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free to move. Figure 11.36 now shows how a particular choice of the fourth point can
lead to the different types of cubic curves. The object therefore is to describe the
regions in the plane, so that, as
p
3
varies over the points of a region, we generate the
same type of singularity. Figure 11.37 is the diagram one obtains if the domain of
the curve (11.133) is restricted to [0,1], where the labels have the following meaning:
Cusp line:
The parabolic curve defined by the equation D=0.
Parabolic point:
A point where the cubic curve degenerates into a quadratic
curve.
See [StoD89] for a more complete analysis.
Other papers on cusps and inflection points are [ManC92a] and [LiCr97].
Figure 11.36.
Cubic curve singularities.
Figure 11.37.
The regions of constant sin-
gularity type.
