results in

Another possibility to compute the circular area is to use the sector formula Leibniz the parametric representation of the circular edge. With, we obtain thus also

Curvature

For the above- derived parameterization of the circle after his arc length results

For the curvature of the circle is obtained, therefore,

The curvature of the loop is constant and the radius of curvature is especially its radius.

In the differential geometry is shown that a plane curve is uniquely determined up to congruence by its curvature. The only plane curves with constant positive curvature are therefore circular arcs. In the limit that the curvature is constant and equal to 0, line segments arise.

Isoperimetrisches problem

Among all surfaces of the Euclidean plane with given perimeter, the circle has the largest area. Conversely, the circle has the smallest perimeter for a given area. In the plane of the circle is therefore the unique solution of the so-called isoperimetric problem. Although this clearly obvious fact was already known to mathematicians in ancient Greece, formal proofs were provided only in the 19th century. Since a curve is sought which maximizes a functional, namely the enclosed area, these are from a modern perspective to a problem of the calculus of variations. A common proof for piecewise continuous curves using the theory of Fourier series.

Generalizations and related subjects

Sphere

It is possible to generalize the code as the object plane in the three dimensional space. Then we obtain the envelope of a sphere. This object is called in mathematics sphere or, more precisely 2-sphere. Analogously, the 2-sphere in n dimensions to n- sphere generalized. In this context it is called the circle also 1- sphere.

Conics

In the planar geometry of the circle can be thought of as a special ellipse with the two focal points with the circle center point coincide. Both semi-axes are equal to the radius of the circle. The circuit is therefore a specific conic: It is formed as a section of a right circular cone with a plane perpendicular to the cone axis. It is thus a special case of a two-dimensional quadric.

This results in another, equivalent definition for circles ( circle of Apollonius ): A circle is the set of all points in the plane, for which the ratio of its distances from two given points is constant. The two points lie on one of the outgoing beam in the distance, respectively, and reciprocally on the polars of each other's point as a pole. Similar definitions are also available for the ellipse ( constant sum ), hyperbolic ( constant difference ) and the Cassini oval (constant product of the distances ).

Circles in synthetic geometry

In the synthetic geometry of circles can be defined solely by an orthogonality in certain affine planes ( for example präeuklidischen levels ) without a notion of distance, by deleting the phrase is used by radius (mean solders set ) to define the circle. This can then be imported in such levels, a weaker notion of " distance " or " length equality" of pairs of points. → See Präeuklidische level.

Drawing in the digital grid

For the drawing of approximate circles in a grid of several algorithms have been developed, see screening of circles. These methods are particularly suitable for computer graphics of concern. For the two-color screening of circles, the basic arithmetic operations are sufficient.