![]() The first five may be inferred from observing that animated sketch. ![]() ![]() This is because P cannot be made to trace the entire line, which is, of course, limitless. Notice that when point P traces a line in the animated sketch, the trace of its inverse is discontinuous. Watch the path of the inverse as point P traces several paths. Let us say that it goes to a point at infinity.Ĭlick on the image below to activate an interactive Java sketch. The only thing unaccounted for is the center of the circle. Every point on the inside goes outside, every point on the outside goes inside, and all of the points on the circle itself stay put. The point is its own inverse.Īn inversion effectively turns the circle inside out. If a point is on the circle of inversion, then so is its inverse. If a point is on the exterior of the circle of inversion, then its inverse is on the interior. If a point is on the interior of the circle of inversion, then its inverse is on the exterior. If point P′ is the inverse of P, then P is the inverse of P′. Look at the equation again and see how many simple conclusions may be drawn from it: The circle is called the circle of inversion, and point O is the center of inversion. If point P is not O, the inverse of P with respect to the circle is the point P′ lying on ray OP such that ( OP)( OP′) = r 2. First, a definition of inversion:Ĭonsider a circle with center O and radius r. This is not a thorough treatment of the subject, but it might do for an introduction or a brush-up. A few of these lessons have employed inversion geometry, so it seemed to make sense to write something on the subject.
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