In Euclidean geometry, Apollonius' problem is to construct all the circles that are tangent to three given circles. Special cases of Apollonius' problem are those in which at least one of the given circles is a point or line, i.e., is a circle of zero or infinite radius. The nine types of such limiting cases of Apollonius' problem are to construct the circles tangent to:
- three points (denoted PPP, generally 1 solution)
- three lines (denoted LLL, generally 4 solutions)
- one line and two points (denoted LPP, generally 2 solutions)
- two lines and a point (denoted LLP, generally 2 solutions)
- one circle and two points (denoted CPP, generally 2 solutions)
- one circle, one line, and a point (denoted CLP, generally 4 solutions)
- two circles and a point (denoted CCP, generally 4 solutions)
- one circle and two lines (denoted CLL, generally 8 solutions)
- two circles and a line (denoted CCL, generally 8 solutions)
In a different type of limiting case, the three given geometrical elements may have a special arrangement, such as constructing a circle tangent to two parallel lines and one circle.
Read more about Special Cases Of Apollonius' Problem: Historical Introduction, Rules of Euclidean Constructions, Preliminary Results, Special Cases With No Solutions, See Also
Famous quotes containing the words special, cases and/or problem:
“When a mother quarrels with a daughter, she has a double dose of unhappiness—hers from the conflict, and empathy with her daughter’s from the conflict with her. Throughout her life a mother retains this special need to maintain a good relationship with her daughter.”
—Terri Apter (20th century)
“We noticed several other sandy tracts in our voyage; and the course of the Merrimack can be traced from the nearest mountain by its yellow sand-banks, though the river itself is for the most part invisible. Lawsuits, as we hear, have in some cases grown out of these causes. Railroads have been made through certain irritable districts, breaking their sod, and so have set the sand to blowing, till it has converted fertile farms into deserts, and the company has had to pay the damages.”
—Henry David Thoreau (1817–1862)
“Like the effects of industrial pollution ... the AIDS crisis is evidence of a world in which nothing important is regional, local, limited; in which everything that can circulate does, and every problem is, or is destined to become, worldwide.”
—Susan Sontag (b. 1933)