SolitaryRoad.com

Website owner:  James Miller


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

                      ANALYSIS OF A CONIC


The general equation of the second degree


1)        f(x, y) = ax2 + 2hxy + by2 + 2gx + 2fy + c = 0


represents one of 9 different conics:

ole.gif





















Any equation of the second degree can be reduced to one of the nine canonical forms by a suitable rotation and translation of the coordinate system. Thus when faced with a particular second degree equation the following questions immediately present themselves:


1] Which of the 9 conics does this equation represent?


2] What is the exact equation of the conic when in canonical form? That is, what is its equation when expressed with respect to the canonical coordinate system?


3] What is the location of the origin of the canonical system?


4] What is the orientation of the canonical coordinate system? In other words, in what directions do the canonical system axes point?



We will now deal with the procedure used in answering these questions. Our starting point is our given equation


2)        f(x, y) = ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 

 

ole1.gif

which represents some conic located somewhere in the plane. In Figure 1 a conic (an ellipse) is shown located at some point in the plane. Figure 1 also shows the original x-y coordinate system along with two other coordinate systems – an intermediate x'- y' coordinate system and the xc - yc 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


ole2.gif


A rotation of this amount will put its axes parallel to the axes of the canonical system.


We proceed as follows:


1. We obtain the equation of the conic as referred to the intermediate x'- y' coordinate system by substituting into equation 2) the expressions for x and y in terms of x' and y' as given by the rotational transformation equations


x = x' cos θ - y' sin θ

y = x' sin θ + y' cos θ


where the θ in these equations is that particular value of θ computed from 3) above.


On expanding and simplifying we then have the equation


4) g(x', y') = 0


as the equation of the conic as referred to the intermediate x'- y' coordinate system.


2. We determine, by some method, the translation required to carry the x'- y' system into the canonical xc - yc system. In other words, we determine the values of h and k where (h, k) is the origin of the canonical xc - yc system with respect to the x'- y' system. In the case of the central conics (ellipses and hyperbolas), this translation corresponds to that translation that will eliminate the x and y terms from the equation.


3. We obtain the equation of the conic as referred to the canonical xc - yc coordinate system by substituting into equation 4) the expressions for x' and y' in terms of xc and yc as given by the translation transformation equations

    

x' = xc + h

y' = yc + k .


This process gives us the answers to our original questions. The mechanics of substituting into equations, expanding, and simplifying can be laborious and fortunately, in the case of the central conics, it can be bypassed. By utilizing certain invariant quantities and employing some abstract results from matrix theory we can answer the questions without going to all that labor.






Analysis procedure..


First we define the following quantities related to the equation


            f(x, y) = ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 :

 



                         ole3.gif

 


                         ole4.gif



                         ole5.gif




            λ1, λ2 – the characteristic roots of matrix e i.e. the roots of the characteristic equation


                                     ole6.gif


                        which can be written as


4)                                λ2 - Iλ + J = 0




Conic identification. We identify the conic by use of the following table:

 Case

ole7.gif  

J

ole8.gif / J

K

            Conic

1

ole9.gif  

> 0

< 0

 

Ellipse

2

ole10.gif  

> 0

> 0

 

Imaginary ellipse

3

0

> 0

 

 

Pair of imaginary lines intersecting in a real point

4

ole11.gif  

< 0

 

 

Hyperbola

5

0

< 0

 

 

A pair of intersecting lines

6

ole12.gif  

0

 

 

Parabola

7

0

0

 

< 0

A pair of parallel lines

8

0

0

 

> 0

A pair of imaginary parallel lines

9

0

0

 

0

A pair of coincident straight lines

 





Analysis of central conics (ellipses and hyperbolas).



Orientation of the canonical coordinate system. The orientation of the canonical system is found by computing the rotation angle θ required to eliminate the xy term. The formula is


                         ole13.gif


Location of the origin of the canonical system. The location of the origin (x0, y0) of the canonical system is given by solving the following system of equations for x0, y0 :


            ax0 + hy0 + g = 0

            hx0 + by0 + f = 0



Equation of the conic in the canonical system. The equation of the conic with respect to the canonical system is


                        λ1x2 + λ2y2 + c' = 0

where

                        c' = gx0 + fy0 + c


and λ1, λ2 are obtained by computing the roots of characteristic equation


            λ2 - Iλ + J = 0 ,


or more explicitly,


            λ2 - (a + b)λ + ab - h2 = 0 



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 ]