Descartes' Theorem - Special Cases

Special Cases

If one of the three circles is replaced by a straight line, then one k_i, say k₃, is zero and drops out of equation (1). Equation (2) then becomes much simpler:

If two circles are replaced by lines, the tangency between the two replaced circles becomes a parallelism between their two replacement lines. For all four curves to remain mutually tangent, the other two circles must be congruent. In this case, with k₂ = k₃ = 0, equation (2) is reduced to the trivial

It is not possible to replace three circles by lines, as it is not possible for three lines and one circle to be mutually tangent. Descartes' theorem does not apply when all four circles are tangent to each other at the same point.

Another special case is when the k_i are squares,

Euler showed that this is equivalent to the simultaneous triplet of Pythagorean triples,

and can be given a parametric solution. When the minus sign of a curvature is chosen,

this can be solved as,

where,

parametric solutions of which are well-known.