Orthogonal trajectories of the given family of circles are no different.
2.
All other points flow along a family of circles which is nested between the two fixed points on the Riemann sphere.
3.
The whole family of circles can be considered as " conics passing through two given points on the line at infinity " at the cost of requiring complex coordinates.
4.
"' Apollonian circles "'are two families of circles such that every circle in the first family intersects every circle in the second family orthogonally, and vice versa.
5.
A circle trapezoid is the region in a circle that lies between two non-crossing chords and a circle trapezoid graph is the intersection graph of families of circle trapezoids on a common circle.
6.
More generally, there is a natural correspondence between circles in the plane and points in three-dimensional projective space; a line in this space corresponds to a one-dimensional continuous family of circles called a pencil.
7.
The conclusion is that the three-parameter family of circles can be treated as a special case of the linear system of conics passing through two given distinct points " P " and " Q ".
8.
In particular, the external tangent lines to two circles are limiting cases of a family of circles which are internally or externally tangent to both circles, while the internal tangent lines are limiting cases of a family of circles which are internally tangent to one and externally tangent to the other of the two circles.
9.
In particular, the external tangent lines to two circles are limiting cases of a family of circles which are internally or externally tangent to both circles, while the internal tangent lines are limiting cases of a family of circles which are internally tangent to one and externally tangent to the other of the two circles.
10.
The circles defined by the Apollonian pursuit problem for the same two points "'A "'and "'B "', but with varying ratios of the two speeds, are disjoint from each other and form a continuous family that cover the entire plane; this family of circles is known as a " hyperbolic pencil ".
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