One of Isaac Newton's many accomplishments was the classification of the cubic curves. In other words, Newton undertook the task of identifying the different types of qualitative behaviour possible by by curves of equations of the following type

where
are fixed parameters and *x* and *y* are variables. IN order
for the equation to define a true cubic (rather than a quadratic) it will
be assumed that at least one *a*, *b*,*c* or *d* is
not zero. Newton found 72 different species of curve, but later investigators
found 6 more and it is now known that there are precisely 78 different
types of cubic curves. What exactly is meant by this assertion? What is
a "species" of curve? It is possible to give precise answers to these questions
but this would lead to technicalities whichit would be best to avoid at
the moment. To show that the answers to these questions are not at all
obvious, it is worth noting that Newton considered all equations of the
form

to have the same species; in spite of the fact that you shoud be able to recall from elementary calculus that these equations can have either zero or two extrema and can tend to either positive or negative infinity. This points to the fact that Newton considered two curves to be of the same species if a simple change of coordinates transforms one to the other. For the purposes of this assignment though, two curves with different numbers of extrema will be considred to be of different species. Furthermore, curves wich are the union of different numbers of disjoint curves will be considered to be different. Other characteristics which can be used to distinguish curves are the number of crossing points, existence and number of closed loops and the number of asymptotic branches.

A useful tool in your investigation will be the `implicitplot`
command which draws graphs of curves defined implicitly. It must be loaded
into Maple as part of the plots package. As an example of how to use this,
the following command draws a representative of one of the 78 types of
cubic curve known as the Folium of Descartes.

However, you should only use the `implicitplot `command when
it is absolutely necessary. The Folium of Descartes really does require
this command but the curve known as The Witch of
Maria Agnesi does not. The following two commands both draw part of
the graph of this curve

What difference is there in time of computation and in the accuracy
of the graphic representation for the preceding two commands? Notice that
the equation for The Witch of Maria Agnesi can easily be transformed so
that it defines *y* as an equation of the variable *x*. Once
this is done it is possible to apply the `plot `command as in the
second of the two examples. Whenever this is possible, the `plot `command
should be used instead of `implicitplot `because `implicitplot
`is far more likely to produce erroneous results.

Begin your investigation by considering various simple, special cases. For example, the case of the cubic function

should be easy to handle. Observe that this does not require the `implicitplot
`command. In fact, Newton was able to show that by making a suitable
change of variables, all cubic equations can be transfored into one of
the following forms

Moreover, 71 of the 78 different species which exist in Newton's scheme
come from the the last type of equation. This suggests that this is a good
place to search for examples by varying the values of the parameters
*a*,
*b*, *c*, *d* and *e*.

Maple's animation capabilites can be used to good effect in this problem. For example the changes which occur as one of the parameters is slowly changed can be displayed in an animation sequence. Can you find a configuration of parameters which allows one to be changed so that a graph with a disconnected closed loop transforms into a graph with a crossing point? What happens if the transformation is allowed to continue?