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Space & Body Centrodes

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I''m having trouble pathing a simple pathing centrode with respect to a moving lamina. Take for instance, a 2d wheel rolling along a line. The space centrode is along the line where the wheel and line make contact, since at any given time, the instantaneous center is the point of the wheel touching the line (where velocity is 0). The body centrode, however, has me stumped. In this example, the textbook identifies it as the circumference of the wheel, with the explanation that "the locus of the instantaneous centers relative the moving lamina is known as the body centrode." Any ideas? What is meant by ''relative'' in this case? Thanks in advance.
blah blah blah.

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I''m starting to think that it may be the path that instantaneous centers follow from the time that they are on the space centrode until the rotate back onto the space centrode. The path would not be a cycloid though, a complete circle, for relative motion...

If I''m right, which I''m not sure I am, then the concept of a body centrode becomes quite meaningless indeed. That would explain the rush of responses...

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I think you are right. Sad none responds whenever it''s a bit technical.

I don''t know how this concept would be useful in physics simulations though. Any thoughts ? My intuition is it could give some answer to timed coldet with such constraints as two infinitely continuous volumes remaining in contact.

In the case of a rolling wheel the contact point is always changing whereas contact points are rather seen as the vertices of meshes in typical ODEs. When you apply typical constraint solvers to a wheel, inaccuracies are not only numerical, they come from a fact a real body point is followed and not a virtual contact point. It''s an algorithmic flaw.

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K just remembered back my physics course and understood the english equivalent terminology.

Body centroid (Cb)is a real material point (or locus).
Space centroids (Cs) is a virtual (mathematical) point.

Right ?


Thus when the body centroid at t matches the space centroid. So
if the wheel rolls perfectly and does not slide :

vel(Cb) = 0
(add the velocity of the wheel center and R*W )

and

vel(Cs) = vel(Wheel Center)

My point was that Cs would be more relevant for collision detection. But actually physics engines would consider the evolution of Cb(tmin) up to Cb(tmax) during a time step [tmin,tmax]. Which results in some imperfections.



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