ellipse

ellipse

An

ellipse

that is centered at the origin is the curve in the plane determined by

(

x

a

)

2

+

(

y

b

)

2

=

1

,

(1)

where

a

,

b

>

0

.

Below is a graph of the ellipse

(

x

3

)

2

+

(

y

2

)

2

=

1

:

The

major axis

of an ellipse is the longest

line segment

whose

endpoints

are on the ellipse. The

minor axis

of an ellipse is the shortest line

segment

through the

midpoint

of the ellipse whose endpoints are on the ellipse.

In the first equation given above, if

a

=

b

, the ellipse reduces to a circle of radius

a

, whereas if

a

>

b

(as in the graph above),

a

is said to be the

major semi-axis

length and

b

the

minor semi-axis

length; i.e. (

http://planetmath.org/Ie

), the lengths of the major axis and minor axis are

2

⁢

a

and

2

⁢

b

, respectively.

More generally, given any two points

p

1

and

p

2

in the (

Euclidean

) plane and any

real number

r

, let

E

be the set of points

p

having the property that the sum of the

distances

from

p

to

p

1

and

p

2

is

r

; i.e.,

E

=

{

p

|

r

=

|

p

-

p

1

|

+

|

p

-

p

2

|

}

.

In terms of the geometric look of

E

, there are three possible scenarios for

E

:

E

=

∅

,

E

=

p

1

⁢

p

2

¯

, the line segment with end-points

p

1

and

p

2

, or

E

is an ellipse. Points

p

1

and

p

2

are called

foci

of the ellipse; the line segments connecting a point of the ellipse to the foci are the

focal radii

belonging to that point. When

p

1

=

p

2

and

r

>

0

,

E

is a circle. Under appropriate

linear transformations

(a

translation

followed by a

rotation

),

E

has an

algebraic

appearance expressed in (1).

In

polar coordinates

, the ellipse is parametrized as

x

⁢

(

t

)

=

a

⁢

cos

⁡

t

,

y

⁢

(

t

)

=

b

⁢

sin

⁡

t

,

t

∈

[

0

,

2

⁢

π

)

.

If

a

>

b

,  then

t

is the

eccentric anomaly

; i.e., the

polar angle

of the point on the circumscribed circle having the same

abscissa

as the point of the ellipse.

Properties

1.

If

a

>

b

,  the foci of the ellipse (1) are on the

x

-axis with distances

a

2

-

b

2

from the origin.  The constant sum of the of a point

p

is equal to

2

⁢

a

.

2.

The

normal line

of the ellipse at its point

p

halves the angle between the focal radii drawn from

p

.

3.

The area of an ellipse is

π

⁢

a

⁢

b

. (See this page (

http://planetmath.org/AreaOfPlaneRegion

).)

4.

The length of the

perimeter

of an ellipse can be expressed using an

elliptic integral

.

Eccentricity

By definition, the

eccentricity

ϵ

(

0

≤

ϵ

<

1

) of the ellipse is given by

ϵ

=

a

2

-

b

2

a

⋅

For

ϵ

=

0

, the ellipse reduces to a circle. Further,

b

=

a

⁢

1

-

ϵ

2

, and by assuming that foci are located on

x

-axis,

p

1

on

x

<

0

and

p

2

on

x

>

0

, then

|

O

-

p

1

|

=

|

O

-

p

2

|

=

ϵ

⁢

a

, where

O

⁢

(

0

,

0

)

is the origin of the rectangular coordinate system.

Polar equation of the ellipse

By translating the

y

-axis towards the focus

p

1

, we have

x

′

=

x

+

ϵ

⁢

a

,

y

′

=

y

,

but from (1) we get

(

x

′

-

ϵ

⁢

a

a

)

2

+

(

y

′

b

)

2

=

1

.

(2)

By using the

transformation

equations to polar coordinates

x

′

=

r

⁢

cos

⁡

θ

,

y

′

=

r

⁢

sin

⁡

θ

,

and through (2) we arrive at the

polar equation

r

(

θ

)

=

(

1

-

ϵ

2

)

⁢

a

1

-

ϵ

⁢

cos

⁡

θ

⋅

(3)

This equation allows us to determine some additional properties about the ellipse:

r

m

⁢

a

⁢

x

:=

r

⁢

(

0

)

=

(

1

+

ϵ

)

⁢

a

,

which is called the

𝑎𝑝ℎ𝑒𝑙𝑖𝑢𝑚

;

r

m

⁢

i

⁢

n

:=

r

⁢

(

π

)

=

(

1

-

ϵ

)

⁢

a

,

which is called the

𝑝𝑒𝑟𝑖ℎ𝑒𝑙𝑖𝑢𝑚

.

Hence, the general definition of the ellipse expressed above shows that

r

m

⁢

i

⁢

n

+

r

m

⁢

a

⁢

x

=

2

⁢

a

and also that the

arithmetic mean

r

m

⁢

i

⁢

n

+

r

m

⁢

a

⁢

x

2

=

a

corresponds to the major semi-axis, while the

geometric mean

r

m

⁢

i

⁢

n

⁢

r

m

⁢

a

⁢

x

=

b

corresponds to the minor semi-axis of the ellipse. Likewise, if

θ

ϵ

is the angle between the

polar axis

x

′

and the radial distance

|

B

-

p

1

|

, where

B

⁢

(

0

,

b

)

is the point of the ellipse over the

y

-axis, then we get the useful equation

cos

⁡

θ

ϵ

=

ϵ

.

Title

ellipse

Canonical name

Ellipse

Date of creation

2013-03-22 15:18:10

Last modified on

2013-03-22 15:18:10

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

33

Author

matte (1858)

Entry type

Definition

Classification

msc 53A04

Classification

msc 51N20

Classification

msc 51-00

Related topic

SqueezingMathbbRn

Related topic

Ellipsoid

Defines

major axis

Defines

minor axis

Defines

major semi-axis

Defines

minor semi-axis

Defines

focus

Defines

foci

Defines

aphelium

Defines

perihelium

Defines

eccentric anomaly

Defines

focal radius

Defines

focal radii

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