A Lego Mindstorms linkage
- Zoltán Kovács
By using GeoGebra, it is not only possible to create the numerical model of the linkage (see the green curve on the right). But it is also possible to compute the curve symbolically (see the red curves). For mathematical reasons GeoGebra also computes a different curve (on the left, also in red): it is the set of positions of the point of another sword, into the opposite direction. By dragging the points, it looks obvious that the green curve on the right is not an ellipse. By computing the equation of the union of the two curves it seems more clear that the curve on the right is indeed not an ellipse. A precise proof can be, however, a bit more difficult since it should be proven that the polynomial in the equation is not reducible, therefore it does not contain the ellipse as a component. Actually, the real model is somewhat more difficult than this simplified approach above. Below you can find the more realistic mathematical model. It delivers a very similar curve which is again of 6th grade, namely, it is a sextic. For comparison, Watt's curve (which arises in connection with his steam engine) is also a sextic.
Credits: My son, Benedek (12) built this robot.