- Purpose

- Reduction in the case of delta not zero.

- Reduction with delta = zero.

- Classification of all affine conic sections

F(x,y,1) = a x^{2}+ 2 b" x y + a' y^{2}+ 2 b' x + 2 b y + a" = 0

x = x' + xIn the new system of axes the equation is_{o}y = y' + y_{o}

a (x' + xTo simplify this, we choose x_{o})^{2}+ 2 b" (x' + x_{o}) (y' + y_{o}) + a' (y' + y_{o})^{2}+ 2 b' (x' + x_{o}) + 2 b (y' + y_{o}) + a" = 0 <=> a x'^{2}+ 2 b" x' y' + a' y'^{2}+ 2(a x_{o}+ b" y_{o}+ b')x' + 2( b" x_{o}+ a' y_{o}+ b)y' + a x_{o}^{2}+ 2 b" x_{o}y_{o}+ a' y_{o}^{2}+ 2 b' x_{o}+ 2 b y_{o}+ a" = 0

/ a xSince delta is not zero, this system has always one unique solution._{o}+ b" y_{o}+ b' = 0 \ b" x_{o}+ a' y_{o}+ b = 0 <=> x_{o}and y_{o}is a solution of the system / F_{x}' (x,y,z) = 0 \ F_{y}' (x,y,z) = 0

With these special values of x_{o} and y_{o}, the equation of the conic section
becomes

a x'^{2}+ 2 b" x' y' + a' y'^{2}+ + a x_{o}+ 2 b" x_{o}y_{o}+ a' y_{o}+ 2 b' x_{o}+ 2 b y_{o}+ a" = 0 <=> a x'^{2}+ 2 b" x' y' + a' y'^{2}+ F(x_{o},y_{o},1) = 0

a xTo reduce this further we use a rather tricky method.^{2}+ 2 b" x y + a' y^{2}+ a" = 0

(From a total different approach of the problem, this method is not tricky at all. See later)

Consider the linear transformation of R x R with matrix

[a b"] [b" a']We calculate the eigenvalues and characteristic vectors. (see Linear transformations )

The eigenvalues are the roots of

|a - r b"| |b" a' - r| = 0 <=> (a - r)(a' - r) - b"b" = 0 <=> rThe discriminant of this quadratic equation is never < 0. Say r^{2}-(a+a')r + aa' -b"^{2}= 0 <=> 2 r -(a+a')r + delta = 0

Denote (cos(t),sin(t)) as coordinates of the characteristic unit vector corresponding with r

[ a b"][cos(t)] [cos(t)] [ b" a'][sin(t)] = rNow we rotate the coordinate system by an angle = t. The transformation matrix is_{1}[sin(t)] <=> acos(t) + b"sin(t) = r_{1}cos(t) b"cos(t) + a'sin(t) = r_{1}sin(t)

[cos(t) -sin(t) 0] M = [sin(t) cos(t) 0] [ 0 0 1]The new cubic matrix C1 of the conic section is given by the formula

T C1 = M C M <=> [cos(t) sin(t) 0][ a b" 0 ][cos(t) -sin(t) 0] [-sin(t) cos(t) 0][ b" a' 0 ][sin(t) cos(t) 0] [ 0 0 1][ 0 0 a"][ 0 0 1] <=> [cos(t) sin(t) 0][acos(t) + b"sin(t) -asin(t)+b"cos(t) 0 ] [-sin(t) cos(t) 0][b"cos(t) + a'sin(t) -b"sin(t)+a'cos(t) 0 ] [ 0 0 1][ 0 0 a"] <=> [cos(t) sin(t) 0][rThe elements marked by (*) are rather difficult to calculate. Here you see how it is done:_{1}cos(t) -asin(t)+b"cos(t) 0 ] [-sin(t) cos(t) 0][r_{1}sin(t) -b"sin(t)+a'cos(t) 0 ] [ 0 0 1][ 0 0 a"] <=> [r_{1}0 (*) 0 ] [0 r_{2}(*) 0 ] [0 0 a"]

