The implicit equation of the tangent surface.
By substituting from the parametrization, we obtain:
y - x² = - u² and
z - 3xy + 2x³ = 2u³.
This leads to the following version of the implicit equation:
4(y - x²)³ +
(z - 3xy + 2x³)² = 0.
We can use this version of the equation to prove that
the twisted cubic, i.e.
y = x² and z = z³,
is the singular locus of the tangent surface. On the other hand,
the terms of degrees 5 and 6 are the same on both sides of the equation, so that the
implicit equation can be simplified to the following degree 4
version:
z² - 6xyz + 4x³z + 4y³
- 3x²y² = 0.
The osculating plane at a point of the twisted cubic,
and the tangent cone (of the tangent surface) at the same point.
For simplicity, let's consider the situation at the origin.
The lowest degree homogeneous component of the defining equation of the
surface is z², so the tangent cone at the origin is
given by the equation z² = 0.
On the other hand,
the parametrization of the curve is given by the vector valued function
f(t) = (t,t²,t³),
so that
f'(0) = (1,0,0)
is a tangent vector at the origin. Furthermore, the osculating plane to
the curve at the origin corresponds to the vector space spanned by
f'(0) and
f"(0) = (0,2,0).
{In general, the principal normal to the curve at the origin
is perpendicular to f'(0)
and lies in the subspace of R³ spanned by
f'(0) and
f"(0) = (0,2,0).
The fact that f"(0) happens to
be perpendicular to f'(0)
is specific to the parameter value t = 0.}
Anyway, the osculating plane at the origin is the plane
z = 0. This means that the tangent cone to the surface
at the origin coincides with the osculating plane to the curve at the
origin -- or more precisely, the tangent cone is equal to the
osculating plane, counted with multiplicity = 2.
In the case of the twisted cubic, one can check directly
that at every point of the curve, the tangent cone to the tangent surface
is equal to the osculating plane of the curve at the same point --
counted with multiplicity 2 of course. The verification is somewhat intricate
and is not presented here. I suspect that
this is a particular
instance of a theorem about space curves, but I don't know a reference for such
a theorem.
Return to the initial view
of the tangent surface of the twisted cubic.
Return to the view
in which all tangent line segments have length = 2.