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Apollonian Families
Apollonius of Perga
is best remembered for his work on conics (he is responsible for the names parabola, ellipse and hyperbola), but he also investigated other families of curves. One such family has particular relevance to inversive geometry.
Definition
If we choose k = 1, then we get the perpendicular bisector of AB, i.e. a line. | ![]()
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Observe that the members of an apollonian family are disjoint, for if a point P
Apollonius's Theorem
Notice that the theorem, shows that the set of all apollonian curves consists
We have already observed that, if the line L is in A(A,B), then
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Definition Suppose that C is a circle with centre O and radius r, and that P is any point other than O. Then the inverse of P with repsect to C is the point P' such that (1) P' lies on the ray OP, and (2) OP.OP' = r2. We denote the inverse by iC(P), or, if C is obvious, by P'. |
P' lies on the ray OP is equivalent to |
Parts two and three of the theorem may be restated as
A circle C belongs to the apollonian family A(A,B) if and only if
Apollonian families have many nice properties. we shall return to some of these |
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