In the first draft of this post, from July 5, I suggested a project to study configurations in space of a ring made of four sticks of known lengths. But in talking with McSteve we realized that the project was really best thought of in terms tetrahedra, so I am modifying and correcting that post.
Consider a ring of four sticks, with points A,B,C,D in order. We can translate any configuration of this ring in space to have A,B,C in the x,y plane; there remains a lot of choice in positioning D in space.
Suppose we assume that the four sticks have known lengths. If we also know the Cayley coordinates (lengths) for AC and BD then the configuration is unique, since tetrahedra are rigid once all six edge lengths are fixed.
One project: make sure we understand the very simple example with lengths AB=2, BC=1, CD=2, and DA=1, with a right angle at B and with right triangle ABC in the x,y plane. Then AC has length \sqrt(5), and we saw that all configurations of the ring in space had point D on a circle (the intersection of spheres centered at A and D). We can see which of these configurations (positions of D) cause two of the sticks to collide.
Maybe let’s investigate how much we can change the given lengths for the ring and get similar phenomena?
McSteve and I noticed that two of the small projects we have discussed so far just involve keeping track of some of the edges of a tetrahedron:
- a ring made of four sticks of known length; and
- a robot arm made of three sticks of known length.
So my next step will be to write up (in TeX) some notes on formulas for edge lengths in tetrahedrons. I will try to post the pdf in our dropbox soon.