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By Renatus Ziegler
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Extra resources for Selected topics in three dimensional synthetic projective geometry
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Coincident correlative fields ρ and σ. (1) Any plane through R (or S) contains (1) Through any point in ρ (or σ) passes either in a non-degenerate conic, two either in a non-degenerate cone of the ranges of points, one range of points, second class, two pencils of planes, one one point, or a field of points belonging pencil of planes, one plane or a bundle of to the surface. planes belonging to the surface. (2) Through R (or S) pass either no, (2) In ρ (or σ) lie either no, one, two or one, two or infinite many lines belong- infinite many lines belonging to the suring to the surface.
Definition An arc is said to be closed if its beginning-element coincides with its end-element. A curve-segment, cone segment, or developable-segment is called a curve, cone, or developable respectively if all its arcs are closed. The definition of space curves and developables implies that the plane-arc of a space curve is a developable and the point-arc of a developable is a space curve. Therefore, these two structures are identical and self-dual. Definition A space curve (or developable) is said to be of the n-th order if there is at least one plane that contains n points of the curve but no plane that contains more than n points.
9a A polarity in a bundle induces an involution of conjugate induces an involution of conjugate polar polar points on any line that is not a planes on any line that is not a selfself-conjugate polar line and an involu- conjugate polar line and an involution tion of conjugate polar lines in any of conjugate polar lines in any plane point that is not a self-conjugate polar that is not a self-conjugate polar plane. point. Proof: Coxeter [1955] p. 68f. Definition A triangle is called a polar triangle if each vertex is the pole of the opposite side.