Projections of The Twisted Cubic
The curve parameterized by (t,t^2,t^3) in affine 3 space
is displayed here. By manipulating the figure with the mouse
the different projections of the twisted cubic curve to the plane
of the screen can be observed. Note that generally speeaking
the projection intersects itself. In special cases the projection
does not have a loop in the curve, but rather a cusp. In
all cases the projection of the twisted cubic to the plane is
a singular cubic curve in the sense that at some point it does
not have a well defined tangent line. If we look along a line
that is tangent to the twisted cubic we see that the projection
has a cusp.
Note some projections don't appear to be singular. This is because
we don't see the whole curve. If we clipped out a larger portion
of 3-space or took an different affine open subset in our picture we
would see the either a loop or a cusp.