An Approach to Mathematic Functions Basics

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

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Section XVIII - Geometric Definition of the Ellipse

Summary for the Section :

• In this Section , we shall do the Geometric Definition for the Ellipse .

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Geometric Definition of the Ellipse :

• Let us now spell-out the definition of the Ellipse as follows :

  STEP ONE :

  • Let us choose at random any two (2) points , distinct-and-different from one-another , on a 2-dimensional plane ,
    • to serve as the 2 [ Focal Points ] for the Ellipse ;
      • as per the diagram on-the-left below .

  

  STEP TWO :

  • The loci of the Ellipse is defined via the following :
    • IF :
      • any point [ Point P ] is a point on the Ellipse ,

      THEN :

      • the sum of :
        • the distance from the [ 1st Focal Point ] to [ Point P ]

          and

        • the distance from the [ 2nd Focal Point ] to [ Point P ]

        is a pre-determined constant .

    And this [ pre-determined constant ] shall remain constant-and-unchanging throughout

    • for all points on the Ellipse .

  Furthermore ,

  • there is a special-and-specific limitation on this [ pre-determined constant ] , namely that :
    • the size of the [ pre-determined constant ] must be greater than { the distance between the 2 [ focal points ] } .

• We then have this sample Ellipse , constructed via the above-said Geometric Definition ,
  • as marked-off as the { the red-color curve } in the diagram on-the-right , above .

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Preparing for the Derivation Process :

• We shall now set up the general Ellipse for our further analysis in the next 2 Sections ,
  • and we shall also set up at-the-same-time the { U-V Axis System }
    • to serve as our primary [ 2 orthogonal-axis system ] for referencing .

  FIRST STEP :

  • We randomly choose any 2 points , distinct-and-separate from one-another , in a 2-dimensional plane ,
    • to serve as our [ Focal Point #1 ] and [ Focal Point #2 ] respectively ;
      • as per the diagram on-the-left below .

    We can then pass a straight-line thru the above-said 2 [ focal points ] ,

    • in the [ Focal Point #2 ] to [ Focal Point #1 ] direction ;

    and :

    • this line shall then serve as the [ U-Axis ] of our [ U-V Axis System ] .

  

  SECOND STEP :

  • We can then identify :
    • the mid-point of the { segment from [ Focal Point #2 ] to [ Focal Point #1 ] } as the point [ Point O ] ,
      • as per the diagram on-the-right above .

    We can then construct a straight-line perpendicular to the [ U-Axis ] passing thru [ Point O ] ,

    • and this line shall then serve as the [ V-Axis ] of our [ U-V Axis System ] .

  • And since [ Point O ] is the mid-point of the { segment connecting [ Focal Point #2 ] and [ Focal Point #1 ] } ,
    • [ Point O ] is in fact equi-distant from the 2 [ focal points ] .

    As such , we can now mark-off the distance from [ Point O ] to either of the [ focal points ]

    • as the distance [ L-sub-F ] , as shown in the diagram on-the-right above ;

      i.e. :

      • 

        

    And the distance between the 2 [ focal points ] is then given by :

    • 

  THIRD STEP :

  • Let us now introduce the [ pre-determined sum ] , denoted as [ S ] , such that :
    • 

    We can then construct the general Ellipse ,

    • in accordance to the afore-said Geometric Definition as detailed above ;
      • as per the diagram on-the-left below .

  

  STEP FOUR :

  • We then identify the 2 Vertices of the Ellipse :
    • as the 2 points-of-intersection between the [ Ellipse ] itself and the [ U-Axis ] ,
      • as marked-off as [ Vertex 1 ] and [ Vertex 2 ] respectively in the diagram on-the-right above .

    We can then also mark-off the distances :

    • 

    • 

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Expressing [ S ] in terms of [ L-V1 ] and [ L-V2 ] :

• Let us now look at the value of [ S ] more closely .

  Then :

  • For [ Vertex 1 ] , we have :
    • 

    • 

    Consequently , for the value of [ S ] being given by :

    • 

    We now have :

    • 

  • For [ Vertex 2 ] , we have :
    • 

    • 

    Consequently , for the value of [ S ] being given by :

    • 

    We now have :

    • 

• Since both [ Vertex 1 ] and [ Vertex 2 ] are points on the same Ellipse ,
  • it then follows that both points do share the same value for the pre-determined sum [ S ] .

  Thus , we can now combine the 2 equations we have above for the value of [ S ] ,

  • to arrive at :
    • 

  And we can see from this equation that :

  • [ L-sub-V1 ] and [ L-sub-V2 ] are in fact equal to one-another .

  Thus , we can now write :

  • 

    where [ L-sub-V ] is now the distance from [ Point O ] to either of the 2 vertices .

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Summarizing ahead of the next 2 Sections :

• In our Geometric Definition of the Ellipse here ,
  • there are only two (2) quantities that need to be assigned specific values ,

    namely :

    • the quantity [ L-sub-F ] and the quantity [ L-sub-V ] ;

  so that :

  • the shape and size of the Ellipse may be fully quantified .

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go to the next section : Section XIX - Equation for the Ellipse - Part I

go to the last section : Section XVII - A Statement of Purpose for Part IV

return to the MATH Functions Basics HomePage

Original dated 2014-12-07