Graphics Programs Reference
In-Depth Information
y
P *
P
r
θ
1
x
Q
Q *
Figure 4.16: Circle Inversion.
along the line that connects P to the origin, we can think of this projection as scaling.
From P =(1 /r, θ ), we obtain x 2 + y 2 =1 /r 2 and this implies
( x, y )
x 2 + y 2
P
x 2 + y 2
P =( x ,y )=
=
= s P
because this relation means that
x 2
( x 2 + y 2 ) 2 +
y 2
( x 2 + y 2 ) 2
1
( x 2 + y 2 ) =1 /r 2 .
x 2 + y 2 =
=
Notice that the scale factor s depends on P , showing that this type of projection is
nonlinear.
Currently, there are several applets on the Internet that make it easy to explore
the properties of circle inversion. This projection has a number of interesting features,
the most important of which are the following:
1. Any circle that intersects the unit circle at right angles is projected to itself.
2. The angle between two projected lines is preserved. Thus, circle inversion is a
conformal projection.
3. Circles that do not pass through the origin are projected into circles (that do
not pass through the origin and generally have a different radius).
4. Similarly, lines that do not pass through the origin are projected into circles
that do pass through the origin (Figure 4.17).
5. A circle centered on the origin is projected to another circle similarly centered.
6. Lines through the origin are projected to themselves (except that the projection
of the origin is undefined).
7.
Thus, ( P ) = P .
The inverse of an inverse is the original point.
(This is
trivial.)
Curves that are their own inverse are called anallagmatic.
 
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