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inversion - Math402 Circle Inversion Let c be a circle on...

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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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