Summary: Problem Set 7
The picture below lies in A2
R. Each of the 3 darker line segments in the picture
represents a steel bar of length 1. The other two line segments have length 2. Let
A = (0, 0), B = (0, 2), C = (x1, y1) andD = (x2, y2). PointsA andB cannotmove
but points C and D can move. The steel bars are connected with hinges. Similar
to the last problem set, 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 (almost) using Macaulay 2 and elimination theory.
Problem 1. Try to 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 more quadratic polynomials. Let I be the ideal generated by the ve
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 2