Special Cases
If one of the three circles is replaced by a straight line, then one ki, 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 ki 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.
Read more about this topic: Descartes' Theorem
Famous quotes containing the words special and/or cases:
“And weren’t there special cemetery flowers,
That, once grief sets to growing, grief may rest:
The flowers will go on with grief awhile,
And no one seem neglecting or neglected?
A prudent grief will not despise such aids.”
—Robert Frost (1874–1963)
“... in all cases of monstrosity at birth anaesthetics should be applied by doctors publicly appointed for that purpose... Every successive year would see fewer of the unfit born, and finally none. But, it may be urged, this is legalized infanticide. Assuredly it is; and it is urgently needed.”
—Tennessee Claflin (1846–1923)