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I am an EE presently writing a book on microwave semiconductors. In one of the common graphs we employ - the Smith chart - we use the bilinear transformation to map rectangular regions to circles in the transformed domain. Does anyone from the math world know of a simple proof to demonstrate the uniqueness of the line connecting the centers of two arbitrary circles that is everywhere orthogonal to the tangent points of the associated contours? And, is there a way to extend this to higher dimensions.

Thanks so much...

John Sevic

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    I don't think you are using bilinear maps to map rectangles to circles, because bilinear maps simply can't do that. Are you maybe using fractional linear maps?2012-11-26
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    ---the line connecting the centers of two arbitrary circles that is everywhere orthogonal to the tangent points of the associated contours?--- Do you mind explaining your terminology a bit? It should be very straightforward business, but the expressions like "a line orthogonal to a point" turn it all into an undecipherable riddle :-).2012-11-26
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    Maybe the lines are field lines, and not actually straight lines? Otherwise there is only one line through the two centers, because a line is completely determined by two points.2012-11-26
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    Thanks for your reply. We define the impedance plane as the semi-infinite region R>0 and -inf2012-11-26
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    Let me expand on my (poor) vocabulary: consider a pair of functions, represented by their contours, that appear as circles after being transformed by the bilinear transform. Each contour from the first function will, at some point, be tangent to a contour from the other function. I am interested in the properties of the line that is everywhere orthogonal to this tangency point. I can plot this line by hand and plot it numerically, and it reminds me of a geodesic (i.e. it's not a straight line, in general). I'm interesting in proving the line is unique. Does this make sense?2012-11-26

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