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

Given a second degree equation

1) F(x,y) = ax² + bxy + cy² + 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 cos² θ + 2 b sin θ cos θ + c sin² θ

            c' = a sin² θ - 2b sin θ cos θ + c cos² 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_c² + c'y_c² + 2(a'h + d')x_c + 2(c'k + e')y_c + f ' = 0

where

            f ' = G(h, k) = a'h² + c^'k² + 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_c² + c'y_c² + 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'h² + c^'k² + 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

            x² - 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'h² + 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_c² + 2e'y_c = 0

or, equivalently,

11)      x_c² + 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_c² - 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'h² + 2d'h + f .

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

            y² - 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^'k² + 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_c² + 2d'x_c = 0

or, equivalently,

17)      y_c² + 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_c² - 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^'k² + 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_c² + f ' = 0

or equivalently,

21)      x_c² + 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^'k² + 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_c² + f ' = 0

or equivalently,

23)      y_c² + 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'h² + 2d'h + f .

To find the origin of the x_c - y_c canonical coordinate system with respect to the original x - y coordinate system in the above five cases we use the coordinate transformation

     x = x' cos θ - y' sin θ

     y = x' sin θ + y' cos θ .

Reference.

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