-a cos(t) sin(t) + b" cosThe equation of the conic section in the new coordinate system is^{2}(t) - b" sin^{2}(t) + a' sin(t) cos(t) = - sin(t) (a cos(t) + b" sin(t) ) + cos(t) (b" cos(t) + a' sin(t) ) = - sin(t) r_{1}cos(t) + cos(t) r_{1}sin(t) = 0 and a sin^{2}(t) - b" sin(t) cos(t) - b" sin(t) cos(t) + a' cos^{2}(t) = a(1 - cos^{2}(t)) - b"sin(t)cos(t) - b"sin(t) cos(t) + a'(1 - sin^{2}(t) ) = a + a' - (a cos^{2}(t) + b" sin(t)cos(t) + b"sin(t)cos(t) + a' sin^{2}(t)) = a + a' - (cos(t)(a cos(t) + b"sin(t))+ sin(t)(b"cos(t) + a'sin(t))) = a + a' - r_{1}= r_{1}+ r_{2}- r_{1}= r_{2}

rThis is the reduced equation of the conic section._{1}x'^{2}+ r_{2}y'^{2}+ a" = 0

3 xWe start with translation of the axes to the point (x^{2}+ 2xy + 3 y^{2}-32 y + 92 = 0

xAfter the translation the equation becomes_{o}and y_{o}is a solution of the system / F_{x}' (x,y,z) = 0 \ F_{y}' (x,y,z) = 0 <=> / 6 x + 2 y = 0 \ 2 x + 6 y = 32 <=> x = -2 and y = 6

a x'This is the equation of the conic section after the translation to (-2,4). Now we start with the rotation from the equation^{2}+ 2 b" x' y' + a' y'^{2}+ F(x_{o},y_{o},1) = 0 <=> 3 x'^{2}+ 2x'y' + 3 y'^{2}- 4 = 0

3 xThe eigenvalues are the roots of^{2}+ 2xy + 3 y^{2}- 4 = 0

|a - r b"| |b" a' - r| = 0 <=> |3-r 1| | 1 3-r| = 0 <=> rThe equation of the conic section after the rotation is^{2}- 6 r + 8 = 0 <=> r_{1}= 4 and r_{2}= 2

rIf you want to calculate the rotation angle t :_{1}x'^{2}+ r_{2}y'^{2}+ a" = 0 <=> 4 x'^{2}+ 2 y'^{2}- 4 = 0 <=> x'^{2}y'^{2}----- + ----- = 1 1 2 and this is an ellipse

[ a b"][cos(t)] [cos(t)] [ b" a'][sin(t)] = r_{1}[sin(t)] <=> acos(t) + b"sin(t) = r_{1}cos(t) b"cos(t) + a'sin(t) = r_{1}sin(t) <=> 3 cos(t) + sin(t) = 4 cos(t) cos(t) + 3 sin(t) = 4 sin(t) <=> cos(t) - sin(t) = 0 <=> tan(t) = 1 choose t = pi/4

F(x,y,1) = a x^{2}+ 2 b" x y + a' y^{2}+ 2 b' x + 2 b y + a" = 0

[a b"] [b" a']We calculate the eigenvalues and characteristic vectors. The eigenvalues are the roots of

|a - r b"| |b" a' - r| = 0 <=> (a - r)(a' - r) - b"b" = 0 <=> rDenote (cos(t),sin(t)) as coordinates of the characteristic unit vector corresponding with r^{2}-(a+a')r + aa' -b"^{2}= 0 <=> r^{2}-(a+a')r + delta = 0 <=> r^{2}-(a+a')r = 0 <=> r_{1}= 0 and r_{2}= a+a'

[ a b"][cos(t)] [cos(t)] [ b" a'][sin(t)] = rNow we rotate the coordinate system by an angle = t. The transformation matrix is_{1}[sin(t)] <=> acos(t) + b"sin(t) = 0 b"cos(t) + a'sin(t) = 0

[cos(t) -sin(t) 0] M = [sin(t) cos(t) 0] [ 0 0 1]The new cubic matrix C1 of the conic section is given by the formula

T C1 = M C MIn the same way as in the case delta not zero, you can show that

[0 0 * ] C1 = [0 * * ] [* * * ]The * stand for real numbers.

