Now, we have a planar node curve with a similar equation:
Z² = Y² - 4Y³,
and we also can substitute
Y = y - x² and
Z = z - 3xy + 2x³ into the equation of this curve. If we do this, we obtain the
following equation of a surface:
(z - 3xy + 2x³)² =
(y - x²)² - 4(y - x²)³.
Exactly as with the implicit equation of the tangent surface, the terms of
degrees 5 and 6 cancel, so that we have a
surface of degree 4.
And since the equations y ‑ x² =
z - 3xy + 2x³ = 0
define a nonsingular space curve (namely the twisted cubic), it follows that
the singular locus of our surface is the twisted cubic.
Our next objective is to find a parametrization
of this surface. We begin by observing that the planar node curve has the
following parametrization:
(Y,Z) = (¼ - u², 2u(¼ - u²)).
{To see how this was obtained, divide the equation of
the node curve by 4Y². This yields
(Z⁄2Y)² =
1⁄4 - Y,
which we use to express
Y in terms of
u := Z⁄2Y.}
Using this parametrization of the curve, we obtain a surface parametrization
by solving the following equations:
y - x² = ¼ - u²
z - 3xy + 2x³ =
1⁄2u
- 2u³.
We find the following parametrization:
(x,u) --> (x, x² - u²
+ 1⁄4, x³
+ 3⁄4x
- 3xu²
+ 1⁄2u
- 2u³).
Note, in particular, that for u = ±½ the
parametrization yields a point of the twisted cubic. One also can use
this parametrization to show that our surface (this
affine model, anyway) has a nonsingular normalization.
This parametrization ( with some
re-scaling) was used in constructing the figure. The
rectangular parameter region
- 5⁄8 ≤ x,u ≤
5⁄8
is shown, with some trimming at two of the corners.
Our final objective is to show how the
parametrization can be modified in order to verify that our surface is indeed
ruled. We set t = x + u, so that
x = t - u. Substituting this into the parametrization and
doing some calculation (mainly taking advantage of
various cancellations), we obtain the following
parametrization:
(t,u) --> (t - u, t² - 2tu
+ 1⁄4, t³
+ 3⁄4t
- 3t²u
- 1⁄4u).
Since all three coordinates are given by expressions that are linear
in u,
it follows that the surface is ruled.