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Computer Graphics : Theory and Practice by Gomes, Jonas; Velho, Luiz; Costa Sousa, Mario

By Gomes, Jonas; Velho, Luiz; Costa Sousa, Mario

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En = (0, 0, . . , 1, 0), en+1 = (0, 0, . . , 0, 1), en+2 = (1, 1, . . , 1, 1) = e1 + · · · + en+1 . 5. Given any projective basis a1 , . . , an+1 , there exists a projective transformation T : RPn → RPn such that T (ei ) = λi ai , i = 1, . . 18) where the λi , i = 1, . . , n+2, are nonzero scalars.

6. Describe ten computer graphics applications in different areas of knowledge. 7. Discuss the use of the four-universe paradigm in other areas of science. Establish, in each case, the abstraction levels and the nature of the elements at each level. 8. Discuss the following: in a computer, every object is discrete. How is the notion of reconstruction useful, if a computer model, at any level, cannot be continuous? Or, perhaps, what does a continuous object mean in the computer? 9. Give examples of an exact and an approximate representation.

Two geometric objects O1 and O2 are said to be congruent if there exists an element g ∈ G such that g(O1 ) = O2 . When we want to emphasize the group G, we use the prefix G, and speak of G-congruence, G-properties, and so on. Klein’s approach can be summarized as follows: Space −→ Transformations −→ Properties −→ Geometry From Klein’s point of view, the study of the geometry defined by a group G consists of identifying the properties invariant under G, describing G-congruence classes, and determining the relation between congruence classes and properties.

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