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By Bruce E. Meserve
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The space dual of P-l is If a and f3 are distinct planes, there is at least one line on both a and f3. Notice that this statement is not the space dual of the above plane dual of P-l. In general, the space dual of a statement is distinct from the space dual of the plane dual of the statement (Exercise 10). The space dual of P-l is a consequence of Exercise 6, Section 2-2. The statements and proofs of the plane and space duals of Postulates P-2 through P-8 are given as Exercises 1 through 8 of this section.
State and discuss the space dual of the statement obtained in Exercise 12. 15. Formulate a principle of duality for projective n-spaces. 2-6 Perspective figures. Each of us makes use of the basic properties of perspective figures in his daily living. For example, we use these properties whenever we use mirrors. Consider an object 0 as a set of points. The points of its apparent image M in the plane of the mirror and the corresponding points of its virtual image V behind the mirror are perspective with respect to the eye of the observer (Fig.
The inserted phrase "on a plane" is appropriate, since any two points are necessarily on a plane and planar duality is concerned only with figures on a plane. The plane dual of P-l is a consequence of Theorem 2-2. The space dual of P-l is If a and f3 are distinct planes, there is at least one line on both a and f3. Notice that this statement is not the space dual of the above plane dual of P-l. In general, the space dual of a statement is distinct from the space dual of the plane dual of the statement (Exercise 10).