ࡱ> -/,|M Abjbj== \FWWyl$"TpD"=~///H=J=J=J=J=J=J=$> @Rn=/E////n=/1~=/1/1/1/@~H=/1/H=/1@/1o8:<,H=~ `!"/< H==0=<R@0@H=/1""   Note that, due to the symmetry of the original shape, it is not entirely necessarily to calculate coordinates of centres of the circles using Rules (2), (3) or (7) above. In fact it is much easier to use the original shape together with the new-found radii in order to calculate the respective centre coordinates.  Inversion Theory Continued An alternative notation that is particularly useful when inverting circles is explained below If R is the radius of the Inverted Image and r is the radius of the Original circle, then the following results hold:  To find the object radius, r, we use  EMBED Equation.DSMT4 , which can be seen to be equivalent to the equation in rule (6) above (using Pythagoras Theorem on denominator), but may be easier to use and recall. To find the distance, d, we use  EMBED Equation.DSMT4 . As is often the case, we will need to find the centre coordinates of the object circles,  EMBED Equation.DSMT4 . We do this by using the two formulae:  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  are the coordinates of the centre of the inverted image circle. Clearly, if the radius of the Inverted Image Circle and coordinates of the centre of the Inverted Image Circle are difficult to find from the diagram, then it will also be very difficult to find the actual radius and centre coordinates of the Object Circle. Therefore, the success of the method of Inversion relies heavily on the skill of the user in choosing the position and size of the Circle of Inversion so as to make the Geometry of the entire Inverted Image as simple as possible. Inversion Exercise:- To find the Radii of the Internal Circles  x  EMBED Equation.DSMT4  a  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Now, using Rule 5 from the Theory Box above we have the result:  EMBED Equation.DSMT4  --(1) So for the smaller Circle:  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Similarly, for the larger Circle:  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Putting these into Eq.(1) quickly yields the result:-  EMBED Equation.DSMT4  Putting these into Eq.(1) quickly yields the result:-  EMBED Equation.DSMT4  In the diagram the red lines are the inversions of the original black lined shape in which we need to find the radii of the internal circles to the square of side length a. Consider the right-angled triangle taken from this picture:-  The hypotenuse can be seen to be  EMBED Equation.DSMT4 . Therefore, x can be found to be  EMBED Equation.DSMT4 . Similarly, the radius of the larger inverted circle is seen as  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Circle of Inversion Inverted Image Object  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Inversion Theory Given a Circle of Inversion, T(k), where the point T is called the Centre of Inversion and k is called the Radius (or Constant) of Inversion, then the following results are determined:- If P is any given point, we can find a point  EMBED Equation.DSMT4  on the line through TP such that  EMBED Equation.DSMT4   EMBED Equation.DSMT4  is known as the Inverse of P. If P and Q are any two points with inverses EMBED Equation.DSMT4 and  EMBED Equation.DSMT4  then  EMBED Equation.DSMT4  As an extension to (1) above it can be shown that:  EMBED Equation.DSMT4  and also from (1) above  EMBED Equation.DSMT4 . Taken together  EMBED Equation.DSMT4   EMBED Equation.DSMT4 . (See (2) above) Where  EMBED Equation.DSMT4  are distances from the Centre of Inversion to the points  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , respectively. The inverse of a Line (Infinite Line) that does NOT pass through Centre of Inversion, T, is an Arc of a Circle (Full Circle) that passes through the T, otherwise if the Line passes through T then it is inverted to a Line, itself through T, and coincident to the Object Line. The inverse of an Arc of a Circle (Full Circle) that does NOT pass through Centre of Inversion, T, is an Arc of a Circle (Full Circle) NOT through T, otherwise it is a line that also does NOT pass through T. Of particular use is the following circle result: If any Circle  EMBED Equation.DSMT4  inverts into the Circle  EMBED Equation.DSMT4  then  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  is the length of a Tangent from T to the Inverse Circle  EMBED Equation.DSMT4 . However, to show how it is done we use the formulae from Rule 7. in the above Theory Box. The centre of the Circle of Inversion is at the top right of the square, therefore we measure x as positive to the left and y positive down, without loss of generality. For the small circle we have  EMBED Equation.DSMT4  Therefore,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  Similarly, for the large circle we have  EMBED Equation.DSMT4  Therefore,  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  It is a simple matter to show further that if we now take the bottom right of the square as our origin that the small circle is  EMBED Equation.DSMT4  up the diagonal and the large circle is  EMBED Equation.DSMT4  up the diagonal.   wx'(pqrsghqr,-DEFGOPg jEHUj> CJUVaJmH sH  jEHUj&> CJUVaJmH sH  jEHUj> CJUVaJmH sH  jfEHUja> CJUVaJmH sH  jUj6U]CJ6]jCJUmHnHsH u2     !y@!"#$%&'()*+,-./0123456789:;<wyypqsNPq,I $`a$$^a$^ $ ha$ h hL^`L & F  $ a$$ a$ghijs y 2 F   L O P Q h i j k m o p q     ( ) @ A ~njh> CJUVaJmH sH  j:EHUjW> CJUVaJmH sH  j7EHUjc`> CJUVaJmH sH  j@5EHUj_> CJUVaJmH sH  jh2EHUj5\> CJUVaJmH sH  jU6] jU j3EHUj> CJUVaJmH sH )       ! 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