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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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    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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I think you should probably hire a mathematician.

However, taking the Smith chart as the standard unit circle in the complex plane, the mapping $ f(z) = \frac{z + 1}{i z - i} $ takes the open disc to the upper half plane. It takes the geodesics meeting $1$ to vertical lines, that is constant real part. It takes the horocycles tangent to the circle at $1$ to horocycles with constant imaginary part. The inverse mapping is $ g(z) = \frac{i z + 1}{i z - 1}. $

I notice some images of a Smith chart take $-1$ as the important point instead.One may simply negate everything in this version to rotate the disc by $180^\circ.$

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    Thanks everyone for your graciousness in helping me out on this. I chatted with a Professor in our math department (here at UC Santa Cruz) and we constructed a simple proof based on convexity. It turns out if one of the contour sets is strictly convex, e.g. a circular level contours, then the trajectory that is everywhere orthogonal to the tangent points is indeed unique. Thanks again...2012-12-05