Apollonius Problem

Some articles on apollonius, problem, apollonius problem:

Problem Of Apollonius - Special Cases - Mutually Tangent Given Circles: Soddy's Circles and Descartes' Theorem
... If the three given circles are mutually tangent, Apollonius' problem has five solutions ... This special case of Apollonius' problem is also known as the four coins problem ... The three given circles of this Apollonius problem form a Steiner chain tangent to the two Soddy's circles ...
Problem Of Apollonius - Solution Methods - Algebraic Solutions
... Apollonius' problem can be framed as a system of three equations for the center and radius of the solution circle ... suggest (incorrectly) that there are up to sixteen solutions of Apollonius' problem ... Therefore, Apollonius' problem has at most eight independent solutions (Figure 2) ...
Problem Of Apollonius - Applications
... The principal application of Apollonius' problem, as formulated by Isaac Newton, is hyperbolic trilateration, which seeks to determine a position from the differences in distances to ... Solutions to Apollonius' problem were used in World War I to determine the location of an artillery piece from the time a gunshot was heard at three different positions, and hyperbolic ... This multilateration problem is equivalent to the three dimensional generalization of Apollonius' problem and applies to global positioning systems such as GPS ...
Problem Of Apollonius
... In Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane (Figure 1) ... Apollonius of Perga (ca. 190 BC) posed and solved this famous problem in his work Ἐπαφαί (Epaphaí, "Tangencies") this work has been lost, but a 4th-century report of his results by Pappus of Alexandria has survived ...
Problem Of Apollonius - Special Cases - Number of Solutions
... The problem of counting the number of solutions to different types of Apollonius' problem belongs to the field of enumerative geometry ... The general number of solutions for each of the ten types of Apollonius' problem is given in Table 1 above ... For illustration, Apollonius' problem has no solution if one circle separates the two (Figure 11) to touch both the solid given circles, the solution circle would have to cross the dashed given circle but ...

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