Circle

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Circle
Circle-withsegments.svg
Circle illustration with circumference (C) in black, diameter (D) in cyan, radius (R) in red, and centre or origin (O) in magenta.
 
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Circle
Circle-withsegments.svg
Circle illustration with circumference (C) in black, diameter (D) in cyan, radius (R) in red, and centre or origin (O) in magenta.

A circle is a simple shape of Euclidean geometry that is the set of all points in a plane that are at a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius. It can also be defined as the locus of a point equidistant from a fixed point.

A circle is a simple closed curve which divides the plane into two regions: an interior and an exterior. In everyday use, 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 the former and the latter is called a disk.

A circle can be defined as the curve traced out by a point that moves so that its distance from a given point is constant.

A circle may also be defined as a special ellipse in which the two foci are coincident and the eccentricity is 0.


Terminology[edit]


Chord, secant, tangent, radius, and diameter
Arc, sector, and segment


History[edit]

The compass in this 13th-century manuscript is a symbol of God's act of Creation. Notice also the circular shape of the halo

The word "circle" derives from the Greek κίρκος (kirkos), itself a metathesis of the Homeric Greek κρίκος (krikos), meaning "hoop" or "ring".[1] The origins of the words "circus" and "circuit" are closely related.

Circular piece of silk with Mongol images
Circles in an old Arabic astronomical drawing.

The circle has been known since before the beginning of recorded history. Natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the wheel, 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 most medieval scholars, and many believed that there was something intrinsically "divine" or "perfect" that could be found in circles.[citation needed]

Some highlights in the history of the circle are:

Tughrul Tower from inside

Analytic results[edit]

Length of circumference[edit]

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

C = 2\pi r = \pi d.\,

Area enclosed[edit]

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

As proved by Archimedes, the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle's circumference and whose height equals the circle's radius,[4] which comes to π multiplied by the radius squared:

\mathrm{Area} = \pi r^2.\,

Equivalently, denoting diameter by d,

\mathrm{Area} = \frac{\pi d^2}{4} \approx 0{.}7854d^2,

that is, approximately 79 percent of the circumscribing square (whose side is of length d).

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

Equations[edit]

Cartesian coordinates[edit]

Circle of radius r = 1, centre (a, b) = (1.2, −0.5)

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

\left(x - a \right)^2 + \left( y - b \right)^2=r^2.

This equation, also known as Equation of the Circle, follows from the Pythagorean theorem applied to any point on the circle: as shown in the diagram to the right, the radius is the hypotenuse of a right-angled triangle whose other sides are of length xa and yb. If the circle is centred at the origin (0, 0), then the equation simplifies to

x^2 + y^2 = r^2.\!\

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

x = a+r\,\cos t,\,
y = b+r\,\sin t\,

where t is a parametric variable in the range 0 to 2π, interpreted geometrically as the angle that the ray from (ab) to (xy) makes with the x-axis. An alternative parametrisation of the circle is:

x = a + r \frac{1-t^2}{1+t^2}\,
y = b + r \frac{2t}{1+t^2}.\,

In this parametrisation, the ratio of t to r can be interpreted geometrically as the stereographic projection of the circle onto the line passing through the centre parallel to the x-axis.

In homogeneous coordinates each conic section with equation of a circle is of the form

ax^2+ay^2+2b_1xz+2b_2yz+cz^2 = 0.\,

It can be proven that a conic section is a circle exactly when it contains (when extended to the complex projective plane) the points I(1: i: 0) and J(1: −i: 0). These points are called the circular points at infinity.

Polar coordinates[edit]

In polar coordinates the equation of a circle is:

r^2 - 2 r r_0 \cos(\theta - \phi) + r_0^2 = a^2\,

where a is the radius of the circle, (r, \theta) is the polar coordinate of a generic point on the circle, and (r_0, \phi) is the polar coordinate of the centre of the circle (i.e., r0 is the distance from the origin to the centre 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 circle). For a circle centred at the origin, i.e. r0 = 0, this reduces to simply r = a. When r0 = a, or when the origin lies on the circle, the equation becomes

r = 2 a\cos(\theta - \phi).\,

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

r = r_0 \cos(\theta - \phi) + \sqrt{a^2 - r_0^2 \sin^2(\theta - \phi)},

the solution with a minus sign in front of the square root giving the same curve.

Complex plane[edit]

In the complex plane, a circle with a centre at c and radius (r) has the equation |z-c|^2 = r^2\,. In parametric form this can be written z = re^{it}+c.

