In mathematics, "'two-center bipolar coordinates "'is a coordinate system, based on two coordinates which give distances from two fixed centers, c _ 1 and c _ 2.
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"' Bispherical coordinates "'are a three-dimensional foci F _ { 1 } and F _ { 2 } in bipolar coordinates remain points ( on the z-axis, the axis of rotation ) in the bispherical coordinate system.
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"' Toroidal coordinates "'are a three-dimensional foci F _ 1 and F _ 2 in bipolar coordinates become a ring of radius a in the xy plane of the toroidal coordinate system; the z-axis is the axis of rotation.
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In general, two disjoint, non-concentric circles can be aligned with the circles of bipolar coordinates; in that case, the radical axis is simply the " y "-axis; every circle on that axis that passes through the two foci intersect the two circles orthogonally.
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Instead of solving for the two hyperbolas, Newton constructs their defined as the set of points that have a given ratio of distances to two fixed points . ( As an aside, this definition is the basis of bipolar coordinates . ) Thus, the solutions to Apollonius'problem are the intersections of a line with a circle.
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On the other hand if you pick some other wonky coordinate system p, q ( it could be bipolar coordinates or something ), the equations will look different, but they will look the same as if you'd used a wonky coordinate system p', q'which is related to p, q by rotation and translation.
