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Inverse Points


InversePoints

Points, also called polar reciprocals, which are transformed into each other through inversion about a given inversion circle C (or inversion sphere). The points P and P^' are inverse points with respect to the inversion circle if

 OP·OP^'=OQ^2=k^2

(Wenninger 1983, p. 2). In this case, P^' is called the inversion pole and the line L through P and perpendicular to OP is called the polar. In the above figure, the quantity k^2 is called the circle power of the point P relative to the circle C.

Inverse points with respect to a triangle are generally understood to use the triangle's circumcircle as the inversion circle (Gallatly 1913).

The point P^' which is the inverse point of a given point P with respect to an inversion circle C may be constructed geometrically using a compass only (Coxeter 1969, p. 78; Courant and Robbins 1996, pp. 144-145).

Inverse points can also be taken with respect to an inversion sphere, which is a natural extension of geometric inversion from the plane to three-dimensional space.


See also

Circle Power, Geometric Construction, Inversion, Inversion Pole, Inversion Circle, Inversion Sphere, Limiting Point, Polar

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References

Courant, R. and Robbins, H. "Geometrical Construction of Inverse Points." §3.4.3 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 144-145, 1996.Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969.Gallatly, W. The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, 1913.Wenninger, M. J. Dual Models. Cambridge, England: Cambridge University Press, 1983.

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Inverse Points

Cite this as:

Weisstein, Eric W. "Inverse Points." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/InversePoints.html

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