Draw Circle of Fixed Diameter Imagej

Simple curve of Euclidean geometry

This commodity is nigh the shape and mathematical concept. For other uses, run across Circle (disambiguation).

Circle
A circumvolve (black), which is measured by its circumference (C), bore (D) in blue, and radius (R) in crimson; its eye (O) is in greenish.
Blazon	Conic section
Symmetry grouping	O(2)
Area	πR2
Perimeter	C = 2πR

A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre; equivalently information technology is the bend traced out by a bespeak that moves in a airplane so that its distance from a given point is constant. The distance between any bespeak of the circle and the centre is chosen the radius. Ordinarily, the radius is required to be a positive number. A circle with ${\displaystyle r=0}$ is a degenerate case. This commodity is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted.

Specifically, a circumvolve is a simple closed curve that divides the plane into ii regions: an interior and an exterior. In everyday utilise, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is simply the boundary and the whole figure is chosen a disc.

A circle may besides be defined every bit a special kind of ellipse in which the two foci are ancillary, the eccentricity is 0, and the semi-major and semi-small-scale axes are equal; or the two-dimensional shape enclosing the almost area per unit of measurement perimeter squared, using calculus of variations.

Euclid's definition

A circle is a plane effigy bounded by ane curved line, and such that all straight lines fatigued from a certain point within it to the bounding line, are equal. The bounding line is chosen its circumference and the bespeak, its centre.

Topological definition

In the field of topology, a circle isn't limited to the geometric concept, but to all of its homeomorphisms. Two topological circles are equivalent if i tin can exist transformed into the other via a deformation of R ³ upon itself (known as an ambient isotopy).^[2]

Terminology

• Annulus: a ring-shaped object, the region divisional by two concentric circles.
• Arc: any connected part of a circle. Specifying two end points of an arc and a heart allows for two arcs that together make upward a full circle.
• Centre: the indicate equidistant from all points on the circumvolve.
• Chord: a line segment whose endpoints lie on the circle, thus dividing a circle into 2 segments.
• Circumference: the length of i circuit along the circumvolve, or the distance around the circle.
• Diameter: a line segment whose endpoints lie on the circle and that passes through the centre; or the length of such a line segment. This is the largest distance between whatsoever two points on the circle. It is a special case of a chord, namely the longest chord for a given circle, and its length is twice the length of a radius.
• Disc: the region of the plane divisional past a circle.
• Lens: the region common to (the intersection of) two overlapping discs.
• Passant: a coplanar straight line that has no point in common with the circle.
• Radius: a line segment joining the centre of a circle with any single point on the circle itself; or the length of such a segment, which is one-half (the length of) a diameter.
• Sector: a region bounded by two radii of equal length with a common center and either of the 2 possible arcs, adamant by this heart and the endpoints of the radii.
• Segment: a region divisional by a chord and one of the arcs connecting the chord's endpoints. The length of the chord imposes a lower purlieus on the diameter of possible arcs. Sometimes the term segment is used just for regions not containing the center of the circle to which their arc belongs to.
• Secant: an extended chord, a coplanar straight line, intersecting a circle in two points.
• Semicircle: one of the two possible arcs determined by the endpoints of a bore, taking its midpoint equally center. In non-technical common usage it may hateful the interior of the 2 dimensional region bounded by a diameter and ane of its arcs, that is technically called a half-disc. A half-disc is a special case of a segment, namely the largest one.
• Tangent: a coplanar direct line that has one single point in common with a circle ("touches the circle at this betoken").

All of the specified regions may exist considered every bit open, that is, not containing their boundaries, or as closed, including their respective boundaries.

Chord, secant, tangent, radius, and bore

History

The compass in this 13th-century manuscript is a symbol of God'due south act of Creation. Notice as well the circular shape of the halo.

The word circumvolve derives from the Greek κίρκος/κύκλος (kirkos/kuklos), itself a metathesis of the Homeric Greek κρίκος (krikos), significant "hoop" or "ring".^[three] The origins of the words circus and excursion are closely related.

Circular piece of silk with Mongol images

The circumvolve has been known since before the kickoff of recorded history. Natural circles would have been observed, such equally the Moon, Sun, and a short plant stem blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the cycle, which, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus.

