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The mapping t 9 from M × G to G, which maps (re, g) E M x G to rh(g), and the mapping t~ from M x P to P, which maps (re,p) E M × P to rh(p), are also referred to as geometric geometry transformation and geometric picture transformation, respectively. 5 Restrictive Transformations Let g be a geometry. Two mappings (id[g) and (id[g) can then be derived, where id is the identity mapping from U to U. The two are named restrictive geometry transformation and restrictive picture ¢ransformation, respectively, and are each thought of as a formalization of clipping.

In what follows, m, g~ and p are members of M, G, and P respectively, whether or not they have any subscript. 2 follow Theorem 1. 2) rao (idig) = (idlm(g)) o m In the latter, m can be replaced by (mlU) and both sides are simplified to (mlg). 3. Theorem 3 validates the next equafities. 13) rh(p2+p,) = S z ( p 2 ) + ~ ( p l ) C H A P T E R 3. FORMAL MODEL OF VISUALIZATION 30 J ~ __------ -q. 2. VISUALIZING NETS 31 The next three equations are related to repfication, where p is a member of P and ( r , ~ .

As defined in the previous section, a restrictive geometry can only be given in pNucleus by the specific geometric primitive 7r~¢~. G-expressions will be described in the next section. The two picture transformations, geometric and restrictive, are represented by the following expressions, respectively: (*gtran*pict ' ' < p i c t u r e > ) (*rtran*pict ' ' < p i c t u r e > ) . 4. The other two transformations are as follows: (*pict*tran ' ' ') (*pict*sum {' }) The invocation of plctorlallzing transformation replicates the pattern given by < p i c t u r e name> along the geometry; the way of replicating is controlled by the control list.

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