The slightly generalised equation pz\overline{z} + gz + \overline{gz} = q for real p, q and complex g is sometimes called a generalised circle. This becomes the above equation for a circle with p = 1,\ g=-\overline{c},\ q=r^2-|c|^2, since |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[edit]

The tangent line through a point P on the circle is perpendicular to the diameter passing through P. If P = (x1, y1) and the circle has centre (a, b) and radius r, then the tangent line is perpendicular to the line from (a, b) to (x1, y1), so it has the form (x1a)x + (y1b)y = c. Evaluating at (x1, y1) determines the value of c and the result is that the equation of the tangent is

(x_1-a)x+(y_1-b)y = (x_1-a)x_1+(y_1-b)y_1\,

or

(x_1-a)(x-a)+(y_1-b)(y-b) = r^2.\!\

If y1b then the slope of this line is

\frac{dy}{dx} = -\frac{x_1-a}{y_1-b}.

This can also be found using implicit differentiation.

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

x_1x+y_1y = r^2,\!\

and its slope is

\frac{dy}{dx} = -\frac{x_1}{y_1}.

Properties[edit]

Chord[edit]

Sagitta[edit]

The sagitta is the vertical segment.
r=\frac{y^2}{8x}+ \frac{x}{2}.

Another proof of this result which relies only on two chord properties given above is as follows. Given a chord of length y and with sagitta of length x, since the sagitta intersects the midpoint of the chord, we know it is part of a diameter of the circle. Since the diameter is twice the radius, the "missing" part of the diameter is (2rx) in length. Using the fact that one part of one chord times the other part is equal to the same product taken along a chord intersecting the first chord, we find that (2rx)x = (y / 2)2. Solving for r, we find the required result.

Tangent[edit]

Theorems[edit]

Secant-secant theorem

Inscribed angles[edit]

Inscribed angle theorem

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


Circle of Apollonius[edit]

Apollonius' definition of a circle: d1 / d2 constant

Apollonius of Perga showed that a circle may also 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.[8][9] (The set of points where the distances are equal is the perpendicular bisector of A and B, a line.) That circle is sometimes said to be drawn about two points.

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

\frac{AP}{BP} = \frac{AC}{BC}.

Analogously, a line segment PD through some point D on AB extended bisects the corresponding exterior angle BPQ where Q is on AP extended. Since the interior and exterior angles sum to 180 degrees, the angle CPD is exactly 90 degrees, i.e., a right angle. The set of points P such that angle CPD is a right angle forms a circle, of which CD is a diameter.

Second, see[10]:p.15 for a proof that every point on the indicated circle satisfies the given ratio.

Cross-ratios[edit]

A closely related property of circles involves the geometry of the cross-ratio of points in the complex plane. If A, B, and C are as above, then the circle of Apollonius for these three points is the collection of points P for which the absolute value of the cross-ratio is equal to one:

|[A,B;C,P]| = 1.\

Stated another way, P is a point on the circle of Apollonius if and only if the cross-ratio [A,B;C,P] is on the unit circle in the complex plane.

Generalised circles[edit]

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

\frac{|AP|}{|BP|} = \frac{|AC|}{|BC|} 

is not a circle, but rather a line.

Thus, if A, B, and C are given distinct points in the plane, then the locus of points P satisfying the above equation is called a "generalised circle." It may either be a true circle or a line. In this sense a line is a generalised circle of infinite radius.

Circles inscribed in or circumscribed about other figures[edit]

In every triangle a unique circle, called the incircle, can be inscribed such that it is tangent to each of the three sides of the triangle.[11]

About every triangle a unique circle, called the circumcircle, can be circumscribed such that it goes through each of the triangle's three vertices.[12]

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.[13]

A cyclic polygon is any convex polygon about which a circle can be circumscribed, passing through each vertex. A well-studied example is the cyclic quadrilateral.

A hypocycloid is a curve that is inscribed in a given circle by tracing a fixed point on a smaller circle that rolls within and tangent to the given circle.

Circle as limiting case of other figures[edit]

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

See also[edit]

References[edit]

  1. ^ krikos, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus
  2. ^ Chronology for 30000 BC to 500 BC. History.mcs.st-andrews.ac.uk. Retrieved on 2012-05-03.
  3. ^ Squaring the circle. History.mcs.st-andrews.ac.uk. Retrieved on 2012-05-03.
  4. ^ Measurement of a Circle by Archimedes
  5. ^ Posamentier and Salkind, Challenging Problems in Geometry, Dover, 2nd edition, 1996: pp. 104–105, #4–23.
  6. ^ College Mathematics Journal 29(4), September 1998, p. 331, problem 635.
  7. ^ Johnson, Roger A., Advanced Euclidean Geometry, Dover Publ., 2007.
  8. ^ Harkness, James (1898). Introduction to the theory of analytic functions. London, New York: Macmillan and Co. p. 30. 
  9. ^ Ogilvy, C. Stanley, Excursions in Geometry, Dover, 1969, 14–17.
  10. ^ Altshiller-Court, Nathan, College Geometry, Dover, 2007 (orig. 1952).
  11. ^ Incircle – from Wolfram MathWorld. Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.
  12. ^ Circumcircle – from Wolfram MathWorld. Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.
  13. ^ Tangential Polygon – from Wolfram MathWorld. Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.

Further reading[edit]

External links[edit]