Early science, particularly geometry and astrology and astronomy, was connected to the divine for almost medieval scholars, and many believed that in that location was something intrinsically "divine" or "perfect" that could exist found in circles.^{[4] [5]}

Some highlights in the history of the circumvolve are:

• 1700 BCE – The Rhind papyrus gives a method to find the expanse of a circular field. The result corresponds to 256 / 81 (3.16049...) as an approximate value of π.^[6]

• 300 BCE – Book iii of Euclid's Elements deals with the backdrop of circles.
• In Plato'southward Seventh Letter of the alphabet in that location is a detailed definition and caption of the circumvolve. Plato explains the perfect circle, and how it is unlike from any drawing, words, definition or caption.
• 1880 CE – Lindemann proves that π is transcendental, effectively settling the millennia-former trouble of squaring the circle.^[7]

Analytic results

Circumference

The ratio of a circle's circumference to its diameter is π (pi), an irrational constant approximately equal to 3.141592654. Thus the circumference C is related to the radius r and diameter d by:

${\displaystyle C=2\pi r=\pi d.\,}$

Area enclosed

Area enclosed by a circumvolve = π × area of the shaded square

Equally proved by Archimedes, in his Measurement of a Circle, the area enclosed past a circle is equal to that of a triangle whose base has the length of the circle's circumference and whose height equals the circumvolve's radius,^[viii] which comes to π multiplied by the radius squared:

${\displaystyle \mathrm {Area} =\pi r^{two}.\,}$

Equivalently, denoting diameter by d,

${\displaystyle \mathrm {Expanse} ={\frac {\pi d^{2}}{4}}\approx 0{.}7854d^{ii},}$

that is, approximately 79% of the circumscribing foursquare (whose side is of length d).

The circle is the airplane curve enclosing the maximum area for a given arc length. This relates the circle to a trouble in the calculus of variations, namely the isoperimetric inequality.

Equations

Cartesian coordinates

Circumvolve of radius r = 1, eye (a,b) = (i.2, −0.v)

Equation of a circle

In an x–y Cartesian coordinate arrangement, the circle with centre coordinates (a, b) and radius r is the set of all points (x, y) such that

${\displaystyle (x-a)^{2}+(y-b)^{two}=r^{2}.}$

This equation, known as the equation of the circle, follows from the Pythagorean theorem practical to whatsoever bespeak on the circle: equally shown in the adjacent diagram, the radius is the hypotenuse of a correct-angled triangle whose other sides are of length |ten − a| and |y − b|. If the circle is centred at the origin (0, 0), then the equation simplifies to

${\displaystyle x^{ii}+y^{2}=r^{two}.}$

Parametric form

The equation can be written in parametric form using the trigonometric functions sine and cosine every bit

${\displaystyle x=a+r\,\cos t,}$

${\displaystyle y=b+r\,\sin t,}$

where t is a parametric variable in the range 0 to twoπ, interpreted geometrically as the angle that the ray from (a,b) to (10,y) makes with the positive x axis.

An alternative parametrisation of the circumvolve is

${\displaystyle x=a+r{\frac {ane-t^{2}}{1+t^{two}}},}$

${\displaystyle y=b+r{\frac {2t}{1+t^{2}}}.}$

In this parameterisation, the ratio of t to r tin be interpreted geometrically as the stereographic projection of the line passing through the centre parallel to the x centrality (come across Tangent half-angle substitution). However, this parameterisation works only if t is fabricated to range not simply through all reals but also to a betoken at infinity; otherwise, the leftmost signal of the circle would be omitted.

iii-betoken form

The equation of the circle determined by three points ${\displaystyle (x_{1},y_{1}),(x_{ii},y_{2}),(x_{3},y_{iii})}$ non on a line is obtained by a conversion of the 3-betoken form of a circle equation:

${\displaystyle {\frac {({\color {green}x}-x_{1})({\color {green}x}-x_{2})+({\color {cherry-red}y}-y_{1})({\color {red}y}-y_{2})}{({\colour {scarlet}y}-y_{i})({\color {dark-green}x}-x_{two})-({\colour {ruby}y}-y_{2})({\color {light-green}x}-x_{1})}}={\frac {(x_{three}-x_{one})(x_{3}-x_{two})+(y_{3}-y_{1})(y_{three}-y_{2})}{(y_{3}-y_{1})(x_{3}-x_{2})-(y_{3}-y_{2})(x_{three}-x_{1})}}.}$

Homogeneous form

In homogeneous coordinates, each conic section with the equation of a circumvolve has the course

${\displaystyle 10^{2}+y^{2}-2axz-2byz+cz^{two}=0.}$

It tin can exist proven that a conic section is a circle exactly when it contains (when extended to the complex projective airplane) the points I(i: i: 0) and J(i: −i: 0). These points are called the round points at infinity.

