SolitaryRoad.com

Website owner:  James Miller


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

CHANGES IN EQUATION COEFFICIENTS PRODUCED BY TRANSLATION AND ROTATION OF THE COORDINATE SYSTEM

Changes in equation coefficients produced by translation. What changes are produced in the coefficients of the general equation of the second degree

             f(x, y) = ax2 + 2bxy + cy2 + 2dx + 2ey + f = 0

by a translation of the coordinate system? Let us observe the results of translating the coordinate system to the point (x0, y0). Let x'-y' be a coordinate system with origin located at point (x0, y0) of the x-y system. We substitute

x = x' + x0

y = y' + y0

into equation 1) above.

The equation becomes

f(x' + x0, y' + y0) = a(x' + x0)2 + 2b(x' + x0)(y' + y0) + c(y' + y0)2 + 2d(x' + x0) + 2e (y' + y0) + f

 

             = ax' 2 + 2bx'y' + cy' 2 + 2(ax0 + by0 + d)x' + 2(bx0 + cy0 + e)y' + ax02 + 2bx0y0 + cy02 + 2dx0 + 2ey0 + f

 

            = ax' 2 + 2bx'y' + cy' 2 + 2(ax0 + by0 + d)x' + 2(bx0 + cy0 + e)y' + f(x0, y0) = 0

 

We see that a translation has no effect on the coefficients of the second degree terms (i.e. the coefficients a, b, c). It only changes the first degree coefficients d and e and the constant term f where


             ole.gif



Changes in equation coefficients produced by rotation. Under a rotation of the coordinate system about its origin by an angle of θ degrees the general equation of the second degree

 

            f(x, y) = ax2 + 2bxy + cy2 + 2dx + 2ey + f = 0


becomes


            a'x' 2 + 2b'x'y' + c'y'2 + 2d'x' + 2e'y' + f ' = 0


where

            a' = a cos2 θ + 2 b sin θ cos θ + c sin2 θ

            b' = (c - a) sin θ cos θ + b (cos2 θ - sin2 θ)

                        = ½ (c - a) sin 2θ + b cos 2θ

            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

            f ' = f

 

where the equations of the transformation are


x = x' cos θ - y' sin θ

y = x' sin θ + y' cos θ .


We note that all of the coefficients of the equation change under the rotation of the coordinate system. Only the constant term f remains unchanged.



Invariance of the quantity b2 - ac under rotation. The quantity b2 - ac is invariant under rotation of coordinates. In other words, if the general equation of the second degree


            f(x, y) = ax2 + 2bxy + cy2 + 2dx + 2ey + f = 0


is transformed by


            x = x' cos θ - y' sin θ

            y = x' sin θ + y' cos θ


to obtain


            a'x' 2 + 2b'x'y' + c'y'2 + 2d'x' + 2e'y' + f ' = 0


then


            b' 2 - a'c' = b2 - ac .



More from SolitaryRoad.com:

The Way of Truth and Life

God's message to the world

Jesus Christ and His Teachings

Words of Wisdom

Way of enlightenment, wisdom, and understanding

Way of true Christianity

America, a corrupt, depraved, shameless country

On integrity and the lack of it

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

Who will go to heaven?

The superior person

On faith and works

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

The teaching is:

On modern intellectualism

On Homosexuality

On Self-sufficient Country Living, Homesteading

Principles for Living Life

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

America has lost her way

The really big sins

Theory on the Formation of Character

Moral Perversion

You are what you eat

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

Cause of Character Traits --- According to Aristotle

These things go together

Television

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

The True Christian

What is true Christianity?

Personal attributes of the true Christian

What determines a person's character?

Love of God and love of virtue are closely united

Walking a solitary road

Intellectual disparities among people and the power in good habits

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

On responding to wrongs

Real Christian Faith

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

Sizing up people

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-discipline

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

We are our habits

What creates moral character?


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