Limiting Point

PonceletsPorismInversion

A point about which inversion of two circles produced concentric circles. Every pair of distinct circles has two limiting points.

PointCircles

The limiting points correspond to the point circles of a coaxal system, and the limiting points of a coaxal system are inverse points with respect to any circle of the system.

To find the limiting point of two circles of radii r and R with centers separated by a distance d, set up a coordinate system centered on the circle of radius R and with the other circle centered at (d,0). Then the equation for the position of the center of the inverted circles with inversion center (x_0,0),

[画像: x^'=x_0+(k^2(x-x_0))/((x-x_0)^2+(y-y_0)^2-a^2), ]

(1)

becomes

x_1^' = [画像:x_0+(k^2(d-x_0))/((d-x_0)^2-r^2)]

(2)

x_2^' = [画像:x_0+(k^2(0-x_0))/((0-x_0)^2-R^2)]

(3)

for the first and second circles, respectively. Setting x_1^'=x_2^' gives

[画像: (d-x_0)/((d-x_0)^2-r^2)=(-x_0)/(x_0^2-R^2), ]

(4)

and solving using the quadratic equation gives the positions of the limiting points as

[画像: x^'=(d^2-r^2+R^2+/-sqrt((d^2-r^2+R^2)^2-4d^2R^2))/(2d). ]

(5)

Explore with Wolfram|Alpha

WolframAlpha

More things to try:

References

Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, & Co., p. 43, 1888.Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, pp. 123 and 130, 1928.

Referenced on Wolfram|Alpha

Limiting Point

Cite this as:

Weisstein, Eric W. "Limiting Point." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LimitingPoint.html

Limiting Point

See also

Explore with Wolfram|Alpha

References

Referenced on Wolfram|Alpha

Cite this as:

Subject classifications