Polar coordinates

In polar coordinates, the equation of a circle is

${\displaystyle r^{2}-2rr_{0}\cos(\theta -\phi )+r_{0}^{two}=a^{2},}$

where a is the radius of the circle, ${\displaystyle (r,\theta )}$ are the polar coordinates of a generic signal on the circle, and ${\displaystyle (r_{0},\phi )}$ are the polar coordinates of the eye of the circumvolve (i.e., r ₀ is the distance from the origin to the heart of the circle, and φ is the anticlockwise angle from the positive x axis to the line connecting the origin to the centre of the circumvolve). For a circle centred on the origin, i.e. r ₀ = 0, this reduces to simply r = a . When r ₀ = a , or when the origin lies on the circle, the equation becomes

${\displaystyle r=2a\cos(\theta -\phi ).}$

In the general case, the equation can be solved for r, giving

${\displaystyle r=r_{0}\cos(\theta -\phi )\pm {\sqrt {a^{2}-r_{0}^{ii}\sin ^{2}(\theta -\phi )}}.}$

Note that without the ± sign, the equation would in some cases describe simply half a circle.

Complex plane

In the complex plane, a circle with a heart at c and radius r has the equation

${\displaystyle |z-c|=r.}$

In parametric grade, this can be written every bit

${\displaystyle z=re^{it}+c.}$

The slightly generalised equation

${\displaystyle pz{\overline {z}}+gz+{\overline {gz}}=q}$

for existent p, q and complex g is sometimes called a generalised circle. This becomes the above equation for a circle with ${\displaystyle p=1,\ g=-{\overline {c}},\ q=r^{2}-|c|^{2}}$ , since ${\displaystyle |z-c|^{2}=z{\overline {z}}-{\overline {c}}z-c{\overline {z}}+c{\overline {c}}}$ . Not all generalised circles are actually circles: a generalised circle is either a (true) circle or a line.

Tangent lines

The tangent line through a point P on the circle is perpendicular to the diameter passing through P. If P = (x _one, y _one) and the circle has heart (a, b) and radius r, then the tangent line is perpendicular to the line from (a, b) to (x ₁, y ₁), so it has the course (x ₁ − a)ten + (y _ane – b)y = c . Evaluating at (x ₁, y ₁) determines the value of c, and the issue is that the equation of the tangent is

${\displaystyle (x_{ane}-a)x+(y_{one}-b)y=(x_{1}-a)x_{1}+(y_{1}-b)y_{i},}$

or

${\displaystyle (x_{1}-a)(x-a)+(y_{1}-b)(y-b)=r^{2}.}$

If y _one ≠ b , and so the slope of this line is

${\displaystyle {\frac {dy}{dx}}=-{\frac {x_{1}-a}{y_{i}-b}}.}$

This tin can also exist establish using implicit differentiation.

When the centre of the circle is at the origin, then the equation of the tangent line becomes

${\displaystyle x_{1}10+y_{one}y=r^{2},}$

and its slope is

${\displaystyle {\frac {dy}{dx}}=-{\frac {x_{1}}{y_{1}}}.}$

Properties

• The circle is the shape with the largest area for a given length of perimeter (see Isoperimetric inequality).
• The circumvolve is a highly symmetric shape: every line through the centre forms a line of reflection symmetry, and information technology has rotational symmetry around the centre for every angle. Its symmetry group is the orthogonal group O(2,R). The grouping of rotations alone is the circle grouping T.
• All circles are like.
  • A circumvolve circumference and radius are proportional.
  • The area enclosed and the square of its radius are proportional.
  • The constants of proportionality are 2π and π respectively.
• The circle that is centred at the origin with radius 1 is called the unit of measurement circle.
  • Thought of as a great circumvolve of the unit sphere, it becomes the Riemannian circle.
• Through any iii points, not all on the aforementioned line, in that location lies a unique circle. In Cartesian coordinates, it is possible to give explicit formulae for the coordinates of the center of the circle and the radius in terms of the coordinates of the three given points. See circumcircle.

