Articulated cube with fixed bar (II)

Topic:Congruence, Constructions, Cube, Geometry, Quadratic Equations

This activity belongs to the GeoGebra book Linkages. This last construction, more complete than the previous one, allows the coincidence of two or more vertices. But there will be some continuity problems. On the one hand, the possible coincidence of two or more vertices alters the freedom of movement of other vertices, as we have seen in the Intersection of two spheres activity. On the other hand, even if two vertices do not coincide, the drag effect prevents it from, for example, returning to the initial configuration (or an isomeric) after rotating E one full turn around O. Thus, the positions of E and A determine the circular paths along which we can move points F, B and D, which will not transmit their motion to either E or A. Points F, B and D determine, in turn, the only possible position (barring isomer) for J. This is true in general, but it can be different, as can be seen from the construction, if two or more vertices coincide.

Note: Depending on the position of points F, B and D, some of them may not be able to cover their entire orbit, as there may be positions where the circumradius of the circle passing through F, B and D is greater than 1 (which would make it impossible for point J to exist).

If we activate a checkbox that forces two vertices to coincide, it may be that the degrees of freedom of the vertices vary. For example, by activating the F=O and B=O checkboxes, point J acquires 1 degree of freedom (changes its color to blue) that it did not have before. If we also activate the checkbox D=O, vertex D loses its freedom, but point J acquires another degree of freedom (and changes its color to green). Whichever configuration is chosen, we can see that the cube never exceeds 6 internal degrees of freedom. In the construction it is possible to choose, in the case that the degree of freedom of the point J is 0, which isomer of J will be shown. The checkbox that activates this option is especially useful when J becomes invisible when it coincides with another vertex, such as U, E or A. Finally, the checkbox J=O allows observing the particular case in which two opposite vertices of the cube coincide. Note that since, in addition to O, point U is also fixed, point E can now be no further than

from point U. Likewise, point A cannot be further from U or E than that measure. These prohibited zones are represented in the construction in the form of dark spherical caps.

Author of the construction of GeoGebra: Rafael Losada

Articulated cube with fixed bar (II)

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