Let be a fixed circle in the Euclidean plane with center and radius . Set for any point of the plane a corresponding point , called the inverse point of with respect to , on the closed ray from through such that the product
has the constant value . This mapping of the plane interchanges the inside and outside of the base circle . The point is the “infinitely distant point” of the plane.
The following is an illustration of how to obtain for a given circle and point outside of . The restricted tangent from to is drawn in blue, the line segment that determines (perpendicular to , having an endpoint on , and having its other endpoint at the point of tangency of the circle and the tangent line) is drawn in red, and the radius is drawn in green.
(-2,-2)(4,2) \psline[linecolor=blue](1,1.732)(4,0) \psline[linecolor=red](1,1.732)(1,0) \psline[linecolor=green](0,0)(1,1.732) \psline(0,0)(4,0) \pscircle(0,0)2 \psdots(0,0)(1,0)(1,1.732)(4,0) \rput[a](0,-0.3) \rput[a](4,-0.3) \rput[l](1.12,1.83) \rput[a](1.0,-0.3) \rput(0.38,0.84) \rput(-1.55,1.6) \rput[b](0,-2). \rput[l](-2,0). \rput[a](0,2).
The picture justifies the correctness of , since the triangles and are similar, implying the proportion whence . Note that this same argument holds if and were swapped in the picture.
Inversion formulae. If is chosen as the origin of and and , then
Note. Determining inverse points can also be done in the
complex plane. Moreover, the mapping is always a
Möbius transformation. For example, if
and , then the mapping is given by defined by .
Properties of inversion
The inversion is involutory, i.e. if , then .
The inversion is inversely conformal, i.e. the intersection angle of two curves is preserved (check the Cauchy–Riemann equations!).
A line through the center is mapped onto itself.
Any other line is mapped onto a circle that passes through the center .
Any circle through the center is mapped onto a line; if the circle intersects the base circle , then the line passes through both intersection points.
Any other circle is mapped onto its homothetic circle with as the homothety center.
- 1 E. J. Nyström: Korkeamman geometrian alkeet sovellutuksineen. Kustannusosakeyhtiö Otava, Helsinki (1948).