Chord

• Chords are equidistant from the centre of a circumvolve if and just if they are equal in length.
• The perpendicular bisector of a chord passes through the center of a circle; equivalent statements stemming from the uniqueness of the perpendicular bisector are:
  • A perpendicular line from the centre of a circle bisects the chord.
  • The line segment through the eye bisecting a chord is perpendicular to the chord.
• If a central angle and an inscribed angle of a circle are subtended by the same chord and on the same side of the chord, then the central angle is twice the inscribed bending.
• If two angles are inscribed on the same chord and on the same side of the chord, then they are equal.
• If two angles are inscribed on the same chord and on reverse sides of the chord, then they are supplementary.
  • For a cyclic quadrilateral, the exterior bending is equal to the interior opposite angle.
• An inscribed bending subtended by a diameter is a correct angle (run across Thales' theorem).
• The bore is the longest chord of the circumvolve.
  • Amidst all the circles with a chord AB in mutual, the circle with minimal radius is the one with diameter AB.
• If the intersection of any two chords divides i chord into lengths a and b and divides the other chord into lengths c and d, and so ab = cd .
• If the intersection of any two perpendicular chords divides one chord into lengths a and b and divides the other chord into lengths c and d, then a ² + b ² + c ² + d ² equals the square of the diameter.^[nine]
• The sum of the squared lengths of any two chords intersecting at correct angles at a given bespeak is the same as that of any other 2 perpendicular chords intersecting at the aforementioned point and is given past 8r ² − fourp ², where r is the circle radius, and p is the distance from the heart point to the indicate of intersection.^[10]
• The altitude from a point on the circumvolve to a given chord times the diameter of the circle equals the product of the distances from the point to the ends of the chord.^{[11] : p.71}

Tangent

• A line drawn perpendicular to a radius through the terminate point of the radius lying on the circumvolve is a tangent to the circle.
• A line drawn perpendicular to a tangent through the point of contact with a circumvolve passes through the centre of the circle.
• Two tangents tin ever be drawn to a circle from any point outside the circumvolve, and these tangents are equal in length.
• If a tangent at A and a tangent at B intersect at the exterior point P, and so denoting the centre every bit O, the angles ∠BOA and ∠BPA are supplementary.
• If AD is tangent to the circle at A and if AQ is a chord of the circle, so ∠DAQ = i / 2 arc(AQ).

Theorems

• The chord theorem states that if ii chords, CD and EB, intersect at A, then Air-conditioning × AD = AB × AE .
• If two secants, AE and AD, also cut the circle at B and C respectively, then AC × AD = AB × AE (corollary of the chord theorem).
• A tangent can be considered a limiting instance of a secant whose ends are coincident. If a tangent from an external point A meets the circumvolve at F and a secant from the external signal A meets the circle at C and D respectively, so AF ⁱⁱ = Ac × Advertizement (tangent–secant theorem).
• The angle between a chord and the tangent at one of its endpoints is equal to one half the bending subtended at the centre of the circle, on the contrary side of the chord (tangent chord angle).
• If the angle subtended past the chord at the centre is 90°, then ℓ = r √2 , where ℓ is the length of the chord, and r is the radius of the circle.
• If two secants are inscribed in the circle as shown at right, and so the measurement of angle A is equal to one half the difference of the measurements of the enclosed arcs ( ${\displaystyle {\overset {\frown }{DE}}}$ and ${\displaystyle {\overset {\pout }{BC}}}$ ). That is, ${\displaystyle 2\angle {CAB}=\angle {DOE}-\angle {BOC}}$ , where O is the center of the circle (secant–secant theorem).

Inscribed angles

An inscribed angle (examples are the blue and greenish angles in the figure) is exactly half the corresponding central angle (crimson). Hence, all inscribed angles that subtend the same arc (pink) are equal. Angles inscribed on the arc (brown) are supplementary. In detail, every inscribed angle that subtends a diameter is a right angle (since the central angle is 180°).

Sagitta

The sagitta is the vertical segment.

The sagitta (as well known as the versine) is a line segment drawn perpendicular to a chord, between the midpoint of that chord and the arc of the circumvolve.

