inversion of plane
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 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.
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
, i.e. (http://planetmath.org/Ie)
and , then the mapping is given by defined by .
Properties of inversion
The inversion is involutory, i.e. if , then .
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).
|Title||inversion of plane|
|Date of creation||2015-06-14 18:40:35|
|Last modified on||2015-06-14 18:40:35|
|Last modified by||pahio (2872)|
|Synonym||mirroring in circle|