Definition
In the words of
Euclid
:
- A
circle
is a
plane figure
contained
by one
line
such that all the
straight lines
falling upon it from one
point
among those lying within the
figure
are equal to one another;
(
The Elements
:
Book $\text{I}$
:
Definition $15$
)
In the words of
Euclid
:
- And the
point
is called the
center of the circle
.
(
The Elements
:
Book $\text{I}$
:
Definition $16$
)
In the above diagram, the
center
is the point $A$.
The
circumference
of a
circle
is the
line
that forms its
boundary
.
In the words of
Euclid
:
- A
diameter of the circle
is any
straight line
drawn through the
center
and terminated in both directions by the
circumference of the circle
, and such a
straight line
also
bisects
the
center
.
(
The Elements
:
Book $\text{I}$
:
Definition $17$
)
In the above diagram, the line $CD$ is a
diameter
.
A
radius
of a
circle
is a
straight line segment
whose
endpoints
are the
center
and the
circumference
of the
circle
.
In the above diagram, the line $AB$ is a
radius
.
An
arc
of a
circle
is any part of its
circumference
.
In the words of
Euclid
:
- A
semicircle
is the
figure
contained
by the
diameter
and the
circumference
cut off by it. And the
center
of the semicircle is the same as that of the
circle
.
(
The Elements
:
Book $\text{I}$
:
Definition $18$
)
A
chord
of a
circle
is a
straight line segment
whose
endpoints
are on the
circumference
of the
circle
.
In the diagram above, the lines $CD$ and $EF$ are both
chords
.
In the words of
Euclid
:
- Equal circles
are those the
diameters
of which are equal, or the
radii
of which are equal.
(
The Elements
:
Book $\text{III}$
:
Definition $1$
)
In the words of
Euclid
:
- In a
circle
straight lines
are said to be
equally distant from the center
when the
perpendiculars
drawn to them from the
center
are equal.
(
The Elements
:
Book $\text{III}$
:
Definition $4$
)
- And that
straight line
is said to be
at a greater
distance
on which the greater
perpendicular
falls.
(
The Elements
:
Book $\text{III}$
:
Definition $5$
)
Let $C$ be a
double napped
right circular cone
whose
base
is $B$.
Let $\theta$ be half the
opening angle
of $C$.
That is, let $\theta$ be the
angle
between the
axis
of $C$ and a
generatrix
of $C$.
Let a
plane
$D$
intersect
$C$.
Let $\phi$ be the
inclination
of $D$ to the
axis
of $C$.
Let $K$ be the
set
of
points
which forms the
intersection
of $C$ with $D$.
Then $K$ is a
conic section
, whose nature depends on $\phi$.
Let $\phi = \dfrac \pi 2$, thereby making $D$
perpendicular
to the
axis
of $C$.
Then $D$ and $B$ are
parallel
, and so $K$ is a
circle
.
Also see
- in
Cartesian coordinates
is $x^2 + y^2 = R^2$
- in
polar coordinates
is $\map r \theta = R$
- parametrically
can be expressed as $x = R \cos t, y = R \sin t$.
- Results about
circles
can be found
here
.
The
adjectival
form of the word
circle
is
circular
.
Sources