Given the length y of a chord and the length x of the sagitta, the Pythagorean theorem can be used to calculate the radius of the unique circumvolve that will fit around the ii lines:

${\displaystyle r={\frac {y^{two}}{8x}}+{\frac {x}{2}}.}$

Another proof of this result, which relies only on 2 chord properties given to a higher place, is equally follows. Given a chord of length y and with sagitta of length 10, since the sagitta intersects the midpoint of the chord, nosotros know that information technology is a part of a diameter of the circle. Since the bore is twice the radius, the "missing" role of the diameter is (iir − 10 ) in length. Using the fact that ane role of i chord times the other part is equal to the same product taken along a chord intersecting the first chord, we find that (twor − x)ten = (y / 2)ⁱⁱ . Solving for r, we find the required result.

Compass and straightedge constructions

There are many compass-and-straightedge constructions resulting in circles.

The simplest and most basic is the construction given the centre of the circumvolve and a point on the circle. Place the fixed leg of the compass on the centre point, the movable leg on the bespeak on the circle and rotate the compass.

Construction with given bore

• Construct the midpoint M of the diameter.
• Construct the circle with center 1000 passing through one of the endpoints of the diameter (it volition also laissez passer through the other endpoint).

Construct a circumvolve through points A, B and C past finding the perpendicular bisectors (reddish) of the sides of the triangle (blue). Only ii of the 3 bisectors are needed to observe the centre.

Structure through three noncollinear points

• Name the points P , Q and R ,
• Construct the perpendicular bisector of the segment PQ .
• Construct the perpendicular bisector of the segment PR .
• Characterization the point of intersection of these two perpendicular bisectors Thou . (They meet because the points are not collinear).
• Construct the circle with centre K passing through one of the points P , Q or R (information technology volition also pass through the other 2 points).

Circle of Apollonius

Apollonius' definition of a circle: d ₁/d _ii constant

Apollonius of Perga showed that a circle may likewise be defined as the set of points in a plane having a constant ratio (other than 1) of distances to two fixed foci, A and B.^{[12] [13]} (The fix of points where the distances are equal is the perpendicular bisector of segment AB, a line.) That circle is sometimes said to be drawn about two points.

The proof is in ii parts. Beginning, one must prove that, given two foci A and B and a ratio of distances, any indicate P satisfying the ratio of distances must fall on a particular circumvolve. Let C be another point, also satisfying the ratio and lying on segment AB. By the angle bisector theorem the line segment PC volition bisect the interior angle APB, since the segments are similar:

${\displaystyle {\frac {AP}{BP}}={\frac {Air-conditioning}{BC}}.}$

Analogously, a line segment PD through some point D on AB extended bisects the respective exterior angle BPQ where Q is on AP extended. Since the interior and outside angles sum to 180 degrees, the bending CPD is exactly ninety degrees; that is, a right angle. The set of points P such that angle CPD is a right bending forms a circle, of which CD is a diameter.

Second, see^{[14] : p.15} for a proof that every point on the indicated circle satisfies the given ratio.

Cross-ratios

A closely related property of circles involves the geometry of the cantankerous-ratio of points in the complex plane. If A, B, and C are as to a higher place, then the circumvolve of Apollonius for these iii points is the drove of points P for which the absolute value of the cross-ratio is equal to ane:

${\displaystyle {\large |}[A,B;C,P]{\large |}=1.}$

Stated another mode, P is a point on the circle of Apollonius if and just if the cross-ratio [A, B; C, P] is on the unit of measurement circle in the circuitous plane.

Generalised circles

If C is the midpoint of the segment AB, then the collection of points P satisfying the Apollonius condition

${\displaystyle {\frac {|AP|}{|BP|}}={\frac {|Air-conditioning|}{|BC|}}}$

is not a circle, but rather a line.

Thus, if A, B, and C are given singled-out points in the plane, then the locus of points P satisfying the to a higher place equation is called a "generalised circle." Information technology may either be a truthful circle or a line. In this sense a line is a generalised circumvolve of infinite radius.

Inscription in or circumscription about other figures

In every triangle a unique circle, chosen the incircle, can be inscribed such that it is tangent to each of the three sides of the triangle.^[fifteen]

About every triangle a unique circumvolve, called the circumcircle, tin exist confining such that it goes through each of the triangle'south iii vertices.^[sixteen]

A tangential polygon, such as a tangential quadrilateral, is any convex polygon within which a circle can be inscribed that is tangent to each side of the polygon.^[17] Every regular polygon and every triangle is a tangential polygon.

