An Approach to Mathematic Functions Basics

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

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Section XX - Equation for the Ellipse - Part II

Summary for the Section :

• In this Section , we shall derive the equation for the Ellipse ,
  • using [ Focal Point #2 ] as the base-point for our [ Polar Co-ordinate System ] .

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The [ Focal-Point based Co-ordinate System ] in this Section :

• Here in this Section , we shall select [ Focal-Point #2 ] , i.e :
  • the point [ Point F2 ] as shown in the diagram on-the-left below ,
    • as the [ base-point ] of our [ Focal-Point based Co-ordinate System ] .

  Then , for any point [ Point P ] on the Ellipse :

  • its position can therefore be uniquely defined by the Position Vector [ Vector F2-P ] ;
    • as shown in the diagram on-the-left below .

  

  And the position-vector [ Vector F2-P ] can now be defined via the two (2) variables :

  • the distance [ R ] :
    • 

  • the angle [ PSI ] :
    • 

      as shown in the diagram on-the-left above .

Preparing for the Derivation Process :

• We shall now mark-off the distance between [ Point F2 ] and the [ Vertex 1 ] as the distance [ R-sub-zero ] ,
  • i.e. :
    • 

  And we see here that :

  • the value of [ R-sub-zero ] is necessarily greater than the [ distance in-between the 2 Focal-Points ] ,

    i.e. :

    • 

      as per the diagram on-the-right above .

• And we can see from this diagram that :
  • 

  Consequently , we have the relation :

  • 

• Let us recall , at this point , that we have already established this relation for the value of [ S ] :
  • 

    towards the end of Section XVIII .

  Thus , we can now re-express [ S ] in terms of [ R-sub-zero ] in this manner :

  • 

    based on the expression we have for [ L-sub-V ] immediately above .

• And with this expression in-hand , we are now ready to proceed with the derivation process ,
  • to follow immediately below .

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Developing the equation for the Ellipse :

• Let us now bring-in this set of 2 diagrams below :

  

  Then , for any point [ Point P ] on the Ellipse , we have :

  • based on the right-angle-triangle [ Triangle F2-Q-P ] in the diagram on-the-right above :
    • 

  • based on the [ Position-Vector F2-P ] :
    • 

• Then , for the original [ Constant-Sum Equation ] for the ellipse , i.e. this equation :
  • 

  we can now write :

  • 

  And on re-arranging terms , we have :

  • 

• Let us now square both sides of the equation to arrive at this :
  • 

  And on further expansion on the right-hand-side , we have :

  • 

    yielding :

  • 

  And on cancelling terms , we have :

  • 

  Further re-arrangements then yield us this equation :

  • 

    yielding :

  • 

• Let us now take a closer look at the terms within the 3 brackets in the above equation , and :
  • we bring-in , from above , this expression for [ S ] in terms of [ R-sub-zero ] and [ L-sub-F ] :
    • 

  Let us now substitute this into each of the 3 brackets , and :

  • For the terms in the 1st bracket , we now have :
    • 

  • For the terms in the 2nd bracket , we now have :
    • 

  • For the terms in the 3rd bracket , we now have :
    • 

  Thus , we can now re-write the original equation above in this manner :

  • 

• And on re-arranging terms , we have :
  • 

    yielding :

  • 

• Let us now re-write the above the above equation in this special format :
  • 

  Therefore :

  • 

    yielding :

  • 

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Introducing the value [ Lamda ] :

• Let us now introduce the value [ Lamda ] such that :
  • 

  We can then write the expression above in this easier-to-read format :

  • 

  And we can now express [ R ] in terms of [ R-sub-zero ] via this equation below :

  • 

• Let us recall , at the point , that the only restriction we have placed on the value of [ R-sub-zero ] is that :
  • 

  Therefore , the range for [ R-sub-zero ] is then :

  • 

  And since the value of [ Lamda ] is defined as :

  • 

    the range for [ Lamda ] here is then :

    • 

• And the final equation we have for the Ellipse here is then :
  • 

  And this is an important equation that we shall be coming back to ,

  • when we look at the { linkage system for the Circle / Ellipse / Parabola / Hyperbola } ,
    • later in Section XXIII of this paper .

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go to the next section : Section XXI - Equation for the Circle

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

return to the MATH Functions Basics HomePage

Original dated 2014-12-07