Graphics Reference
In-Depth Information
and
()
=
()
=- -
()
-
()
-
xt
t
2
3
(14.12)
yt
t
12
t
18
t
...
The fact that there is more than one solution indicates that we are at a singular point
because only one place exists at a regular point.
In general, the equations like (14.10) that one uses to solve (14.9) for a
i
and b
i
can
actually be put into the form
(
)
—
()
m
()
=
f
p
0
•
g
t
c
,
fm
,
where g(t) = (x(t), y(t)). One deals with them in a way similar to how equations (13.18)
and (13.19) are handled in Section 13.5.2. At nonsingular points where we have a
unique
solution we trace f using an approximation to g(t), say by a cubic approxi-
mation by using the first three terms of the series, and then use a Newton-Raphson
method to converge to a point on the curve.
So far there is nothing new. Things get interesting when we approach a singular-
ity on
C
. Assume that the singularity is at the origin. We need to find a transforma-
tion that will map our curve into a singularity-free curve.
Removing a Singularity at the Origin.
We use the quadratic transformations
Tu
x
y
Tux r
:
=
:
=
(14.13)
1
2
y
x
v
=
vy
=
.
One can show that T
1
has the following properties (with similar properties for T
2
):
(1) The y-axis gets mapped to infinity.
(2) Away from the y-axis T
1
is one-to-one.
(3) The tangent lines of
C
at the origin will get mapped to distinct tangent lines
at distinct regular points of the curve defined by g.
Furthermore, it is known (see Section 10.12 in [AgoM05] and in particular Lemma
10.12.15) that the singularity-free curve g(u,v) = 0 we are seeking can be obtained by
applying a finite number of transformations of the form T
1
or T
2
. If our curve has no
vertical tangent lines, then we use T
1
, otherwise we use T
2
. The transformations
resolve our singularity into a finite number of points on the curve defined by g.
We shall clarify the singularity removal process with an example. First, recall that
it is easy to compute the equation of any transformed implicitly defined object. In our
case, the transformations T
1
or T
2
map the curve
C
into a curve that has equation
(
)
=
(
)
=
(
)
=
1
1
-
(
)
(
)
=
2
1
-
(
)
guv
,
fT
uv
,
0
or
guv
,
fT
uv
,
0
,
respectively, and the inverses of T
1
and T
2
are defined by
