Another Look at the Linkage
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Given a circle in the plane, you now study a transformation of the plane associated with this circle called inversion through the circle. In other words, given a point F in the plane, there is another point F' called the inverse of F. The picture below shows a circle of inversion, a point F and its inverse F'. Although inversion may seem an artificial construction, it has many powerful applications in mathematics and physics.
Here is how you determine the inverse of F through a circle with center A and radius r. (Refer to the picture.) Draw a ray starting at A passing through F. The inverse F' of F is the point on the ray such that:
Another way to write this: If R is the point of intersection of the circle and the ray, then:
(AF')= [(AR)/(AF)] (AR)
Now use Sketchpad to construct the inverse of a given point through a given circle. When you are done, make the process into a script. Doing so will save you a lot of time.
Investigate the following set of questions. Think about them in terms of the definition of inversion as well as by investigation in Sketchpad.
As a point goes around a circle, its inverse traces out a shape. The shape traced out is called the inverse of the circle. Investigate the following questions about the inverses of circles and lines. Again, try to answer the questions first by thinking about them and then by using Sketchpad. Use the answers to fill out the chart below.
|Shape inverted||Inverse of shape|
|Circle through A|
|Circle not through A|
|Line through A|
|Line not through A|
Through every two points goes a line. Through every three points goes a circle: its center is the intersection of the lines that bisect the segments between the points. Use this and your table to do the following:
Test your construction: as a point goes around the circle, does its inverse always lie on the shape you constructed?
Author: Evelyn Sander
Comments to: firstname.lastname@example.org
Created: Jun 09 1996 --- Last modified: Jun 11 1996