This model is based on [2]. This is the most precise and concise model I could find. For example, there is no need for "line segments" or "circle arcs" which are only there for aesthetic reasons.
The model of computation of Euclid can be found in the first three postulates of his Elements:
Postulate 1 To draw a straight line from any point to any point.
Postulate 2
Postulate 3 

A common mistake is to think that the compass can be used to transfer distances. Recall that Euclid's compass is an idealized tool, and one must not infer too much from the physical object.
Euclid's compass is often called a "collapsing compass" (no screw!), and it can be conceptualized as follows:
Hopefully, as shown by Book I Proposition 2, it is possible to transfer a distance (by using more than one step).
For a good exposition of the relationship between the collapsing compass and Euclid's second proposition, see this page (it has a link to a postscript version of [5]).
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