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 Discussion: All Topics Topic: Common Tangent Construction

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 Subject: RE: Common Tangent Construction Author: Alan Cooper Date: May 11 2006
On May 10 2006, Tup wrote:
> I don't know as accuracy has anything to do with geometric
> constructions.

In principle, it doesn't. ie The proof that a specified object is constructable
does not depend on an accurate version of the construction actually being
feasible. But fine carpenters, navigators, and many others know that Euclidean
constructions actually do correspond to practical operations. So, while I
wouldn't argue that the common tangent is something for which Euclidean methods
are best, the question of feasibility of a construction is nonetheless
potentially of interest.

Aside from practical considerations, another important use of constructions is
in the demonstration or proof of geometrical facts, and this is generally easier
to follow if all aspects of the required diagram can be made visible at the same
time.

The practical construction of a common tangent by finding its intersection with
the line of centres becomes infeasible when the radii are too close to one
another because of the extreme distance of the intersection point, and this also
makes it impossible to get a good single view of all aspects of the
construction. Certainly this deficiency is also a virtue in that it demonstrates
the importance of sometimes considering a situation from different scales at the
same time, but I took Mathman's original question to be asking for a
construction of the common tangent in which all relevant details would be
visible on any page sufficient to contain both circles.

While it is true that the difference circle becomes invisibly small when the
radii are almost equal, this does not actually reduce the accuracy of the
construction so long as one keeps in mind the fact that the point of tangency is
also on the circle through the two centres (so when the difference circle is too
small to be used, the perpendicular to the line of centres will be accurate to
within an error of the same magnitude as the radius of the tiny circle).
-Alan