An Approach to Mathematic Functions Basics

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

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Section XXIII - Linkage System for the Circle / Ellipse / Parabola / Hyperbola

Summary for the Section :

• In this Section , we shall do the { Linkage System for the Circle / Ellipse / Parabola / Hyperbola }
  • using a single type of equation , namely equations of the format :
    • 

      expressing [ R ] in terms of the angle [ Angle PSI ] in { Polar Co-ordinates } .

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Bringing-in the Equations :

• Let us now bring-in the equations from various previous Sections for a quick comparison :
  • For the Ellipse in Section XX , we have :
    • 

  • For the Circle in Section XXI , we have :
    • 

  • For the Ellipse in Section XIX , we have :
    • 

  • For the Parabola in Section XXII , we have :
    • 

  • For the Hyperbola in Section XIII , we have :
    • 

  • For the Hyperbola in Section XIV , we have :
    • 

• QUESTION :
  • Do we have a single type of equation covering the shapes of the Circle / Ellipse / Parabola / Hyperbola ?

  And the ANSWER is YES :

  • one type of equation here will do it all :
    • 

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Bringing in the Graphic Diagrams :

• Let us now take a quick look at the Curves associated with the various values of [ lamda ] :

  For [ lamda = -0.5 ] and [ lamda = 0.0 ] , we have :

  

  For [ lamda = +0.5 ] and [ lamda = +1.0 ] , we have :

  

  For [ lamda = +1.414214 ] and [ lamda = -1.414214 ] , we have :

  

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Consolidating the Graphic Diagrams :

• Let us now do a single Graphic Diagram for :
  • the range of [ lamda ] varying from [ -1 ] to [ plus infinity ]

    combined with

  • the range of [ lamda ] varying from from [ minus infinity ] to [ -1 ] .

  

• Very interesting diagram indeed but we are missing something here , namely :
  • the straight-line as marked-off via the [ yellow-color dotted-line ] in the diagram above .

  And the equation for this straight-line here is :

  • 

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Explaining why the Straight-Line does below here :

• Let us quickly explain why this straight-line might possibly belong to this { family of curves } here .

  And we bring-in again the core equation for the said { family of curves } , i.e. :

  • 

  Let us now put this equation into this special format below :

  • 

    yielding :

  • 

  Let us now take the limit on the right-hand-side for { [ One-over-Lamda ] approaching [ zero ] } :

  • 

  This then yields us this relation :

  • 

• Let us bring back the equation for the [ yellow-color straight-line ] :
  • 

    for a quick comparison .

  Rather similar indeed !

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The Question on Discontinuity :

• The QUESTION here is then :
  • whether the [ yellow-color straight-line ] represented a discontinuity ?

  And we shall give further thoughts on this in the next Section .

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go to the next section : Section XXIV - The Reasons for bringing-in the Orbit of a Satellite

go to the last section : Section XXII - Equation for the Parabola

return to the MATH Functions Basics HomePage

Original dated 2014-12-07