A circadian polygon is any convex polygon about which a circle can exist circumscribed, passing through each vertex. A well-studied example is the cyclic quadrilateral. Every regular polygon and every triangle is a cyclic polygon. A polygon that is both cyclic and tangential is called a bicentric polygon.

A hypocycloid is a bend that is inscribed in a given circle by tracing a stock-still betoken on a smaller circle that rolls within and tangent to the given circle.

Limiting case of other figures

The circle can exist viewed as a limiting case of each of various other figures:

• A Cartesian oval is a set of points such that a weighted sum of the distances from whatever of its points to two stock-still points (foci) is a abiding. An ellipse is the case in which the weights are equal. A circle is an ellipse with an eccentricity of nil, meaning that the two foci coincide with each other as the centre of the circle. A circle is also a unlike special case of a Cartesian oval in which one of the weights is naught.
• A superellipse has an equation of the class ${\displaystyle \left|{\frac {x}{a}}\right|^{n}\!+\left|{\frac {y}{b}}\right|^{n}\!=ane}$ for positive a, b, and n. A supercircle has b = a . A circumvolve is the special case of a supercircle in which n = 2.
• A Cassini oval is a fix of points such that the product of the distances from any of its points to two fixed points is a constant. When the two stock-still points coincide, a circumvolve results.
• A curve of abiding width is a figure whose width, defined as the perpendicular altitude between 2 distinct parallel lines each intersecting its purlieus in a single indicate, is the aforementioned regardless of the management of those two parallel lines. The circle is the simplest case of this blazon of figure.

In other p-norms

Illustrations of unit circles (see likewise superellipse) in different p -norms (every vector from the origin to the unit circle has a length of one, the length being calculated with length-formula of the corresponding p ).

Defining a circle as the set of points with a fixed distance from a point, different shapes can be considered circles under different definitions of distance. In p-norm, distance is determined by

${\displaystyle \left\|x\correct\|_{p}=\left(|x_{1}|^{p}+|x_{2}|^{p}+\dotsb +|x_{northward}|^{p}\correct)^{1/p}.}$

In Euclidean geometry, p = 2, giving the familiar

${\displaystyle \left\|x\right\|_{ii}={\sqrt {|x_{1}|^{ii}+|x_{2}|^{two}+\dotsb +|x_{due north}|^{2}}}.}$

In taxicab geometry, p = 1. Taxicab circles are squares with sides oriented at a 45° bending to the coordinate axes. While each side would have length ${\displaystyle {\sqrt {ii}}r}$ using a Euclidean metric, where r is the circle's radius, its length in taxicab geometry is 2r. Thus, a circle's circumference is 8r. Thus, the value of a geometric analog to ${\displaystyle \pi }$ is four in this geometry. The formula for the unit circle in taxicab geometry is ${\displaystyle |ten|+|y|=one}$ in Cartesian coordinates and

${\displaystyle r={\frac {1}{|\sin \theta |+|\cos \theta |}}}$

in polar coordinates.

A circle of radius 1 (using this altitude) is the von Neumann neighborhood of its center.

A circle of radius r for the Chebyshev altitude (L_∞ metric) on a plane is also a foursquare with side length 2r parallel to the coordinate axes, so planar Chebyshev distance tin exist viewed every bit equivalent by rotation and scaling to planar taxicab distance. However, this equivalence between L_ane and 50_∞ metrics does non generalize to higher dimensions.

Locus of constant sum

Consider a finite ready of ${\displaystyle n}$ points in the plane. The locus of points such that the sum of the squares of the distances to the given points is abiding is a circle, whose center is at the centroid of the given points.^[18] A generalization for higher powers of distances is obtained if under ${\displaystyle north}$ points the vertices of the regular polygon ${\displaystyle P_{north}}$ are taken.^[19] The locus of points such that the sum of the ${\displaystyle (2m)}$ -thursday power of distances ${\displaystyle d_{i}}$ to the vertices of a given regular polygon with circumradius ${\displaystyle R}$ is constant is a circle, if

${\displaystyle \sum _{i=1}^{north}d_{i}^{2m}>nR^{2m}}$ , where ${\displaystyle m}$ =ane,2,…, ${\displaystyle n}$ -1;

whose center is the centroid of the ${\displaystyle P_{n}}$ .

