Definition:Circle

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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 $15$)


Circle.png

Center

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 $A$.


Circumference

The circumference of a circle is the line that forms its boundary.


Diameter

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.


Radius

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.


Arc

An arc of a circle is any part of its circumference.


Semicircle

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 $18$)


Chord

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.


Equality

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 $1$)


Equidistant from Center

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 $5$)


Intersection with Cone

ConicSections-All.png

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 $\phi$.


ConicSections-Circle.png

Let $\phi = \dfrac \pi 2 - \theta$, 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 $r \left({\theta}\right) = R$
parametrically can be expressed as $x = R \cos t, y = R \sin t$.


  • Results about circles can be found here.


Sources