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	<U><BIG>Calculation of the Arc of the Ellipse.</BIG></U>



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	There is no "formula" to calculate the 'circumference' of a ellipse so the only way to do such a thing is to approximate it by a large number of very small straight lines.  This, of course, is very tedious but computers don't mind this at all.  It is necessary to calculate the length of the ellipse arc from point m to point p in order to find how far the small circle has rolled and to do this we simply have to add up [Integrate] the total length of these very small straight lines.

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	Assume, then, that the 'Circumference' is made up of a large number of Small, straight-line segments, centred on point p  (where the circle touches the ellipse).  The slope of the tangent to the ellipse at this point is dy/dx, which is equal to the slope of the curve at that point ie. dy/dx.



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	Refering to the above diagram,  o-p is a line joining the centre of the ellpise o to point p on the circumference which makes an angle  q with the x (major) axis and has a length R.  It can be shown* that the length of the increment of arc  d<B>s</B>  can be calculated from the value of R for any given value of  q  and the rate at which R changes with  q .

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	The relevant equation is:-  	________________

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	                                              d<B>s  = </B> <B>R&sup2;  +  ( dR / d</B>q )<B>&sup2;</B>  . dq       (Approximatly - the smaller     	dq , the better the approximation. ) 



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	<B>dR /d</B>q <B> can be found by expressing R&sup2; in terms of  </B>q.



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	ie.  R&sup2; =  b&sup2; (1+ tan&sup2; q ) / (b&sup2;/a&sup2;  +  tan&sup2; q )	[See appendix C]

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	* See Appendix B or "Mathmatics for Engineers &amp; Scientists"  Page 176.

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	Differentating gives :- 

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                                dR/dq  =  <B>b&sup2; (2 tan </B>q<B> (b&sup2;/a&sup2; - 1)/(b&sup2;/a&sup2; + tan&sup2; </B>q<B>  )&sup2; )   [See Appendix C]</B>



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	A suitable sequence for calculating the position of the point m on the rolling circle is given 					               	below :-

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	1)  Calculate tn = tan q   &  <B>tsq = tan&sup2; </B>q



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	2) Calculate  D = (b&sup2; / a&sup2;  +  tsq )



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	3)  Calculate  R&sup2; =  b&sup2; (1 + tsq ) / D    [ and R ]



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	4)  Calculate  dRdT  =  2 b&sup2; tn (1 + tsq ) / D

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			     ___________

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	5) Calculate   d<B>s </B> =  R&sup2; + dRdT&sup2;  . dq



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	6)  Increment  s



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	7)  Calculate  x &amp; y  ie.   x = R cos q  ,     y = R sin q            Plot  point xy if required.



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	8)  Calcuate  tan b      =  -a&sup2; y / b&sup2; x



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	9)  Calculate   b



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	10)  Calculate  co-ordinates of centre of rolling circle  c.

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		ie.  xc = x + r sin b            yc = y + r cos b               r = Radius of rolling circle.



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	11)  Calculate the angle  g  through which the circle has rolled. Its arc will be equal to the  arc 

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                      of  the ellipse from m to p .

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				ie.   g r = s   or  g  = s/r



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	12)  Calculate the position of the tracing point m w.r.t. point c. 

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			ie.  xm = xc + kr sin ( g - b )

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			      ym = yc -  kr cos ( g - b )



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	13)  Plot position of point m .



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	14)   Increment  q.



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	15)   Repeat loop.         

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The equation of an ellipse in Cartesian co-ordinates is given by:-

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			x&sup2; / a&sup2;  +  y&sup2; / b&sup2;  =  1



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[	This gives dy/dx <I>=  (</I>d<B>y/</B>d<B>x)  =  -b&sup2;x/a&sup2;y]</B>

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