ࡱ> LNKVlM )bjbj== \WWS"l$0<0<0<0<<k====Cnnn$! ARnfnnn;u=Co#%;u;u;un=C;un;u;u~:b,>Cx= <30<uo  ;0kRp;u       The sequence the first four internal Tangent Circles and their Inverted Images are shown below.    Consider just the Inverted Image and ignore, for clarity, the Original Arch structure.          Inversion Exercise (3):- Continued (6)                              Inversion Exercise (3):- Continued (3) Of  EMBED Equation.DSMT4   EMBED Equation.DSMT4   Here then we see that each of the Inverted Circles has a radius of  EMBED Equation.DSMT4 . In order that we may apply either of Rule 6 or 7 from the above Theory Box to determine both the radii and centres of the Original Tangent Circles, we must first find either  EMBED Equation.DSMT4  or  EMBED Equation.DSMT4 , respectively. Using the above Rule 8 with the bottom partial Arch and first Inverted Circle with radii  EMBED Equation.DSMT4 , gives:  EMBED Equation.DSMT4  Therefore, we have  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Therefore, the radius of the Top Tangent Circle is  EMBED Equation.DSMT4 . Similarly, for the coordinates of the centre of the Top Tangent Circle, relative to the Centre of the Circle of Inversion are given by:-  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Therefore, the coordinates of the centre of the Top Tangent Circle is given by:  EMBED Equation.DSMT4  NB. It seems clear that the easiest method will be to use the Theory from Rule 7. above in the Theorem Box on page 2, rather than Rule 6.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Of Inversion Theory Continued (2) 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   EMBED Equation.DSMT4  Inversion Theory (1) 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 . In this problem our aim is to use Inversion Theory to find the radii and centres of the family of circles sandwiched between the outer Arch and the inner Semicircle shown below. Also shown below, after some trial and error, is a position and size of the Circle of Inversion that gives us a relatively simple geometry of the Inverted Image (though there may be other equally valid positions and sizes. d Inversion Exercise (3):- Continued (4) Of We have chosen the Centre of the Circle of Inversion as the bottom right point of the outer Arch and the Radius of the Circle of Inversion as a, the width of the Arch (or the diameter of the internal Semicircle). The Inverted Images of the outer Arches, the Semicircle and the Base of the Arch are shown in red. From the SketchPad file, drag the random points that lie on either the Arch or its Inverted Image to show exactly how the inversion process has worked on each separate object. Before we continue a result is needed:- Rule (8) Distance between Centres of Two Circles tangential to each other and a common line.      By Pythagoras we see that:  EMBED Equation.DSMT4 , where d is the horizontal distance between centres. Therefore,  EMBED Equation.DSMT4  Shown here is how the actual Top, Tangent Circle in the Arch will be Inverted. Over the page we see further Side Tangent Circles.   Inversion Exercise (3):- Continued (5) Of  EMBED Equation.DSMT4  In this case we have:  EMBED Equation.DSMT4   EMBED Equation.DSMT4 . Therefore, the radius of the Top Tangent Circle is  EMBED Equation.DSMT4 . Also, the centre coordinates are found from:  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . Therefore, the coordinates of the centre of the Top Tangent Circle is given by:  EMBED Equation.DSMT4    EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  In this case we have:  EMBED Equation.DSMT4   EMBED Equation.DSMT4 . Therefore, the radius of the Top Tangent Circle is  EMBED Equation.DSMT4 . Also, the centre coordinates are found from:  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . Therefore, the coordinates of the centre of the Top Tangent Circle is given by:  EMBED Equation.DSMT4  Inversion Exercise (3):- Continued (7) Of In this case we have:  EMBED Equation.DSMT4   EMBED Equation.DSMT4 . Therefore, the radius of the Top Tangent Circle is  EMBED Equation.DSMT4 . Also, the centre coordinates are found from:  EMBED Equation.DSMT4 ,  EMBED Equation.DSMT4 . Therefore, the coordinates of the centre of the Top Tangent Circle is given by:  EMBED Equation.DSMT4  In general then, for n Internal, Tangent Circles and their associated Inverted Image Circles, we observe the following patterns:   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  The radius of the nth Internal, Tangent Circle,  EMBED Equation.DSMT4 , is given by:  EMBED Equation.DSMT4 , and the location of the Centre Coordinates, with respect to the Centre of the Circle of Inversion, is given by:  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  That is: which are each valid for  EMBED Equation.DSMT4 , the case  EMBED Equation.DSMT4 giving the Top Tangent Circle.  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