# Can three elbows generate any point in space?

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• petersng
petersng
TL;DR Summary
Can three elbows generate any point in space?
Consider 3 elbows. One extremity is connected to a plane surface and may rotate 360° freely. The available extremity may be connected to another elbow only if the both faces "face" each other (normal vector aligned). Once one connect a second elbow, a second liberty degree is added, since one have two rotation axes.
Consider now three elbows connected. Is one able to reach any point in space with that configuration? Every normal vector possible in that space also generated in that configuration (i.e., could i potentially turn my albows so to connect a 4th elbow?).

Consider the length on elbow as bein L.

Hello. Your question is kind of odd. Would you happen to have an illustration?

I think you only need one turn? Assuming your arms can have arbitrary length, the first arm can reach the (x,y,0) point corresponding to any (x,y,z), then you just make a single turn up or down and attach another arm of length z.

Are the lengths of the arms restricted in any way? E.g. the set of points you can reach with three unit length arms might be interesting

petersng said:
TL;DR Summary: Can three elbows generate any point in space?

One extremity is connected to a plane surface and may rotate 360° freely.
That is a rather ambiguous description, could you clarify?

Is that rotation around the attachment point a rotation,
1) about the attachment point with the arm always parallel to the attachment plane
2) about the axis of the arm (as in palm-down vs palm-up)
3) 3-dimensional such that the distal end of the arm can touch any point on a hemisphere whose center is at the attachment point.

If both 2) and 3) are true for all of the joints, I believe (but have not proved) that the answer is Yes. (though there may be an inaccessible area near some of the joints)

Interesting gedankenexperiment!
(https://www.merriam-webster.com/dictionary/gedankenexperiment)

Cheers,
Tom

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