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By Derrick Norman Lehmer

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Extra info for An Elementary Course in Synthetic Projective Geometry

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To find d' in S' corresponding to a given line d of S we note the point L where d meets u. Join L to M and get the point N where this line meets u'. L and N are corresponding points on u and u', and d' must therefore pass through N. The intersection P of d and d' is thus another point on the locus. In the same manner any number of other points may be obtained. 65. The lines u and u' might have been drawn in any direction through A (avoiding, of course, the line a for u and the line a' for u'), and the center of perspectivity M would be easily obtainable; 66.

Not until the notion of an infinitely distant point is brought in is any mention made of distances or directions. We have been able to make all of our constructions up to this point by means of the straightedge, or ungraduated ruler. A construction made with such an instrument we shall call a linear construction. It requires merely that we be able to draw the line joining two points or find the point of intersection of two lines. 41. Parallels and mid-points. It might be thought that drawing a line through a given point parallel to a given line was only a special case of drawing a line joining two points.

For the construction is reversible, and if we are given the middle point of a given segment, we can construct linearly a line parallel to that segment. [24] 26 [25] An Elementary Course in Synthetic Projective Geometry Thus, given that B is the middle point of AC, we may draw any two lines through A, and any line through B cutting them in points N and L. Join N and L to C and get the points K and M on the two lines through A. Then KM is parallel to AC. The bisection of a given segment and the drawing of a line parallel to the segment are equivalent data when linear construction is used.

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