The figure shows the tangent surface of a particular rational curve of degree 4.
Thus, the surface is the union of the tangent lines of that curve. The curve
is given parametrically by:
t > (x,y,z) = (t, t^{2},
t^{4}),
so that the surface is parametrically by:
(t,u) > (t+u, t^{2} + 2tu, t^{4}
+ 4t^{3}u).
The 4^{th} degree rational curve is lightly sketched in
dark blue on the surface.
 
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In the portion shown here, we have 1 < t <
1, while the range of uvalues varies with t
in such a way that a tangent line segment of length = 2 is
shown for each value of t. The midpoint of each
tangent line segment is at a point of our degree 4 rational curve.
 ¿ How is this surface different from the tangent surface of the
twisted cubic? The most visible difference is that the surface
crosses itself  and this is something that does not happen
for the tangent surface of the twisted cubic. Thus, for
t ≠ 0, the tangent line to our 4^{th} degree
rational curve at
(x,y,z) = (t,t^{2},t^{4})
intersects the tangent line at
(x,y,z) = (t,(t)^{2},(t)^{4}). {The intersection of the two tangent lines occurs at
(0,t^{2},3t^{4}).} As
t varies, we obtain a line of intersection points. Two sheets
of the surface cross each other along this line.
 For comparison:
click here to view
the tangent surface of the twisted cubic.
 ¿ Is this curve a generic projection? Of course,
the answer depends on what we mean by "generic". Our curve certainly is a
projection of the rational normal curve in 4space, given parametrically
{in the affine version} by
t > (t,t^{2},t^{3},t^{4}).
And it certainly is nonsingular; moreover, these properties extend to the
projective closure. So far, so good.
On the other hand, the
projective closure does not have a well defined osculating plane at its
(unique) point at infinity. {Proof omitted.} To see why this could
disqualify our curve from being a generic projection, we just have to observe
that if a nonsingular curve in nspace, with n ≥ 4,
has a well defined osculating plane at every point, then its projection from a
sufficiently general center is a nonsingular curve in 3space which
has a well defined osculating plane at every point.
{Indeed, the union of the osculating planes of the curve in nspace
is a 3dimensional variety, and thus not all of nspace when
n ≥ 4.}
