An Elementary Course in Synthetic Projective Geometry 0 1
An Elementary Course in Synthetic Projective Geometry 0 1
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Weexamine now the possibility of two forms related in this way, having anelement or elements that correspond to themselves. We have seen, indeed, that if _B_ and _D_ are harmonic conjugates with respect to _A_ and _C_, then the point-row described by _B_ is projective to the point-rowdescribed by _D_, and that _A_ and _C_ are self-corresponding points. Consider more generally the case of two pencils perspective to each otherwith axis of perspectivity _u’_ (Fig. 9). Cut across them by a line _u_.... We get thus two projective point-rows superposed on the same line _u_, anda moment’s reflection serves to show that the point _N_ of intersection_u_ and _u’_ corresponds to itself in the two point-rows. Also, the point_M_, where _u_ intersects the line joining the centers of the two pencils, is seen to correspond to itself. It is thus possible for two projectivepoint-rows, superposed upon the same line, to have two self-correspondingpoints. Clearly _M_ and _N_ may fall together if the line joining thecenters of the pencils happens to pass through the point of intersectionof the lines _u_ and _u’_.
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To quickly assess the difficulty of the text, read a short excerpt:
Weexamine now the possibility of two forms related in this way, having anelement or elements that correspond to themselves. We have seen, indeed, that if _B_ and _D_ are harmonic conjugates with respect to _A_ and _C_, then the point-row described by _B_ is projective to the point-rowdescribed by _D_, and that _A_ and _C_ are self-corresponding points. Consider more generally the case of two pencils perspective to each otherwith axis of perspectivity _u’_ (Fig. 9). Cut across them by a line _u_.... We get thus two projective point-rows superposed on the same line _u_, anda moment’s reflection serves to show that the point _N_ of intersection_u_ and _u’_ corresponds to itself in the two point-rows. Also, the point_M_, where _u_ intersects the line joining the centers of the two pencils, is seen to correspond to itself. It is thus possible for two projectivepoint-rows, superposed upon the same line, to have two self-correspondingpoints. Clearly _M_ and _N_ may fall together if the line joining thecenters of the pencils happens to pass through the point of intersectionof the lines _u_ and _u’_.
What to read after An Elementary Course in Synthetic Projective Geometry 0 1?
You can find similar books in the "Read Also" column, or choose other free books by Lehmer to read onlineMoreLess
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