Attach a string to a point on a curve. Extend the string so that it is tangent to the curve at the point of attachment. Then wind the string up, keeping it always taut. The locus of points traced out by the end of the string is called the involute of the original curve, and the original curve is called the evolute of its involute. This process is illustrated above for a circle.

Although a curve has a unique evolute, it has infinitely many involutes corresponding to different choices of initial point. An involute can also be thought of as any curve orthogonal to all the tangents to a given curve.

The equation of the involute is

(1)

where is the tangent vector

(2)

and is the arc length

(3)

This can be written for a parametrically represented function as

			(4)
			(5)

The following table lists the involutes of some common curves, some of which are illustrated above.

curve	involute
astroid involute	astroid 1/2 times as large
cardioid involute	cardioid 3 times as large
catenary involute	tractrix
circle catacaustic	limaçon
circle involute	a spiral
cycloid involute	equal cycloid
deltoid involute	deltoid 1/3 times as large
ellipse involute	unnamed curve
epicycloid involute	smaller epicycloid
hypocycloid involute	similar hypocycloid
logarithmic spiral involute	another logarithmic spiral
nephroid involute	Cayley's sextic or nephroid 2 times as large
semicubical parabola involute	half a parabola

REFERENCES:

Cundy, H. and Rollett, A. "Roulettes and Involutes." §2.6 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 46-55, 1989.

Dixon, R. "String Drawings." Ch. 2 in Mathographics. New York: Dover, pp. 75-78, 1991.

Gray, A. "Involutes." §5.4 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 103-107, 1997.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 40-42 and 202, 1972.

Lockwood, E. H. "Evolutes and Involutes." Ch. 21 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 166-171, 1967.

Pappas, T. "The Involute." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 187, 1989.

Yates, R. C. "Involutes." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 135-137, 1952

بر گرفته از سایت mathworld.wolfram.com

For a unit circle with parametric equations

			(1)
			(2)

the negative pedal curve with respect to the pedal point is

			(3)
			(4)

Therefore if the point is inside the circle (), the negative pedal is an ellipse, if , it is a single point, if the point is outside the circle (), the negative pedal is a hyperbola.

REFERENCES:

Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, p. 157, 1967.

CITE THIS AS:

Eric W. Weisstein. "Circle Negative Pedal Curve." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CircleNegativePedalCurve.html

© 1999 CRC Press LLC, © 1999-2005 Wolfram Research, Inc. | Terms of Use

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

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

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

(1)

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

(2)

but since , this becomes simply

(3)

The circumference-to-diameter ratio 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 increases its perimeter by ), and also scales by . This ratio is denoted (pi), and has been proved transcendental.

Knowing , 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

(4)

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

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

(5)

From calculus, the area follows immediately from the formula

(6)

again using polar coordinates.

A circle has also be viewed as the limiting case of a regular polygon with inradius and circumradius as the number of sides approaches infinity. This then gives the circumference as

			(7)
			(8)

and the area as

			(9)
			(10)

which are equivalently since the radii and converge to the same radius as .

Unfortunately, geometers and topologists adopt incompatible conventions for the meaning of "-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 sphere the 2-sphere, while topologists refer to it as the 1-sphere and denote it .

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 dimensions for 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 centered on is

(11)

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

(12)

The circle having as a diameter is given by

(13)

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

			(14)
			(15)

The circle can also be parameterized by the rational functions

			(16)
			(17)

but an elliptic curve cannot.

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

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

			(18)
			(19)
			(20)

The Cesàro equation is

(21)

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

(22)

is a circle of radius centered at origin,

(23)

is circle of radius centered at , and

(24)

is a circle of radius centered on .

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

(25)

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

(26)

where and (since there is no cross term). completing the square gives

(27)

The center can then be identified as

			(28)
			(29)

and the radius as

(30)

where

			(31)
			(32)
			(33)
			(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

(35)

with (Kimberling 1998, p. 219).

The center of a circle given by equation (◇) is given by

			(36)
			(37)
			(38)

(Kimberling 1998, p. 222).

In exact trilinear coordinates , the equation of the circle passing through three noncollinear points with exact trilinear coordinates , , and is

(39)

(Kimberling 1998, p. 222).

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

REFERENCES:

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 125 and 197, 1987.

Casey, J. "The Circle." Ch. 3 in A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 96-150, 1893.

Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, 1971.

Courant, R. and Robbins, H. What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 74-75, 1996.

Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973.

Coxeter, H. S. M. and Greitzer, S. L. "Some Properties of the Circle." Ch. 2 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 27-50, 1967.

Dunham, W. "Archimedes' Determination of Circular Area." Ch. 4 in Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 84-112, 1990.

Eppstein, D. "Circles and Spheres." http://www.ics.uci.edu/~eppstein/junkyard/sphere.html.

Hilbert, D. and Cohn-Vossen, S. Geometry and the Imagination. New York: Chelsea, p. 1, 1999.

Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 3, 1948.

Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

Lachlan, R. "The Circle." Ch. 10 in An Elementary Treatise on Modern Pure Geometry. London: Macmillian, pp. 148-173, 1893.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 65-66, 1972.

MacTutor History of Mathematics Archive. "Circle." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Circle.html.

Pappas, T. "Infinity & the Circle" and "Japanese Calculus." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 68 and 139, 1989.

Pedoe, D. Circles: A Mathematical View, rev. ed. Washington, DC: Math. Assoc. Amer., 1995.

Yates, R. C. "The Circle." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 21-25, 1952