An Approach to Mathematic Functions Basics

by Frank C. Fung ( 1st published in December, 2014. )

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Section IX - The Hyperbola in Cartesian Co-ordinates

Summary for the Section :

• In this Section , we shall take an initial first look at the Hyperbola ,
  • a special class of [ 2-dimensional curves ] obeying a specific set of defining criteria ,
    • and therefore possessing a specific set of special properties .

  We shall start-off here initially with the Hyperbola in the Cartesian Plane ,

  • governed by the equation :
    • 

  We shall then proceed to re-orient the [ 2 orthogonal-axis co-ordinate system ] ,

  • in preparation for the Geometric Definition of the Hyperbola ,
    • coming up in the next Section to follow .

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The Hyperbola in the 2-D Cartesian Plane :

• In general , the 2-dimensional Cartesian Plane is defined via its 2 orthogonal axis ,
  • usually the [ X-Axis ] and the [ Y-Axis ] respectively .

  And the [ governing equation ] for the Hyperbola in this 2-dimensional [ X-Y Axis System ]

  • is usually written as :
    • 

    or alternately in this format :

    • 

      when the variable [ y ] is expressed as a function of the variable [ x ] .

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Looking at a sample Hyperbola in Cartesian Co-ordinates :

• We then have this sample hyperbola obeying the relation :
  • 

    or alternately :

  • 

    as per the diagram on-the-left below .

  

  And we take special note here that :

  • for [ x = 0 ] , the value of [ y = 1/x ] is undefined ;

    and therefore :

    • a discontinuity does occur at [ x = 0 ] .

  And because of this discontinuity , it is necessary for us , for our purpose at-hand :

  • to treat
    • [ the curve to-the-left of the Y-Axis ] as marked-off in cyan-color above ,

    as separate from

    • [ the curve to-the-right of the Y-Axis ] as marked-off in blue-color above .

• And for the remainder of this Section and further Sections-to-come on the Hyperbola ,
  • we shall be focusing our attention primairly on this [ blue-color curve ] alone ;
    • ( as per the diagram on-the-right above )

    leaving-out and disregarding the [ cyan-color curve ] for the time-being .

  And this is done for a special reason :

  • It is essential-and-necessary for us to preserve the continuity in the curve we shall be analyzing ,

    so that :

    • the equations we develop therefrom for the Hyperbola will also be continuous in nature ,

      and

    • the risks of being thrown-off by kinks and/or discontinuities are therefore always kept at a minimum .

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Preparing for the Geometric Definition :

• Let us bring-in again the diagram for the [ blue-color curve ] from above and :
  • rotate [ the X-Y Axis together with the curves ] clockwise thru an angle of 45 degrees
    • to arrive at the diagram on-the-right below .

  

  And for our analysis to follow in the next 7 Sections on the Hyperbola ,

  • we shall be primarily using the [ U-V Axis System ] ,
    • as marked-off in the diagram on-the-right below .

  

  And we do take note here that :

  • this new set of [ U-V Axis ] are at-an-angle-of-[ 45 degrees ] to the original set of [ X-Y Axis ] .

• Let us now move-on to do the Geometric Definition of the Hyperbola , next .

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go to the next section : Section X - The Geometric Definition of the Hyperbola

go to the last section : Section VIII - A Statement of Purpose for Part III

return to the MATH Functions Basics HomePage

Original dated 2014-12-07