This curve is called a capricornoid, because it’s supposed to resemble the sign of Capricorn in the zodiac, though I don’t see how. Its equation is
so it’s described by a quartic equation in two variables.
What makes this curve fun is that it has two different kinds of singularities. At the point
Further up, at the point
there is a crunode, meaning that two different branches of the curve cross each other, but are not tangent. The word ‘crunode’ is a bit old-fashioned: usually people call a point like this an ordinary double point.
The simplest example of a tacnode is the curve
This consists of a parabola and an upside-down parabola that are tangent at the origin: the tacnode is at the origin.
The basic example of an ordinary double point for a curve in the plane is
This consists of two lines that cross at the origin: the ordinary double point is at the origin. This is, in some obvious sense, a simpler singularity than the tacnode. But can we make this obvious sense precise?
is said to have singularity of type An at the origin. If we take we get
which has an ordinary double point at the origin: this is another way to draw two lines crossing. If we take we get
which is the simplest example of a spinode. Again this term is a bit old-fashioned: these days people call such a singularity a cusp.
And if we take we get
which is the simplest example of a tacnode!
So, there’s a systematic way in which these singularities are getting more complicated:
• crunode (ordinary double point):
• spinode (cusp)
and the series goes on forever.
I drew the capricornoid using the SageMathCloud. Since I’m not very good at computing, I found myself cursing a few times in the process. For example, this system seems to fall asleep if you don’t use it fast enough. When it falls asleep, it forgets that you’ve declared variables—that is, told it that variables are variables. Then it spews out obscure error messages.
But a more mathematically interesting reason for cursing is that it’s a bit hard to draw a tacnode! I said that the curve shown above is a picture of
but I was lying. It’s actually a picture of
If I draw a picture of
I get this:
This doesn’t include the origin, so we don’t get the tacnode—due to some discretization effect or small numerical errors, I guess. This says something interesting about what happens to a curve with a tacnode when you perturb it slightly!
All this is part of my slow and much belated plan to learn algebraic geometry.
Puzzle. Can you get the equation of the capricornoid by taking the equation of an ellipse
and the equation of another ellipse
and multiplying them to get