Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematic

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Circle

DOWNLOAD Mathematica Notebook  Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematic_第1张图片

A circle is the set of points in a plane that are equidistant from a given point O. The distance r from the center is called the radius, and the point  is called the center. Twice the radius is known as the diameter d=2r. The angle a circle subtends from its center is a full angle, equal to 360 degrees or 2pi radians.

A circle has the maximum possible area for a given perimeter, and the minimum possible perimeter for a given area.

The perimeter C of a circle is called the circumference, and is given by

 C=pid=2pir.
(1)

This can be computed using calculus using the formula for arc length in polar coordinates,

 C=int_0^(2pi)sqrt(r^2+((dr)/(dtheta))^2)dtheta,
(2)

but since , this becomes simply

 C=int_0^(2pi)rdtheta=2pir.
(3)

The circumference-to-diameter ratio C/d for a circle is constant as the size of the circle is changed (as it must be since scaling a plane figure by a factor sincreases its perimeter by s), and d also scales by s. This ratio is denoted pi (pi), and has been proved transcendental.

Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematic_第2张图片

Knowing C/d, the area of the circle can be computed either geometrically or using calculus. As the number of concentric strips increases to infinity as illustrated above, they form a triangle, so

 A=1/2(2pir)r=pir^2.
(4)

This derivation was first recorded by Archimedes in Measurement of a Circle (ca. 225 BC).

Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematic_第3张图片

If the circle is instead cut into wedges, as the number of wedges increases to infinity, a rectangle results, so

 A=(pir)r=pir^2.
(5)

From calculus, the area follows immediately from the formula

 A=int_0^(2pi)dthetaint_0^rrdr=(2pi)(1/2r^2)=pir^2,
(6)

again using polar coordinates.

A circle can also be viewed as the limiting case of a regular polygon with inradius r and circumradius  as the number of sides n approaches infinity (a figure technically known as an apeirogon). This then gives the circumference as

C lim_(n->infty)2rntan(pi/n)=2pir
(7)
= lim_(n->infty)2Rnsin(pi/n)=2piR,
(8)

and the area as

= lim_(n->infty)nr^2tan(pi/n)=pir^2
(9)
= lim_(n->infty)1/2nR^2sin((2pi)/n)=piR^2,
(10)

which are equivalently since the radii r and R converge to the same radius as n->infty.

Unfortunately, geometers and topologists adopt incompatible conventions for the meaning of "n-sphere," with geometers referring to the number of coordinates in the underlying space and topologists referring to the dimension of the surface itself (Coxeter 1973, p. 125). As a result, geometers call the circumference of the usual circle the 2-sphere, while topologists refer to it as the 1-sphere and denote it S^1.

The circle is a conic section obtained by the intersection of a cone with a plane perpendicular to the cone's symmetry axis. It is also a Lissajous curve. A circle is the degenerate case of an ellipse with equal semimajor and semiminor axes (i.e., with eccentricity 0). The interior of a circle is called a disk. The generalization of a circle to three dimensions is called a sphere, and to n dimensions for n>=4 a hypersphere.

The region of intersection of two circles is called a lens. The region of intersection of three symmetrically placed circles (as in a Venn diagram), in the special case of the center of each being located at the intersection of the other two, is called a Reuleaux triangle.

In Cartesian coordinates, the equation of a circle of radius a centered on (x_0,y_0) is

 (x-x_0)^2+(y-y_0)^2=a^2.
(11)

In pedal coordinates with the pedal point at the center, the equation is

 pa=r^2.
(12)

The circle having P_1P_2 as a diameter is given by

 (x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0.
(13)

The parametric equations for a circle of radius a can be given by

x = acost
(14)
y asint.
(15)

The circle can also be parameterized by the rational functions

x = (1-t^2)/(1+t^2)
(16)
= (2t)/(1+t^2),
(17)

but an elliptic curve cannot.

Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematic_第4张图片

The plots above show a sequence of normal and tangent vectors for the circle.

The arc length s, curvature kappa, and tangential angle phi of the circle with radius a represented parametrically by (◇) and (◇) are

s(t) =
(18)
kappa(t) = 1/a
(19)
phi(t) = t/a.
(20)

The Cesàro equation is

 kappa=1/a.
(21)

In polar coordinates, the equation of the circle has a particularly simple form.

 r=a
(22)

is a circle of radius a centered at origin,

 r=2acostheta
(23)

is circle of radius a centered at , and

 r=2asintheta
(24)

is a circle of radius a centered on (0,a).

The equation of a circle passing through the three points (x_i,y_i) for i=1, 2, 3 (the circumcircle of the triangle determined by the points) is

 |x^2+y^2 x y 1; x_1^2+y_1^2 x_1 y_1 1; x_2^2+y_2^2 x_2 y_2 1; x_3^2+y_3^2 x_3 y_3 1|=0.
(25)

The center and radius of this circle can be identified by assigning coefficients of a quadratic curve

 ax^2+cy^2+dx+ey+f=0,
(26)

where a=c and b=0 (since there is no xy cross term). Completing the square gives

 a(x+d/(2a))^2+a(y+e/(2a))^2+f-(d^2+e^2)/(4a)=0.
(27)

The center can then be identified as

x_0 -d/(2a)
(28)
y_0 = -e/(2a)
(29)

and the radius as

 r=sqrt((d^2+e^2)/(4a^2)-f/a),
(30)

where

a |x_1 y_1 1; x_2 y_2 1; x_3 y_3 1|
(31)
d = -|x_1^2+y_1^2 y_1 1; x_2^2+y_2^2 y_2 1; x_3^2+y_3^2 y_3 1|
(32)
e = |x_1^2+y_1^2 x_1 1; x_2^2+y_2^2 x_2 1; x_3^2+y_3^2 x_3 1|
(33)
f = -|x_1^2+y_1^2 x_1 y_1; x_2^2+y_2^2 x_2 y_2; x_3^2+y_3^2 x_3 y_3|.
(34)

Four or more points which lie on a circle are said to be concyclic. Three points are trivially concyclic since three noncollinear points determine a circle.

In trilinear coordinates, every circle has an equation of the form

 (lalpha+mbeta+ngamma)(aalpha+bbeta+cgamma)+k(abetagamma+bgammaalpha+calphabeta)=0
(35)

with k!=0 (Kimberling 1998, p. 219).

The center alpha_0:beta_0:gamma_0 of a circle given by equation (35) is given by

alpha_0 = l+kcosA-ncosB-mcosC
(36)
beta_0 m-ncosA+kcosB-lcosC
(37)
gamma_0 = n-mcosA-lcosB+kcosC
(38)

(Kimberling 1998, p. 222).

In exact trilinear coordinates (alpha,beta,gamma), the equation of the circle passing through three noncollinear points with exact trilinear coordinates (alpha_1,beta_1,gamma_1), , and (alpha_3,beta_3,gamma_3) is

 |abetagamma+bgammaalpha+calphabeta alpha beta gamma; abeta_1gamma_1+bgamma_1alpha_1+calpha_1beta_1 alpha_1 beta_1 gamma_1; abeta_2gamma_2+bgamma_2alpha_2+calpha_2beta_2 alpha_2 beta_2 gamma_2; abeta_3gamma_3+bgamma_3alpha_3+calpha_3beta_3 alpha_3 beta_3 gamma_3|=0
(39)

(Kimberling 1998, p. 222).

An equation for the trilinear circle of radius R with center alpha_0:beta_0:gamma_0 is given by Kimberling (1998, p. 223).

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