The equation of the conic section in the new coordinate system is of the form

a'yFrom this form we start with a translation^{2}+ 2 b' x + 2 b y + a" = 0

a'yIf you calculate DELTA, you'll find^{2}+ 2 b' x + 2 b y + a" = 0

DELTA = - a' b'Since the conic section is not degenerated, a' and b' are not zero.^{2}

We translate the axes to the point (x

x = x' + xIn the new system of axes the equation is_{o}y = y' + y_{o}

a'(y' + yBecause a' and b' are not zero, we can always choose x_{o})^{2}+ 2 b' (x' + x_{o}) + 2 b (y' + y_{o}) + a" = 0 <=> a' y'^{2}+ 2 b'x' + 2(a' y_{o}+ b)y' + a' y_{o}^{2}+ 2 b' x_{o}+ 2 b y_{o}+ a" = 0

a' ythen the reduced equation is_{o}+ b = 0 and a' y_{o}^{2}+ 2 b' x_{o}+ 2 b y_{o}+ a" = 0

a' y'This is an equation of a parabola.^{2}+ 2 b'x' = 0

9 xdelta = 0.^{2}-24 xy + 16 y^{2}-2x + 4y + 5 = 0

One of the eigenvalues is r

Denote (cos(t),sin(t)) as coordinates of the unit characteristic vector corresponding with r

[ a b"][cos(t)] [cos(t)] [ b" a'][sin(t)] = r_{1}[sin(t)] <=> acos(t) + b"sin(t) = 0 b"cos(t) + a'sin(t) = 0 <=> 9 cos(t) - 12 sin(t) = 0 -12 cos(t) + 16 sin(t) = 0 then tan(t) = 3/4 ; cos(t)= 4/5 ; sin(t) = 3/5 [ 4/5 -3/5 0] M = [ 3/5 4/5 0] [ 0 0 1] we find T [ 0 0 2/5 ] C1 = M C M = [ 0 25 11/5] [2/5 11/5 5 ] The equation after the rotation is 2 25 y + 4/5 x + 22/5 y + 5 = 0 Now the translation x = x' + x_{o}y = y' + y_{o}4 22 25 (y'+ y_{o})^{2}+ - (x'+ x_{o}) + -- (y'+ y_{o}) + 5 = 0 5 5 <=> 25y'^{2}+ (4/5)x' + (50y_{o}+ 22/5)y' + 25y_{o}^{2}+ (4/5)x_{o}+ (22/5)y_{o}+ 5=0 We choose x_{o}and y_{o}such that (50y_{o}+ 22/5) = 0 and 25y_{o}^{2}+ (4/5)x_{o}+ (22/5)y_{o}+ 5=0 We find 751 11 xo = - ---- yo = - ---- 125 125 The reduced equation is 25 y^{2}+ (4/5) x = 0

a xThe ideal points of that conic section are the solutions of^{2}+ 2 b" x y + a' y^{2}+ 2 b' x z + 2 b y z + a" z^{2}= 0

/ |a xWe see that the ideal points of the given conic section are the same points as these of the lines^{2}+ 2 b" x y + a' y^{2}+ 2 b' x z + 2 b y z + a" z^{2}= 0 | \ z = 0 <=> / | a x^{2}+ 2 b" x y + a' y^{2}= 0 | \ z = 0

a xFrom the theory about degenerated affine conic sections, we know the relationship between the nature and number of the ideal points and delta. So, about the given general conic section, we can say:^{2}+ 2 b" x y + a' y^{2}= 0

- if delta > 0

The conic section has two conjugate imaginary ideal points.

Since an ellipse has two conjugate imaginary ideal points, we generalize and say that all conic sections with delta > 0 are ellipses. - if delta = 0

The conic section has two coinciding ideal points.

Since a parabola has two coinciding ideal points, we generalize and say that all conic sections with delta = 0 are parabolas. - if delta < 0

The conic section has two different ideal points.

Since a hyperbola has two different ideal points, we generalize and say that all conic sections with delta < 0 are hyperbolas .

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