SolitaryRoad.com

Website owner: James Miller

[ Home ] [ Up ] [ Info ] [ Mail ]

REDUCTION OF A GENERAL SECOND DEGREE EQUATION TO ONE OF THE 9 CANONICAL FORMS

Given a second degree equation

1) F(x,y) = ax^{2} + 2bxy + cy^{2} + 2dx + 2ey + f = 0

we wish to reduce it to canonical form. It may represent any one of the following 9 conics:

Our equation represents some conic as referred to an x -y coordinate system. See Figure 1. Here

a conic (an
ellipse in this case) is shown located at some point in
the plane. Figure 1 shows the original x-y coordinate
system along with two other coordinate systems – an
intermediate x'- y' coordinate system and the x_{c }- y_{c}
canonical coordinate system. We wish to know the
location and orientation of the canonical coordinate
system and the exact equation of the conic as referred
to that system. Figure 1 shows the x'- y' coordinate
system as a system obtained by rotating the x-y
system by θ degrees about its origin. θ represents the
rotation required to eliminate the xy term in the original equation as computed from the formula

A rotation of this amount will put its axes parallel to the axes of the canonical system. Upon performing the rotation the equation of the conic in the x'-y' system becomes

2) G(x',y') = a'x'^{2} + c'y^{'2} + 2d'x' + 2e'y' + f = 0

where

a' = a cos^{2} θ + 2 b sin θ cos θ + c sin^{2} θ

c' = a sin^{2} θ - 2b sin θ cos θ + c cos^{2} θ

d' = d cos θ + e sin θ

e' = e cos θ - d sin θ

Let us now assume that the origin of the x_{c }- y_{c} canonical coordinate system is located at
coordinates (h, k) of the x'-y' system and let us do a translation to the x_{c }- y_{c} system. The
equation of the conic in the x_{c }- y_{c} system would then be given by

3) a'(x_{c} + h)^{2} + c'(y_{c} + k)^{2} + 2d'(x_{c} + h) + 2e'(y_{c} + k) + f = 0

or, on removing parenthesis and collecting similar terms,

4) a'x_{c}^{2} + c'y_{c}^{2} + 2(a'h + d')x_{c} + 2(c'k + e')y_{c} + f ' = 0

where

f ' = G(h, k) = a'h^{2} + c^{'}k^{2} + 2d'h + 2e'k + f

We consider the following five cases:

Case I. Neither a' nor c' of equation 4) are zero. This case corresponds to canonical forms 1 - 5. Taking

h = - d'/a'

k = - e'/c'

we eliminate the terms in x_{c} and y_{c} and obtain the equation

5) a'x_{c}^{2} + c'y_{c}^{2} + f ' = 0

If f ' 0 , we can rewrite 5) as

6)

which is one of the canonical forms 1, 2 or 4.

If f ' = 0 , equation 5) can be written as

7)

which corresponds to canonical form 3 or 5.

The origin of the x_{c }- y_{c} canonical coordinate system is at location (h, k) of the x'-y' system where

h = - d'/a'

k = - e'/c' .

The value of f ' is given by

f ' = a'h^{2} + c^{'}k^{2} + 2d'h + 2e'k + f

Case II. a' 0, c' = 0, e' 0 in equation 4). This case corresponds to a parabola of the form

x^{2} - 4py = 0 .

We will translate the x'-y' system to its final position in a two-step process, first translating in the
x' direction to eliminate the x_{c} term of equation 4), and then translating in the y' direction. For
the first translation let

h = - d'/a'

k = 0

and equation 4) becomes

8) a'x''^{2} + 2e'y'' + f ' = 0

or, equivalently,

9) a'x''^{2} + 2e'(y'' + f ' /2e') = 0

where

f ' = a'h^{2} + 2d'h + f

h = - d'/a' .

Equation 9) is the equation of the conic as referred to an x''- y'' coordinate system with origin at point (- d'/a', 0) of the x'- y' system and axes parallel to the corresponding x'-y' axes.

Now we substitute

x_{c} = x''

y_{c} = y'' + f '/2e'

into equation 9), an act which is equivalent to a translation in the y'' direction by an amount

k' = - f '/2e' , a translation that puts us into the x_{c }- y_{c} canonical coordinate system .

This gives

10) a'x_{c}^{2} + 2e'y_{c} = 0

or, equivalently,

11) x_{c}^{2} + 2e'y_{c}/a' = 0

which is the equation of our parabola in canonical form.

Letting

12) p = - e'/2a'

equation 11) can be written as

13) x_{c}^{2} - 4py_{c} = 0 ,

the equation of a parabola in canonical form.

