circle

circle

A

circle

is the locus of points which are equidistant from some

fixed point

. It is in the plane is determined by a

center

and a

radius

. The center is a point in the plane,
and the length of the radius is a positive real number, the radius being a

line segment

from the center to the

circumference

. The circle consists of all points whose

distance

from the center equals the radius.

Another way of defining the circle is thus: Given

A

and

P

as two points and

O

as another point, the circle with center

O

is the locus of points

X

with

O

⁢

X

congruent

to

A

⁢

P

. (Hilbert, 1927)

Notice there is no definition of distance needed to make that definition
and so it works in many

geometries

, even ones with no distance function.
Hilbert uses it in his Foundations of Geometry book
Also used by Forder in his Foundations of

Euclidean geometry

book, c 1927

In this entry, we only work with the standard

Euclidean
norm

in the plane. A

circle has one center

only.

A circle determines a

closed curve

in the plane, and this curve is called the

perimeter

or

circumference

of the circle. If the radius of a circle is

r

, then the length of the
perimeter is

2

⁢

π

⁢

r

. Also, the area of the circle is

π

⁢

r

2

. More precisely, the interior of the perimeter has
area

π

⁢

r

2

. The

diameter

of a circle is defined as

d

=

2

⁢

r

.

The circle is a special case of an

ellipse

. Also, in three

dimensions

, the analogous geometric object to a circle
is a sphere.

1

Let us next derive an analytic equation for a circle in

Cartesian coordinates

(

x

,

y

)

. If the circle has center

(

a

,

b

)

and radius

r

>

0

, we obtain the following condition for the points of the sphere,

(

x

-

a

)

2

+

(

y

-

b

)

2

=

r

2

.

(1)

In other words, the circle is the set of all points

(

x

,

y

)

that satisfy the above equation. In the special case that

a

=

b

=

0

, the equation is simply

x

2

+

y

2

=

r

2

. The

unit circle

is the circle

x

2

+

y

2

=

1

.

It is clear that equation

1

can always be reduced to the form

x

2

+

y

2

+

D

⁢

x

+

E

⁢

y

+

F

=

0

,

(2)

where

D

,

E

,

F

are real numbers. Conversely, suppose that we are given an equation of the above form where

D

,

E

,

F

are
arbitrary real numbers. Next we derive conditions for these constants, so that equation (

2

) determines
a circle

[

2

]

. Completing the squares yields

x

2

+

D

⁢

x

+

D

2

4

+

y

2

+

E

⁢

y

+

E

2

4

=

-

F

+

D

2

4

+

E

2

4

,

whence

(

x

+

D

2

)

2

+

(

y

+

E

2

)

2

=

D

2

-

4

⁢

F

+

E

2

4

.

There are three cases:

1.

If

D

2

-

4

⁢

F

+

E

2

>

0

, then equation (

2

) determines
a circle with center

(

-

D

2

,

-

E

2

)

and radius

1

2

⁢

D

2

-

4

⁢

F

+

E

2

.

2.

If

D

2

-

4

⁢

F

+

E

2

=

0

, then equation (

2

) determines
the point

(

-

D

2

,

-

E

2

)

.

3.

If

D

2

-

4

⁢

F

+

E

2

<

0

, then equation (

2

) has no (real)
solution in the

(

x

,

y

)

- plane.

2

The circle in polar coordinates

Using

polar coordinates

for the plane, we can parameterize the circle. Consider the circle with center

(

a

,

b

)

and radius

r

>

0

in the plane

ℝ

2

. It is then natural to introduce polar coordinates

(

ρ

,

ϕ

)

for

ℝ

2

∖

{

(

a

,

b

)

}

by

x

⁢

(

ρ

,

ϕ

)

=

a

+

ρ

⁢

cos

⁡

ϕ

,

y

⁢

(

ρ

,

ϕ

)

=

b

+

ρ

⁢

sin

⁡

ϕ

,

with

ρ

>

0

and

ϕ

∈

[

0

,

2

⁢

π

)

. Since we wish to parameterize the circle, the point

(

a

,

b

)

does not pose a problem; it is not part of the circle. Plugging these expressions for

x

,

y

into
equation (

1

) yields the condition

ρ

=

r

.
The given circle is thus parameterization by

ϕ

↦

(

a

+

ρ

⁢

cos

⁡

ϕ

,

b

+

ρ

⁢

sin

⁡

ϕ

)

,

ϕ

∈

[

0

,

2

⁢

π

)

. It follows that a circle is a closed curve in the plane.