In the case of the equilateral triangle, the loci of the constant sums of the second and quaternary powers are circles, whereas for the square, the loci are circles for the constant sums of the 2nd, fourth, and 6th powers. For the regular pentagon the constant sum of the eighth powers of the distances volition be added and so forth.

Squaring the circle

Squaring the circumvolve is the problem, proposed by ancient geometers, of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge.

In 1882, the task was proven to be impossible, as a outcome of the Lindemann–Weierstrass theorem, which proves that pi (π) is a transcendental number, rather than an algebraic irrational number; that is, information technology is not the root of whatever polynomial with rational coefficients. Despite the impossibility, this topic continues to be of involvement for pseudomath enthusiasts.

Significance in art and symbolism

From the time of the earliest known civilisations – such equally the Assyrians and ancient Egyptians, those in the Indus Valley and forth the Yellow River in Red china, and the Western civilisations of ancient Greece and Rome during classical Antiquity – the circle has been used directly or indirectly in visual art to convey the artist'south message and to express certain ideas. Yet, differences in worldview (behavior and civilisation) had a great bear upon on artists' perceptions. While some emphasised the circle's perimeter to demonstrate their autonomous manifestation, others focused on its middle to symbolise the concept of cosmic unity. In mystical doctrines, the circle mainly symbolises the infinite and cyclical nature of existence, merely in religious traditions it represents heavenly bodies and divine spirits. The circle signifies many sacred and spiritual concepts, including unity, infinity, wholeness, the universe, divinity, balance, stability and perfection, among others. Such concepts have been conveyed in cultures worldwide through the use of symbols, for example, a compass, a halo, the vesica piscis and its derivatives (fish, middle, aureole, mandorla, etc.), the ouroboros, the Dharma bike, a rainbow, mandalas, rose windows and and then forth.^[20]

Encounter also

References

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3. ^ krikos Archived 2013-eleven-06 at the Wayback Automobile, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
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9. ^ Posamentier and Salkind, Challenging Problems in Geometry, Dover, 2d edition, 1996: pp. 104–105, #4–23.
10. ^ College Mathematics Journal 29(4), September 1998, p. 331, problem 635.
11. ^ Johnson, Roger A., Advanced Euclidean Geometry, Dover Publ., 2007.
12. ^ Harkness, James (1898). "Introduction to the theory of analytic functions". Nature. 59 (1530): 30. Bibcode:1899Natur..59..386B. doi:x.1038/059386a0. S2CID 4030420. Archived from the original on 2008-10-07.
13. ^ Ogilvy, C. Stanley, Excursions in Geometry, Dover, 1969, 14–17.
14. ^ Altshiller-Court, Nathan, College Geometry, Dover, 2007 (orig. 1952).
15. ^ Incircle – from Wolfram MathWorld Archived 2012-01-21 at the Wayback Automobile. Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.
16. ^ Circumcircle – from Wolfram MathWorld Archived 2012-01-20 at the Wayback Machine. Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.
17. ^ Tangential Polygon – from Wolfram MathWorld Archived 2013-09-03 at the Wayback Machine. Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.
18. ^ Apostol, Tom; Mnatsakanian, Mamikon (2003). "Sums of squares of distances in m-space". American Mathematical Monthly. 110 (half-dozen): 516–526. doi:10.1080/00029890.2003.11919989. S2CID 12641658.
19. ^ Meskhishvili, Mamuka (2020). "Cyclic Averages of Regular Polygons and Ideal Solids". Communications in Mathematics and Applications. 11: 335–355. arXiv:2010.12340.
20. ^ Abdullahi, Yahya (Oct 29, 2019). "The Circle from Due east to Westward". In Charnier, Jean-François (ed.). The Louvre Abu Dhabi: A World Vision of Art. Rizzoli International Publications, Incorporated. ISBN9782370741004.

Further reading

• Pedoe, Dan (1988). Geometry: a comprehensive course . Dover. ISBN9780486658124.
• "Circle" in The MacTutor History of Mathematics archive

External links

Wikiquote has quotations related to: Circles

• "Circle", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Circle at PlanetMath.
• Weisstein, Eric Westward. "Circle". MathWorld.
• "Interactive Java applets". for the properties of and unproblematic constructions involving circles
• "Interactive Standard Form Equation of Circle". Click and drag points to see standard form equation in action
• "Munching on Circles". cut-the-knot

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Source: https://en.wikipedia.org/wiki/Circle

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