1. Inversion Transformation
Lisa Berry
Stephanie Kimber

2. Inversion
Consider a circle with center O and radius r. 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’)=r2.
The circle is called the circle of inversion.
The point O is the center of inversion

3. Begin Part One of Activity 1

4. Activity 1 Questions: Where does P’ go when P is inside the circle?
When P is outside the circle?
When P is on the circle?
When P is on the circle center O?

5. Basic Conclusions drawn from (OP)*(OP’) =r2
If (OP)(OP’) = r2, then (OP’)(OP) = r2
If point P’ is the inverse of P, then P is the inverse of P’
If OP < r, then OP’>r
If a point is on the interior of the circle of inversion, then its inverse is on the exterior.
If OP>r, then OP’<r
If a point of the exterior of the circle of inversion, then its inverse is on the interior
If OP=r, then OP’=r
If a point is on the circle of inversion, then so is the inverse. This is an invariant point. The point is its own inverse.

6. (OP)*(OP’)=r2
OPQ ~ OQP’ by AA So
By algebra we get
OP*OP’ = OQ2

7. Verify OP*OP’=r2 by completing Part Two of Activity 1 Begin Activity 2, Part 1

8. Activity 2 Questions Part 1
As point P travels along the line, what do you notice about
the trace of P’?
What does this mean in terms of a line and its inverse?

9. Answers Part 1 As point P travels along the line
the trace forms a circle.
This means that the inverse
of a line is a circle.

10. Begin Part Two of Activity 2

11. Part 2
Questions
Now that the line goes through the center of the circle, what do you observe about the line’s inverse?
What does this mean in terms of a line and its inverse if the line goes through the center of inversion?

12. Answers Part 2 When the line goes through the
center of the circle, the trace of
P’ is on the line.
The inverse of a line that goes
through the center of
inversion is the line itself.

13. Begin Part Three of Activity 2

14. Activity 2 Questions Part 3 What is the inverse of a circle?
How does the location of circle A change its inversion?
How does the size of circle A change its inversion?
What is the inversion of a circle whose edge falls on the inversion center?

15. Answers Part 3 The inverse of a circle is a circle.
As a circle gets closer to the inversion
center, the inversion circle gets larger
and further away from the circle of
inversion.
As circle A gets smaller/larger, the
inversion circle get smaller/larger.
Now the inversion of the circle A is a line.

16. Activity 2 Part 4 The inverse of a circle that
is orthogonal to the circle of
inversion is the circle itself

17. Inversion Preserves Angle Measure

18. Inversion Preserves Angle Measure

19. Inversion Preserves Angle Measure

20. Inversion Preserves Angle Measure

21. Inversion Preserves Angle Measure

22. Inversion Preserves Angle Measure

23. Summary Here are six propositions that exist:
The inverse of a line (not through the center of inversion) is a circle through the center of inversion
A line through the center of inversion is its own inverse.
The inverse of a circle (not through the center of inversion) is a circle.
The inverse of a circle through the center of inversion is a line
The circle orthogonal to the circle of inversion is its own inverse.
Angles are preserved in inversion.

24. THE END