ࡱ> (*'M ~bjbj== \<WWO.l$>LD>?PPPPZK1K1K1???????$9@ YBR(?K1..K1K1K1(?2PZ=?222K1P Z?2K1?2v2)::z>,v?ZD )>vg1> ?S?0?>RB+2B?2>>               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.  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 . Inversion Exercise (2):- To find the Radii of the Internal Circle Putting these into Eq.(1) quickly yields the result:-  EMBED Equation.DSMT4    EMBED Equation.DSMT4  a  EMBED Equation.DSMT4   EMBED Equation.DSMT4  r T  EMBED Equation.DSMT4   EMBED Equation.DSMT4  x Now using Rule 1.from the Theory Box above we have that:  EMBED Equation.DSMT4 . Therefore,  EMBED Equation.DSMT4 , which is the x coordinate of the centre of the required circle. Alternatively, using the formulae from Rule 7 above gives:  EMBED Equation.DSMT4  Therefore,  EMBED Equation.DSMT4  However, since we have the tangent length from the Centre of Inversion without any effort due to the final geometry of the Inverted form, then this last step is perhaps using a hammer to crack a nut! Now, using Rule 5. from the Theory Box above we have the result:  EMBED Equation.DSMT4  --(1) So for the enclosed Circle:  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4   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 circle to the square of side length a.     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