An Approach to Mathematic Functions Basics

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

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Section XXI - Equation for the Circle

Summary for the Section :

• In this Section , we shall simply express the Equation for the Circle in a very special format :
  • i.e. , this format :
    • 

  And this is the format that we shall be using for the Circle ,

  • when we do the { linkage system for the Circle / Ellipse / Parabola / Hyperbola } in Section XXIII .

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Defining of the Circle :

• Let us first bring in the Circle ,
  • as per the diagram on-the-left below .

  And we identify here on this diagram :

  • the [ Center of the Circle ] as the point [ Point O ] ,

    and

  • the [ Radius of the Circle ] as shown .

  

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

  • its position can be uniquely defined by the Position Vector [ Vector O-P ] ;
    • as shown in the diagram on-the-right above .

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

  • the distance [ R ] :
    • being the [ distance O-P ] ;

  • the angle [ PSI ] :
    • being the angle in-between the [ Position Vector O-P ] and the [ U-Axis ] .
      • as shown in the diagram on-the-right .

• Let us now identify , for our purpose at hand , the point [ Point Q ] as shown in the diagram on-the-right above ,
  • with [ Point Q ] here being an intersection-point for the Circle and the U-Axis as shown .

  We then define the distance [ R-sub-zero ] :

  • as the distance between the center-point [ Point O ] and [ Point Q ] ,
    • as shown above .

• And the equation for the Circle here is then :
  • 

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The Special Format for the equation :

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

  And we bring-in , at this point , the 2 equations for the Ellipse for a quick comparison ,

  • for the ellipse in Section XIX , we have :
    • 

  • for the ellipse in Section XX , we have :
    • 

  These 3 equations then set the stage for our Linkage System in Section XXIII ,

  • but we still need to do the Parabola , next , to create the bridge onto the Hyperbola's .

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

go to the last section : Section XX - Equation for the Ellipse - Part II

return to the MATH Functions Basics HomePage

Original dated 2014-12-07