The origin of the x_{c }- y_{c} canonical coordinate system is at location (h, k') of the x'-y' system where

h = - d'/a'

k' = - f '/2e'

and

f ' = a'h^{2} + 2d'h + f .

Case III. a' = 0, c' 0, d' 0 in equation 4). This case corresponds to a parabola of the form

y^{2} - 4px = 0 .

As in the previous case we will translate the x'- y' system to its final position in a two-step
process. We will first translate in the y' direction to eliminate the y_{c} term of equation 4), and then
translate in the x' direction. For the first translation let

h = 0

k = - e'/c'

and equation 4) becomes

14) c'y''^{2} + 2d'x'' + f ' = 0

or, equivalently,

15) c'y''^{2} + 2d'(x'' + f ' /2d') = 0

where

f ' = c^{'}k^{2} + 2e'k + f

k = - e'/c' .

Equation 15) is the equation of the conic as referred to an x''- y'' coordinate system with origin at point (0, - e'/c') of the x'- y' system and axes parallel to the corresponding x'-y' axes.

Now we substitute

x_{c} = x'' + f '/2d'

y_{c} = y''

into equation 15), an act which is equivalent to a translation in the x'' direction by an amount

h' = - f '/2d' , a translation that puts us into the x_{c }- y_{c} canonical coordinate system .

This gives

16) c'y_{c}^{2} + 2d'x_{c} = 0

or, equivalently,

17) y_{c}^{2} + 2d'y_{c}/c' = 0

which is the equation of our parabola in canonical form.

Letting

18) p = - d'/2c'

equation 11) can be written as

19) y_{c}^{2} - 4px_{c} = 0 ,

the equation of a parabola in canonical form.

The origin of the x_{c }- y_{c} canonical coordinate system is at location (h', k) of the x'-y' system where

h' = - f '/2d'

k = - e'/c'

and

f ' = c^{'}k^{2} + 2e'k + f

Case IV. a' 0, c' = 0, e' = 0 in equation 4). This case corresponds to one of the canonical forms 7, 8 or 9. Taking

h = - d'/a'

k = 0

we obtain the equation

20) a'x_{c}^{2} + f ' = 0

or equivalently,

21) x_{c}^{2} + f '/a' = 0

which corresponds to one of the equations 7, 8 or 9.

The origin of the x_{c }- y_{c} canonical coordinate system is at location (h, k) of the x'-y' system where

h = - d'/a'

k = 0

and

f ' = c^{'}k^{2} + 2e'k + f .

Case V. a' = 0, c' 0, d' = 0 in equation 4). This case corresponds to one of the canonical forms 7, 8 or 9. Taking

h = 0

k = - e'/c'

we obtain the equation

22) c'y_{c}^{2} + f ' = 0

or equivalently,

23) y_{c}^{2} + f '/c' = 0

which corresponds to one of the canonical forms 7, 8 or 9.

The origin of the x_{c }- y_{c} canonical coordinate system is at location (h, k) of the x'-y' system where

h = 0

k = - e'/c'

and

f ' = a'h^{2} + 2d'h + f .

Reference.

Mathematics, Its Content, Methods and Meaning. Vol. I, p. 210 - 213

More from SolitaryRoad.com:

Jesus Christ and His Teachings

Way of enlightenment, wisdom, and understanding

America, a corrupt, depraved, shameless country

On integrity and the lack of it

The test of a person's Christianity is what he is

Ninety five percent of the problems that most people have come from personal foolishness

Liberalism, socialism and the modern welfare state

The desire to harm, a motivation for conduct

On Self-sufficient Country Living, Homesteading

Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.

Theory on the Formation of Character

People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest

Cause of Character Traits --- According to Aristotle

We are what we eat --- living under the discipline of a diet

Avoiding problems and trouble in life

Role of habit in formation of character

Personal attributes of the true Christian

What determines a person's character?

Love of God and love of virtue are closely united

Intellectual disparities among people and the power in good habits

Tools of Satan. Tactics and Tricks used by the Devil.

The Natural Way -- The Unnatural Way

Wisdom, Reason and Virtue are closely related

Knowledge is one thing, wisdom is another

My views on Christianity in America

The most important thing in life is understanding

We are all examples --- for good or for bad

Television --- spiritual poison

The Prime Mover that decides "What We Are"

Where do our outlooks, attitudes and values come from?

Sin is serious business. The punishment for it is real. Hell is real.

Self-imposed discipline and regimentation

Achieving happiness in life --- a matter of the right strategies

Self-control, self-restraint, self-discipline basic to so much in life

[ Home ] [ Up ] [ Info ] [ Mail ]