I would expect the 'preferred velocity' of a circular motion to become a infinite amount of straight lines myself Brain

Ending as they start. Then again, there is always a possibility to see all circular motions in SpaceTime as being 'straight lines', when excluding gravity, if we accept that interpretation. But I can't see it myself, thinking about it, though.

Maybe there is some mathematical translation somewhere on the net? Preferably with a visual representation too :)It would be interesting to see what the moons orbit would translate into without gravity and 3-D, if you see how I mean here?

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On the third tentacle, isn't all orbits accelerations?

I'm not sure how Einstein defined it suddenly?

The Newtonian definition is a constant 'acceleration', right. But isn't it also a geodesic, and therefore according to the theory of relativity, the 'easiest', least energy consuming path, gravitationally speaking?

I need too look at that. Both ideas make sense to me, but they clash.

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No they don't clash. They are mirrors to each other. Newton thinks of it as a 'flat' space, and Einstein of a 'curved/distorted'. In a flat space a orbit must be represented of a willingness to move in a straight line, like rotating with a ball on a string, letting it go. But then we have 'gravity' that forms 'space' and if we could color its gradients depending on 'gravity strength' we would have a lot of shades representing the way those gradients bend/distort. But I still find it really hard to translate a orbit into a straight line mentally.

How the he* would it look? Ignoring relative motion, unfolding gravity.