y-x^{2}=0

z-x^{3}=0

that defines a curve in (x,y,z) 3-space
called the **twisted cubic curve**, whose projection onto
the yz-plane is shown below. You can click on the graph of the projection
to generate an MPEG movie (308K) of the curve in 3-space.

If we try to eliminate the variable x from those
equations, we can produce the equation z^{2}-y^{3}=0, which is
the curve with a cusp in the yz-plane:

If you project the twisted cubic curve orthogonally onto the
yz-plane, this curve with a cusp is exactly what you hit.
It should at least be clear that the equation z^{2}-y^{3} produced
by elimination of the variable x is satisfied by all points
in the projection, since the projection map takes a point
(x,y,z) satisfying the first two equations (and hence also
satisfying z^{2}-y^{3}) and maps it to (y,z).

More generally, if we are given a system of equations

f_{1}(x_{1},...,x_{n})=0

f_{2}(x_{1},...,x_{n})=0

etc.

they define a subset V of the n-space with coordinates
x_{1},..., x_{n},
Then any equations g(x_{k},x_{k+1},...,x_{n})=0 obtained
from the f_{i}'s by elimination of the variables
x_{1},...,x_{k-1}
will be satisfied by all points in the projection of V onto
the (n-k+1)-subspace with coordinates x_{k},x_{k+1},...,x_{n}.
This is part of what we mean by "elimination = projection".
For a more detailed discussion of the case where the f_{i}
are polynomials, and a more precise statement about the relation
between elimination and projection in this case, see
Cox, Little, and O'Shea Sections 3.1 and 3.2.

How did we decide which points (x,y,z) on the torus to project into
the plane? If you stare at the picture and think about it,
you'll see that we chose to project the points where the tangent plane to the
torus is parallel to the z-axis. We will call this set the
**z-profile** of the surface.

To find equations in x,y that contain the z-profile, we must eliminate the variable z from the equations in x,y,z that define points on the torus that have tangent plane parallel to the z-axis. So first we need to figure out what those equations are.

- Explain how to calculate the normal vector at a point P on the surface f(x,y,z)=0 using facts about tangent planes and normals to level sets of a function that you learned in multivariable calculus. (Hint: the gradient vector (df/dx, df/dy, df/dz) of f evaluated at P is involved somehow).
- Once you have the normal vector at P, how can you describe the condition that the tangent plane is parallel to the z-axis?
- Put this together to explain why the points in the z-profile should
be the projections to the xy-plane of the solutions (x,y,z) of
f(x,y,z)=0

d/dz f(x,y,z)=0

This suggests that if f(x,y,z) is a polynomial, we could compute the equation for the z-profile of the surface f(x,y,z)=0 using a Gröbner basis computation in Maple to eliminate variables from the two equations above. This is exactly what we did for the tilted torus.

We would like to be able to compute an equation satisfied by the points of the envelope if we are given the equation F(t,x,y)=0. The key observation is this: the

This then suggests that when F(t,x,y) is a polynomial, we can follow the recipe for computing the t-profile of the surface F(t,x,y) to find the equations satisfied by the points of the envelope. That is, we eliminate t (using a Gröbner basis computation) from the equations

F(t,x,y)=0

d/dt F(t,x,y)=0

Here is the computation for the family of circles we discussed earlier:

>F:=(x-t^2)^2+(y-t^3/3)^2-(t^3/3)^2; 2 2 3 2 6 F := (x - t ) + (y - 1/3 t ) - 1/9 t > GB:=gbasis({F,diff(F,t)},[t,x,y],plex);Only the 7th (last) polynomial in the resulting Gröbner basis did not involve t, so by factoring it we obtain the equations for the envelope curves:

> factor(GB[7]); 2 2 3 2 2 2 2 4 y (x + y ) (64 x + 3 y x + 72 y x - 144 y + 3 y )The resulting implicit equation for the envelope can then be plotted to obtain the red curves in the 4th figure on this page.

- Can you explain why the ``extra" factor y shows up in the envelope equation? What does it correspond to in the picture of the envelope?
- Does the extra factor x
^{2}+y^{2}in the envelope equation correspond to something you can see in the picture?

For a different (and more detailed) discussion of envelopes, see Cox, Little, and O'Shea Section 3.4. For an even deeper discussion, see Bruce and Giblin.

Vic Reiner <reiner@math.umn.edu> Frederick J. Wicklin <fjw@geom.umn.edu> Last modified: Thu Apr 25 08:20:55 1996