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, t2,
t4),
so that the surface is parametrically by:
(t,u) --> (t+u, t2 + 2tu, t4
+ 4t3u).
The 4th 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 u-values 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 4th degree
rational curve at
(x,y,z) = (t,t2,t4)
intersects the tangent line at
(-x,y,z) = (-t,(-t)2,(-t)4). {The intersection of the two tangent lines occurs at
(0,-t2,-3t4).} 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 4-space, given parametrically
{in the affine version} by
t --> (t,t2,t3,t4).
And it certainly is non-singular; 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 non-singular curve in n-space, with n ≥ 4,
has a well defined osculating plane at every point, then its projection from a
sufficiently general center is a non-singular curve in 3-space which
has a well defined osculating plane at every point.
{Indeed, the union of the osculating planes of the curve in n-space
is a 3-dimensional variety, and thus not all of n-space when
n ≥ 4.}
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