Graphics Reference
In-Depth Information
(a)
(b)
(c)
(d)
(e)
Figure 10.10.
Different types of singularities.
Analysis.
See Figure 10.10(b). The origin, now an isolated point of the real curve,
is an ordinary double point with tangent lines X +
i
Y = 0 and X -
i
Y = 0.
X
3
- Y
2
10.6.8. Example.
= 0
Analysis.
See Figure 10.10(c). The origin is a double point but not ordinary. We have
a “cusp.”
(X
2
+ Y
2
)
3
- 4X
2
Y
2
10.6.9. Example.
= 0
Analysis.
See Figure 10.10(d). The origin is a point of multiplicity four. There are
two double tangents.
Y
6
- X
3
Y
2
- X
5
10.6.10. Example.
= 0
Analysis.
See Figure 10.10(e). There is a triple tangent and two simple tangents at
the origin.
The curve in example 10.6.8 is a special case of the well-known family of curves
p
-
XY
0,
(10.43)
where p, q > 1. These curves capture a whole class of singularities that can be indexed
by the pair of integers (p,q). In general though, singular points of plane curves are
much more complicated than that and cannot be catalogued that simply. A general
