Automatic derivation of the locus equation
In this section we illustrate how it is possible to automatically derive the locus equation of some relevant moving point in a linkage. Our observation is Watt’s linkage, introduced in 1784, which had a crucial role in the industrial revolution.
Watt's linkage
James Watt was more proud of the parallel motion than of any other invention he has ever made. Despite of looking good for the first glance, a closer look on the trace of the central moving point shows that this movement is far from being straight. Dynamic geometry tools, including the very first versions of Geometer’s Sketchpad, Cabri and Cinderella were already able to demonstrate the motion of the pencil point approximately. For the non-expert, it may be however still a question if there is probably a short path of the motion which is completely linear, and by appropriately scaling Watt’s machine it is perhaps achievable to have an arbitrary long straight motion.
To get an exact answer to decide the question "how straight is the Watt linkage", when not considering the numerical experiments (which can still lead to a false impression), we end up that some symbolic computations may be needed to obtain a closed formula. The first attempts to get the exact implicit curve equation by using the construction steps of the figure were implemented in the GDI software tool. Later, the same idea was further developed in LADucation. However these tools are not maintained any longer, their know-how has already been implemented and improved in GeoGebra’s Automated Reasoning Tools (ART).
By using GeoGebra ART this can be checked as follows:
- Construct a GeoGebra model of the linkage.
- Obtain a precise polynomial equation of the movement by entering LocusEquation[T,M]. Here we remark that the plot of the equation (in red) fully contains l, but it is a superset of the locus. (By using complex algebraic geometry is not possible to separate the extra parts from the real locus.) The obtained equation (here ) does not have any linear factors which means that no part of the curve can be totally straight.
For this latter consequence we used a generalization of the fundamental theorem of algebra, namely, Bézout’s theorem. That is, the number of intersection points of two plane algebraic curves which do not share a common component is at most equal to the product of their degrees. Now let a = 0 be the equation of the obtained locus curve and s = 0 a straight line, and let us assume that a and s have no common factors, that is, they do not share a common component. Indirectly, assuming that their intersection a ∩ s has infinitely many points, we obtain that the product of the degrees of a and s must be ∞. Since deg(a) < ∞ and deg(s) = 1 we reached a contradiction.
In general it may be challenging to decide if a polynomial can be factorized or it is irreducible. For a safe answer one needs to factorize over the reals. In most computer algebra systems (including GeoGebra) factorization is implemented only over the integers.