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:
“He’s leaving Germany by special request of the Nazi government. First he sends a dispatch about Danzig and how 10,000 German tourists are pouring into the city every day with butterfly nets in their hands and submachine guns in their knapsacks. They warn him right then. What does he do next? Goes to a reception at von Ribbentropf’s and keeps yelling for gefilte fish!”
—Billy Wilder (b. 1906)
“... 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)
“The family environment in which your children are growing up is different from that in which you grew up. The decisions our parents made and the strategies they used were developed in a different context from what we face today, even if the “content” of the problem is the same. It is a mistake to think that our own experience as children and adolescents will give us all we need to help our children. The rules of the game have changed.”
—Lawrence Kutner (20th century)