Unformatted text preview: Math402 Circle Inversion Let c be a circle on the Euclidean Plane with the center 0 and the radius 'r. The inversion in c is a transformation of the plane (or more precisely, of the plane with either the
center of the circle removed or the point 00 added) which takes a point P to the point P’ deﬁned
as follows (cf. Deﬁnition 2.35 in the book): One can think of an inversion as a “reﬂection in a circle”. In particular the inversion takes points
inside of c to points outside, preserves points on c, and is reﬂexive, 216. if P goes to P’, then P’
goes to P. Here are some basic properties of inversions (you’ll have to provide illustrations to them in the
homework). Some of these properties have been proved in class and others in the homework. o Inversion takes the inside of the circle c to the outside and vice versa. The center 0 goes
to 00. o Inversion is reﬂexive: (P’)’ = P. o Inversion preserves the circle 0 point—Wise. o Inversion takes circles and lines (tie. circles of inﬁnite radii) into circles and lines. However
it can take a circle into a line or a line into a circle. 0 A line through the center of inversion O is preserved under the inversion. o A circle containing 0 becomes a line not containing 0 after inversion. A line not containing
0 becomes a circle containing 0. o A circle not containing 0 becomes another circle not containing 0 after inversion. o A circle perpendicular to c is preserved under inversion. o Angles between curves are preserved under inversion. o Distances between points are not preserved under inversion. Inversion is not an Euclidean transformation (an isomorphism of the plane as a model of Euclidean
Geometry), because it does not take lines to lines and does not preserve distances. ...
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 Spring '09
 malkin

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