3

Three point formula for the circle

Suppose we are given three points on a circle, say

(

x

1

,

y

1

)

,

(

x

2

,

y

2

)

,

(

x

3

,

y

3

)

.
We next derive expressions for the

parameters

D

,

E

,

F

in terms of these points.
We also derive equation (

3

), which gives an equation for a circle in terms
of a

determinant

.

First, from equation (

2

), we have

x

1

2

+

y

1

2

+

D

⁢

x

1

+

E

⁢

y

1

+

F

=

0

,

x

2

2

+

y

2

2

+

D

⁢

x

2

+

E

⁢

y

2

+

F

=

0

,

x

3

2

+

y

3

2

+

D

⁢

x

3

+

E

⁢

y

3

+

F

=

0

.

These equations form a linear set of equations for

D

,

E

,

F

, i.e.,

(

x

1

y

1

1

x

2

y

2

1

x

3

y

3

1

)

⋅

(

D

E

F

)

=

-

(

x

1

2

+

y

1

2

x

2

2

+

y

2

2

x

3

2

+

y

3

2

)

.

Let us denote the matrix on the left hand side by

Λ

. Also, let us assume
that

det

⁡

Λ

≠

0

. Then, using Cramer’s rule, we obtain

D

=

-

1

det

⁡

Λ

⁢

det

⁡

(

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

)

,

E

=

-

1

det

⁡

Λ

⁢

det

⁡

(

x

1

x

1

2

+

y

1

2

1

x

2

x

2

2

+

y

2

2

1

x

3

x

3

2

+

y

3

2

1

)

,

F

=

-

1

det

⁡

Λ

⁢

det

⁡

(

x

1

y

1

x

1

2

+

y

1

2

x

2

y

2

x

2

2

+

y

2

2

x

3

y

3

x

3

2

+

y

3

2

)

.

These equations give the parameters

D

,

E

,

F

as

functions

of the
three given points.
Substituting these equations into equation (

2

) yields

(

x

2

+

y

2

)

⁢

det

⁡

(

x

1

y

1

1

x

2

y

2

1

x

3

y

3

1

)

-

⁡

x

⁢

det

⁡

(

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

)

-

⁡

y

⁢

det

⁡

(

x

1

x

1

2

+

y

1

2

1

x

2

x

2

2

+

y

2

2

1

x

3

x

3

2

+

y

3

2

1

)

-

⁡

det

⁡

(

x

1

y

1

x

1

2

+

y

1

2

x

2

y

2

x

2

2

+

y

2

2

x

3

y

3

x

3

2

+

y

3

2

)

=

0

.

Using the

cofactor expansion

, we can now write the equation for the circle

passing through

(

x

1

,

y

1

)

,

(

x

2

,

y

2

)

,

(

x

3

,

y

3

)

as

[

3

,

4

]

det

⁡

(

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

.

(3)

References

1

D. Hilbert,

Foundations of Geometry

Chicago: The Open Court Publishing Co. (1921): 163

2

J. H. Kindle,

Schaum’s Outline Series: Theory and problems of plane of Solid

Analytic Geometry

,
Schaum Publishing Co., 1950.

3

E. Weisstein, Eric W. Weisstein’s world of mathematics,

http://mathworld.wolfram.com/Circle.html

entry on the circle.

4

L. Råde, B. Westergren,

Mathematics Handbook for Science and Engineering

,
Studentlitteratur, 1995.

Title

circle

Canonical name

Circle

Date of creation

2013-03-22 13:36:23

Last modified on

2013-03-22 13:36:23

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

14

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 51-00

Synonym

circular

Related topic

SqueezingMathbbRn

Related topic

CurvatureOfACircle

Defines

unit circle

Defines

radius

Defines

radii

Defines

perimeter

Defines

circumference

Defines

three point formula for the circle

Defines

center

Generated on Fri Feb 9 21:50:02 2018 by

LaTeXML