CHAPTER 3
INVERSIONS – NON-RIGID TRANSFORMATIONS
1. Let AB be a chord of a circle and let AT be any ray from A. Then AT is tangent to the
circle at A  BAT is equal to the angle subtended by the intercepted arc at the
circumference
2. Let P be a point outside a given circle q. Let PT be a straight line, and let PAB be a
secant with chord AB. Then PT is tangent to the circle q at T  PA  PB  PT 2
3. Let q be a circle with center C and radius r. Two points P & P are said to be symmetrical
with respect to q (or P is symmetrical with P ) 
 C , P & P  are collinear;
 C is outside PP 
 CP  CP   r 2
4. N.B.
 P is symmetrical with P   P  is symmetrical with P
 P is symmetrical with itself  P lies on the circle q
 No point is symmetrical with C
5. Given a point inside a circle (not the center), construct the corresponding symmetrical
point.
6. Definition: The Inversion Function
 Let q be a circle with center C and radius r.
 Let P & P be symmetrical with respect to q
 The inversion function I C , r is the function such that I C ,r ( P )  P 
7. N.B.
 The inversion function is undefined at C.
 The inversion is an involution
 The fixed points of the inversion function are those that lie on the circle.
8.
a.
b.
c.
d.
Theorem: The inversion function maps
Straight lines containing C onto themselves
Straight lines not containing C onto circles through C
Circles through C onto straight lines not containing C
Circles not through C onto circles not through C
9. Examples:
 Consider the inversion I O , 4 where O is the origin. Let m = - 4. Then I O , 4 (m) is a
circle that contains O. Observe that I O , 4 fixes (- 4, 0), i.e. (- 4, 0) lies on this
circle. The center of I O , 4 (m) lies on the x-axis. Thus, this circle is of radius 2 with
center at (- 2, 0).

If q is the circle centered at (0, 2) with a radius of 1 unit, then I O , 4 (q) is a circle
bisected by the y-axis. The intercepts of q with the y-axis are (0, 1) and (0, 3) and
these are transformed by I O , 4 to the points (0, 16/3) and (0, 16).

What is the inversion that transforms the circle of radius 2 that is centered at the
origin into the line y = 6?
10. Inversions preserve measures of angles and reverse the orientation of angles.
11. Transformations that preserve angle measures are said to be conformal.
12. Note that inversions do not transform rectilineal angles to rectilineal angles.
13. Theorem: Inversions are conformal transformations of the plane.
14. Theorem: Let q be a circle with center C and radius r, and let p be any other circle. Then,
the inversion I C , r fixes the circle p iff the circles p and q are orthogonal.
EXERCISES:
1. If O denotes the origin, to what point or curve does the inversion I O , 5 transform the
following sets:
a. The point (-2, 3)
b. The line x  y  5
c. The circle centered at (0, 3) with radius 1
2. For each of the following pairs of curves, decide whether there exists an inversion that
transforms one onto the other. Identify the inversion if it exists
a. The y-axis and the straight line x = 2.
b. The circle x 2  y 2  16 and the straight line x = 2
c. The circles x 2  y 2  16 and ( x  4) 2  y 2  16
d. The circles x 2  y 2  16 and ( x  34) 2  y 2  900
3. Let p and q be circles with unequal radii. Prove that there is an inversion that transforms
p onto q
4. Prove that in polar coordinates for any point on the curve r  f ( ) , the angle  from
r
the radius vector CP to the tangent line at P is given by tan( ) 
r
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Assignment 6 (due Nov 6th 2008)
Do exercises 1c, 2d, and 4. above.
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