conic section

conic section

Definitions

In

Euclidean

3-space, a

conic section

, or simply a conic, is the

intersection

of a plane with a

right circular double cone

.

But a conic can also defined, in several

equivalent

ways,
without using an enveloping 3-space.

In the

Euclidean plane

, let

d

be a line and

F

a point not on

d

. Let

ϵ

be a positive real number.
For an arbitrary point

P

, write

|

P

⁢

d

|

for the

perpendicular

(or shortest)

distance

from

P

to the line

d

.
The set of all points

P

such that

|

P

⁢

F

|

=

ϵ

⁢

|

P

⁢

d

|

is a conic
with

eccentricity

ϵ

,

focus

F

, and

directrix

d

.

An

ellipse

,

parabola

, or

hyperbola

has eccentricity

<

1

,

=

1

, or

>

1

respectively.
For a parabola, the focus and directrix are unique.
Any ellipse other than a circle, or any hyperbola, may be defined by
either of two focus-directrix pairs; the eccentricity is the same for both.

The definition in terms of a focus and a directrix leaves out the case
of a circle; still, the circle can be thought of as a limiting case:
eccentricity zero, directrix at

infinity

, and two

coincident

foci.

The chord through the given focus,

parallel

to the directrix,
is called the

latus rectum

; its length is traditionally
denoted by

2

⁢

l

.

Given a conic

σ

which is the intersection of
a circular cone

C

with a plane

π

, and given a focus

F

of

σ

,
there is a unique sphere

tangent

to

π

at

F

and tangent also to

C

at all points of a circle.
That sphere is called the

Dandelin sphere

for

F

.
(Consider a spherical ball resting on a table.
Suppose that a point source of light, at some point above the table
and outside the ball, shines on the ball.
The margin of the shadow of the ball is a conic,
the ball is one of the Dandelin spheres of that conic,
and the ball meets the table at the focus corresponding to that sphere.)

Degenerate conics;

coordinates

in 2 or 3

dimensions

The intersection of a plane with a cone may consist of a
single point, or a line, or a pair of lines.
Whether we should regard these sets as conics is a matter
of convention, but in general they are not so regarded.

In the Euclidean plane with the usual

Cartesian coordinates

,
a conic is the set of solutions of an equation of the form

P

⁢

(

x

,

y

)

=

0

where

P

is a polynomial of the second degree over

ℝ

.
For a degenerate conic,

P

has

discriminant

zero.

In three dimensions, if a conic is defined as the intersection
of the cone

z

2

=

x

2

+

y

2

with a plane

α

⁢

x

+

β

⁢

y

+

γ

⁢

z

=

δ

then, assuming

γ

≠

0

, we can eliminate

z

to get a
polynomial for the curve in terms of

x

and

y

only; a
linear change of variables will then give Cartesian coordinates,
within the plane, for the given conic.
If

γ

=

0

we can eliminate

x

or

y

instead, with the same
result.

Conics and physics

Kepler revolutionized astronomy by describing simple mathematical laws for planetary motion, which Newton then derived from his laws of motion using calculus, which he invented for the purpose (although several other people invented calculus at more or less the same time). One of Kepler’s laws was that planets move on ellipses with the Sun at one focus. More generally, objects under the influence of the Sun’s gravity (and no other forces) move in conic sections with the Sun at one focus. Comets moving on parabolae or hyperbolae do not return; comets moving on elongated ellipses return.

To work with conic sections in such an astronomical

context

, it is very useful to have a description in terms of

polar coordinates

centered at one focus. Measuring the angle

θ

from the closest approach to the focus, an ellipse with semi-major axis

a

can be described by

r

⁢

(

θ

)

=

a

⁢

(

1

-

ϵ

2

)

1

-

ϵ

⁢

cos

⁡

θ

.

A parabola or hyperbola can also be described in an essentially

similar

way, although they do not have a semi-major axis. If we call the length of the latus rectum

2

⁢

l

, then any conic can be described as

r

⁢

(

θ

)

=

l

1

-

ϵ

⁢

cos

⁡

θ

.

Observe that as

ϵ

goes to zero we approach a circle; as

ϵ

goes to

1

we approach a parabola; and for

ϵ

>

1

we obtain a hyperbola. (Normally one does not consider the case

ϵ

<

0

, as this simply amounts to choosing a different place to

measure

angle).

Conics in a

projective plane

Conic sections can be defined in a projective plane, even though,
in such a plane, there is no notion of angle nor any notion of distance.
In fact there are several equivalent ways to define it.
The following elegant definition
was discovered by von Staudt:
A conic is the set of self-conjugate points of a hyperbolic

polarity

.
In a little more detail, the polarity is a pair of

bijections

f

:

P

→

L

g

:

L

→

P

where

P

is the set of points of the plane,

L

is the set of lines,

f

maps

collinear points

to concurrent lines, and

g

maps concurrent lines
to collinear points.
The set of

fixed points

of

g

∘

f

is a conic, and

f

⁢

(

x

)

is the tangent
to the given conic at the given point

x

.

A projective conic has no focus, directrix, or eccentricity, for in a
projective plane there is no notion of distance (nor angle).
Indeed all projective conics are alike; there is no distinction between a parabola and a hyperbola, for example.

If we visualize the projective plane as the set of all lines in three-space, then a conic is precisely a cone. An

embedding

of the plane into the projective plane, in this context, is just the selection of a plane not

passing through

the origin. Then we identify a projective “point” (a line through the origin) with its intersection with this plane. Points whose line does not intersect this plane are considered

points at infinity

. With this

interpretation

, it is clear why all conic sections are essentially the same in the projective plane: if we choose a

plane cutting

the cone perpendicularly, we obtain a circle; if we tip the plane, we get an ellipse; as we tip it further, at a certain point there is exactly one point at infinity on the cone, giving a parabola; finally we get two branches,

connected

by two points at infinity, that is, a hyperbola. The only

difference

is how we choose to embed the plane in the projective plane.

Title

conic section

Canonical name

ConicSection

Date of creation

2013-03-22 13:08:45

Last modified on

2013-03-22 13:08:45

Owner

drini (3)

Last modified by

drini (3)

Numerical id

18

Author

drini (3)

Entry type

Definition

Classification

msc 51N20

Synonym

conic

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Hyperbola2

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Parabola2

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AnalyticGeometry

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IntersectionOfQuadraticSurfaceAndPlane

Related topic

SimplestCommonEquationOfConics

Related topic

PropertiesOfEllipse

Defines

ellipse

Defines

parabola

Defines

hyperbola

Defines

focus

Defines

eccentricity

Defines

directrix

Defines

latus rectum

Defines

Dandelin sphere

Defines

degenerate conic

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