Summary: Problem Set 6
The picture below lies in A2
R. Each of the 3 line segments in the picture
represents a steel bar of length 1. Let A = (0, 0), B = (0, 2), C = (x1, y1) and
D = (x2, y2). Points A and B cannot move but points C and D can move. The
steel bars are connected with hinges. Point M is exactly in the middle of the
bar. It turns out that as the three bars move into every allowable position, the
point M sweeps out a curve. This curve is an irreducible ane variety given as
V (F) for some polynomial F R[X, Y ]. In the problems following the picture,
you will compute F using Macaulay 2 and elimination theory.
Problem 1. Give a rough sketch of the curve traced out by M.
Now we will construct an ideal, I, in R[X, Y, x1, x2, y1, y2] which represents all
allowable congurations and the corresponding positions of M. Each bar yields a
constraint on the variables x1, x2, y1, y2, this yields 3 quadratic polynomials. Let
M = (X, Y ) and write down the coordinates of M in terms of x1, x2, y1, y2, this
yields 2 linear polynomials. Let I be the ideal generated by the three quadratic
polynomials and the two linear polynomials.
Problem 2. Write out the equations for I.
We would like to know all of the allowable values of X and Y . This corresponds
to computing J = I R[X, Y ]. If you carry out this computation in Macaulay