Graphics Programs Reference
In-Depth Information
shows that triangles
OPQ
and
OP
∗
Q
∗
are similar (notice that they have a common
angle), which, in turn, implies that angles
OPQ
and
OQ
∗
P
∗
are equal. Since the former
is a right angle, the latter must be also. However, point
Q
is an arbitrary point on
L
,
so angle
OQ
∗
P
∗
equals 90
◦
for any point
Q
on
L
, showing that the projection
Q
∗
lies
on a circle that passes through the origin
O
and has a diameter
OP
∗
. The projection
of
P
is
P
∗
, and the projection of the origin is the point (or points) at infinity. Line
L
of
Figure 4.18 passes inside the unit circle. For lines outside this circle, the diagram looks
different but the proof is identical.
Exercise 4.7:
Use similar arguments to prove feature 3.
Exercise 4.8:
The discussion so far has assumed inversion with respect to the unit
circle. Given a circle
C
of radius
R
about the origin, show how to project a point
P
with respect to it.
Figure 4.19 shows a simple geometric construction of the inverse of a point
P
.In
part (a) of the figure,
P
is inside the circle. Line
L
1
is constructed from the center
through
P
and continues outside the circle. Line
L
2
is then constructed perpendicular
to
L
1
.Point
A
is the intersection of
L
2
with the circle. A tangent
L
3
to the circle is
constructed at
A
,and
P
∗
is placed at the intersection of the tangent and
L
1
.Part(b)
shows the similar construction when
P
is outside the circle.
A
A
L
3
L
3
L
2
L
2
R
P
P
*
L
1
L
1
P
P
*
P
0
(a)
(b)
Figure 4.19: Construction of Circle Inversion.
Figure 4.19b illustrates another feature of circle inversion. Up to now, we assumed
that the inversion is about a unit circle centered on the origin. Given a circle of radius
R
, the two triangles
PP
0
A
and
P
∗
P
0
A
are similar, implying that
P
0
P
/R
=
R/
P
0
P
∗
or
R
2
=
P
0
P
×
P
0
P
∗
.Thequantity
R
2
is termed the
circle power
.Theinverse
P
∗
of a
point
P
with respect to an inversion circle of radius
R
centered at
P
0
is given by
P
∗
=
P
0
+
R
2
P
−
P
0
2
.
|
P
−
P
0
|
As is common with nonlinear projections, it is possible to come up with many
variants of circle inversions. For example, project point (
r, θ
)to(1
/r,
180
◦
+
θ
). An
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