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.