You talk about just two polygons (left and right tangents), but your sample shows the ellipses as well. Also, the points of tangency are not likely to match up, so the tangent segments between ellipses will not actually meet. But you show only tangents between adjacent ellipses, drawing an open polyline rather than a polygon. Polygons are closed figures, and to make a polygon out of the tangents would imply incuding the tangents between the first and last ellipse. There is some discrepancy between your description of what you want and the sample diagram. Originally, in order to allow internal tangents to be found, I had code that effectively ignored the direction of the ellipse and chose a standard direction for the line (considered as a set of points).

The ellipses are also directional (following the normal convention of counterclockwise) and the equations for the tangent define the tangent line that goes in the same direction as the ellipse. The representation of a line as the length and direction of the normal results in defining a directed line - a line with a direction, not just a set of points. The reason the complex number version of the sheet produces only external tangents is that in that version I removed the code (the if statement) that allowed for internal tangents (which, contrary to jmG's remarks, have nothing to do with secants). That fixes numerous problems, and also adds the NaN constant. You should go to the Mathcad download area and get the upgrade to 11.2a as soon as possible.

Where Tangents Meet by NOT A BOOK