Graphics Reference
In-Depth Information
(1) Determine the tangent line at the origin of the branch of f that we are cur-
rently on. Suppose that it has slope equal to m.
(2) Compute the singularity-free curve g(u,v) = 0 using a composite T of the quad-
ratic transformations T
i
as described above. Let
q
= T(
p
).
(3) Trace g from
q
to a point
q
¢ that is a small distance past (0,m) and map these
traced points back to f using the inverse of T.
We will now be at the point
p
¢=T
-1
(
q
¢), which is past the singularity of f and we con-
tinue tracing f as before until we get to a new singularity.
Three issues were brushed over in the description of our algorithm: determining
when we are approaching a singularity, moving it to the origin, and making sure that
we do not invert our tracing direction when we move to tracing g.
Finding Singularities.
Our singular points are defined by the constraints f = f
x
= f
y
= 0. With infinite precision there would be no problem, but without that it turns out
that we should use the condition number of the matrix
f
-
f
Ê
Á
ˆ
˜
x
y
f
f
y
x
to determine when we are getting close to a singularity. [Hoff89] suggests several
approaches. Two possible iterative approaches are using a least squares method or
some sort of constrained minimization. Two possible direct approaches for finding
roots are to use resultants or Gröbner bases.
Moving Singularities to the Origin.
The problem here is that we need our singu-
larity to be at the origin to apply our quadratic transformations, but we may have to
apply a series of these and they are sensitive to numeric errors in the transformed
functions.
Preserving the Tracing Direction.
Since the vector —f = (f
x
,f
y
) is normal to the
curve, the orthogonal vector
v
= (-f
y
,f
x
) will be tangent to it and will be the default
direction in which to start traversing f. This choice is motivated by the fact that the
ordered basis (—f,
v
) induces the standard orientation of
R
2
because
—
f
v
Ê
Ë
ˆ
¯
det
>
0
.
Intuitively this corresponds to preferring a counterclockwise direction, as for example
in the case of the unit circle defined by f(x,y) = x
2
+ y
2
- 1.
Definition.
We call (-f
y
,f
x
) the
standard trace direction for f
.
Now the actual tracing of a curve is done using a parameterization g(t) = (x(t),y(t))
as defined by equations (14.9). The parameter t in the parameterization induces a
direction on the curve that will agree with our choice if
f
ab
—
Ê
Ë
ˆ
¯
d
g
=
det
>
0
.
f,
(
)
,
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