quadratic curves

quadratic curves

We want to determine the graphical representant of the general bivariate

quadratic equation

A

⁢

x

2

+

B

⁢

y

2

+

2

⁢

C

⁢

x

⁢

y

+

2

⁢

D

⁢

x

+

2

⁢

E

⁢

y

+

F

=

0

,

(1)

where

A

,

B

,

C

,

D

,

E

,

F

are known

real numbers

and

A

2

+

B

2

+

C

2

>

0

.

If

C

≠

0

, we will rotate the

coordinate system

, getting new

coordinate

axes

x

′

and

y

′

, such that the equation (1) transforms into a new one having no more the

mixed term

x

′

⁢

y

′

.  Let the

rotation

angle be

α

to the anticlockwise (positive)

direction

so that the

x

′

- and

y

′

-axes form the angles

α

and

α

+

90

∘

with the original

x

-axis, respectively.  Then there is the

x

=

x

′

⁢

cos

⁡

α

-

y

′

⁢

sin

⁡

α

y

=

x

′

⁢

sin

⁡

α

+

y

′

⁢

cos

⁡

α

between the new and old coordinates (see

rotation matrix

).  Substituting these expressions into (1) it becomes

M

⁢

x

′

⁣

2

+

N

⁢

y

′

⁣

2

+

2

⁢

P

⁢

x

′

⁢

y

′

+

2

⁢

G

⁢

x

′

+

2

⁢

H

⁢

y

′

+

F

=

0

,

(2)

where

{

M

=

A

⁢

cos

2

⁡

α

+

B

⁢

sin

2

⁡

α

+

C

⁢

sin

⁡

2

⁢

α

,

N

=

A

⁢

sin

2

⁡

α

+

B

⁢

cos

2

⁡

α

-

C

⁢

sin

⁡

2

⁢

α

,

2

⁢

P

=

(

B

-

A

)

⁢

sin

⁡

2

⁢

α

+

2

⁢

C

⁢

cos

⁡

2

⁢

α

.

(3)

It’s always possible to determine

α

such that

(

B

-

A

)

⁢

sin

⁡

2

⁢

α

=

-

2

⁢

C

⁢

cos

⁡

2

⁢

α

,  i.e. that

tan

⁡

2

⁢

α

=

2

⁢

C

A

-

B

for

A

≠

B

and

α

=

45

∘

for the case

A

=

B

.  Then the term

2

⁢

P

⁢

x

′

⁢

y

′

vanishes

in (2), which becomes, dropping out the apostrophes,

M

⁢

x

2

+

N

⁢

y

2

+

2

⁢

G

⁢

x

+

2

⁢

H

⁢

y

+

F

=

0

.

(4)

•

If none of the

coefficients

M

and

M

equal zero, one can remove the first degree terms of (4) by first writing it as

M

⁢

(

x

+

G

M

)

2

+

N

⁢

(

y

+

H

N

)

2

=

G

2

M

+

H

2

N

-

F

and then translating the origin to the point

(

-

G

M

,

-

H

N

)

,  when we obtain the equation of the form

M

⁢

x

2

+

N

⁢

y

2

=

K

.

(5)

If

M

and

M

have the same sign (

http://planetmath.org/SignumFunction

), then in that (5) could have a counterpart in the plane, the sign must be the same as the sign of

K

; then the counterpart is the

ellipse

(

http://planetmath.org/Ellipse2

)

x

2

(

|

K

/

M

|

)

2

+

y

2

(

|

K

/

N

|

)

2

=

1

.

If

M

and

N

have opposite signs and

K

≠

0

,  then the curve (5) correspondingly is one of the

hyperbolas

(

http://planetmath.org/Hyperbola2

)

x

2

(

|

K

/

M

|

)

2

-

y

2

(

|

K

/

N

|

)

2

=

±

1

,

which for

K

=

0

is reduced to a pair of

intersecting lines

.

•

If one of

M

and

N

, e.g. the latter, is zero, the equation (4) may be written

M

⁢

(

x

+

G

M

)

2

+

2

⁢

H

⁢

y

+

F

-

G

2

M

=

0

i.e.

M

⁢

(

x

+

G

M

)

2

+

2

⁢

H

⁢

(

y

+

M

⁢

F

-

G

2

2

⁢

H

⁢

M

)

=

0

.

Translating now the origin to the point

(

-

G

M

,

G

2

-

M

⁢

F

2

⁢

H

⁢

M

)

the equation changes to

M

⁢

x

2

+

2

⁢

H

⁢

y

=

0

.

(6)

For

H

≠

0

,  this is the equation

y

=

-

M

2

⁢

H

⁢

x

2

of a

parabola

, but for

H

=

0

,  of a

double line

x

2

=

0

.

The kind of the quadratic curve (1) can also be found out directly from this original form of the equation. Namely, from the formulae (3) between the old and the new coefficients one may derive the

connection

M

⁢

N

-

P

2

=

A

⁢

B

-

C

2

(7)

when one first adds and subtracts them obtaining

M

+

N

=

A

+

B

,

M

-

N

=

(

A

-

B

)

⁢

cos

⁡

2

⁢

α

+

2

⁢

C

⁢

sin

⁡

2

⁢

α

,

2

⁢

P

=

(

A

-

B

)

⁢

sin

⁡

2

⁢

α

+

2

⁢

C

⁢

cos

⁡

2

⁢

α

.

Two latter of these give

(

M

-

N

)

2

+

4

⁢

P

2

=

(

A

-

B

)

2

+

4

⁢

C

2

,

and when one subtracts this from the equation

(

M

+

N

)

2

=

(

A

+

B

)

2

,
the result is (7), which due to the choice of

α

is simply

M

⁢

N

=

A

⁢

B

-

C

2

.

(8)

Thus the curve

A

⁢

x

2

+

B

⁢

y

2

+

2

⁢

C

⁢

x

⁢

y

+

2

⁢

D

⁢

x

+

2

⁢

E

⁢

y

+

F

=

0

is, when it is real,

1.

for

A

⁢

B

-

C

2

>

0

an ellipse (

http://planetmath.org/Ellipse2

),

2.

for

A

⁢

B

-

C

2

<

0

a hyperbola (

http://planetmath.org/Hyperbola2

) or two intersecting lines,

3.

for

A

⁢

B

-

C

2

=

0

a parabola (

http://planetmath.org/Parabola2

) or a double line.

References

1

L. Lindelöf

:

Analyyttisen geometrian oppikirja

.  Kolmas painos.  Suomalaisen Kirjallisuuden Seura, Helsinki (1924).

Title

quadratic curves

Canonical name

QuadraticCurves

Date of creation

2013-03-22 17:56:26

Last modified on

2013-03-22 17:56:26

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Topic

Classification

msc 51N20

Synonym

graph of quadratic equation

Related topic

ConicSection

Related topic

TangentOfConicSection

Related topic

OsculatingCurve

Related topic

IntersectionOfQuadraticSurfaceAndPlane

Related topic

PencilOfConics

Related topic

SimplestCommonEquationOfConics

Defines

double line

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