By Derrick Norman Lehmer
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Extra resources for An Elementary Course in Synthetic Projective Geometry
If, moreover, we project the figure to any point in space, we shall get a cone, standing on a circular base, generated by two projective axial pencils which are the projections of the pencils at S and S'. Cut across, now, by any plane, and we get a conic section which is thus exhibited as the locus of intersection of two projective pencils. It thus appears that a conic section is a point-row of the second order. It will later 74. Conic through five points 49 appear that a point-row of the second order is a conic section.
Pascal's theorem furnishes an elegant solution of the problem of drawing a conic through five given points. To construct a sixth point on the conic, draw through the point numbered 1 an arbitrary line (Fig.
Four harmonic lines 21 33. Four harmonic lines. We are now able to extend the notion of harmonic elements to pencils of rays, and indeed to axial pencils. For if we define four harmonic rays as four rays which pass through a point and which pass one through each of four harmonic points, we have the theorem Four harmonic lines are cut by any transversal in four harmonic points. 34. Four harmonic planes. We also define four harmonic planes as four planes through a line which pass one through each of four harmonic points, and we may show that Four harmonic planes are cut by any plane not passing through their common line in four harmonic lines, and also by any line in four harmonic points.
An Elementary Course in Synthetic Projective Geometry by Derrick Norman Lehmer