SolitaryRoad.com

Website owner: James Miller

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

THE FUNCTION f(x, y) AND ASSOCIATED MATRICES e AND E

The function z = f(x, y) associated with the equation f(x, y) = 0. Associated with the general equation of the second degree

1) ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0

is the function

2) f(x, y) = ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c

to which we now wish to direct our attention.

Let us first make some general observations in connection with the equation

ax^{2} + bx + c = 0

which is the single variable analogue of equation 1) above. Associated with this equation is the function

f(x) = ax^{2} + bx + c

or, equivalently,

y = ax^{2} + bx + c

whose graph is always a parabola which is
symmetric about some vertical axis. See Fig. 1. Its trace on the x axis corresponds to the
solution set of the equation ax^{2} + b x + c = 0 . A translation of the coordinate system by the
distance d in the x direction, so as to position the
origin over the parabola axis, will eliminate the
term in x giving an equation of the form

y = ax^{2} + q

with the coordinate system then at the position shown in Fig. 2. It can be shown that

The constant term q is equal to the distance of the vertex above origin (the value of y at x = 0). In the figure q is negative.

Now we have a completely analogous situation when we go from an equation in a single variable f(x) = 0 to the equation in two variables

ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0

with which is associated the function

f(x, y) = ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c

or, equivalently,

3) z = ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c .

The function 3) is always a paraboloid (either an elliptic or hyperbolic paraboloid) which is symmetric about some vertical axis in 3-space. See Figures 3 and 4. The equation f(x, y) = 0 corresponds to the trace of z = f(x, y) in the xy-plane. The trace in that plane is one of the 9 conics, a parabola, ellipse, hyperbola, or one of the limiting cases.

A rotation of the coordinate system about the z-axis through the correct angle will always eliminate the xy term in 3) above. What angle? The same angle that eliminates the xy term in the equation f(x, y) = 0. The same rotational transformation that eliminates the xy term in the equation f(x, y) = 0 eliminates the xy term in the parabolic function z = f(x, y).

In the case where f(x, y) = 0 represents a central conic (ellipse, hyperbola) a translation by the proper amounts in the x and y directions will eliminate the terms in x and y of 3) above.

What translation will do this? The same translation that will eliminate the x and y terms in the equation

ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0 .

Intuitively, this amounts to translating the coordinate system in such a way as to position its origin on the axis of the paraboloid. The terms in x and y disappear and our function takes the form

z = ax^{2} + 2hxy + by^{2} + q .

Our coordinate system is then in the position shown in Fig. 5. The constant term q is equal to the distance of the vertex above origin (in the figure q is negative).

In general, when we do the translations and rotations to eliminate the x, y and xy terms in the equation f(x, y) = 0 it can be helpful to think of what is happening in terms of the parabolic function z = f(x, y).

Matrices e and E. The function

f(x, y) = ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c

can also be written in the form

or in the form

4) f(x, y) = (ax + hy + g)x + (hx + by + f)y + (gx + fy + c)

Associated with f(x, y) are two symmetric matrices

which will be seen to come directly from representation 3) above. These two matrices play an central role in the analysis of the general equation of the second degree. We will refer to them later.

Quadratic form. Of importance in connection with the function f(x, y) is the quadratic form

F(x, y) = ax^{2} + 2hxy + by^{2}

which can be written as

or in matrix form as

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 ]