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:
“Friendship is learned by watching and listening to you. If she sees that your friends are people you like and trust and don’t pretend with—people who suit you—she probably won’t pick friends who just pass by, or people who can help her or improve her status. If you treat friends in a special way, if you are kinder, more generous, more sympathetic, more forgiving with friends, she probably will be, too.”
—Stella Chess (20th century)
“In most cases a favorite writer is more with us in his book than he ever could have been in the flesh; since, being a writer, he is one who has studied and perfected this particular mode of personal incarnation, very likely to the detriment of any other. I should like as a matter of curiosity to see and hear for a moment the men whose works I admire; but I should hardly expect to find further intercourse particularly profitable.”
—Charles Horton Cooley (1864–1929)