A first course in projective geometry . FiG. 123. § 7. We conclude this chapter with a theorem of which§ 5 (h), Chap. XIV., is a particular case. The envelope of the joins of corresponding points of twohomographic. ranges on a conic is a conic having double con-tact with the given conic. Let A, B, ... A, B, ... be the homographic ranges and XY thecross-axis. Then AB, AB meet at P on XY. The polar of P with respect to the conic passes through Othe point of intersection of AA and BB (Chap. VIII. ^ 7 (VIII.)),also through K the point of intersection of the tangents at Xand Y (def. of polar). .. i
A first course in projective geometry . FiG. 123. § 7. We conclude this chapter with a theorem of which§ 5 (h), Chap. XIV., is a particular case. The envelope of the joins of corresponding points of twohomographic. ranges on a conic is a conic having double con-tact with the given conic. Let A, B, ... A, B, ... be the homographic ranges and XY thecross-axis. Then AB, AB meet at P on XY. The polar of P with respect to the conic passes through Othe point of intersection of AA and BB (Chap. VIII. ^ 7 (VIII.)),also through K the point of intersection of the tangents at Xand Y (def. of polar). .. it is KO. .. the pencils K(PXOY), O(PAKB) are harmonic, and conse-quently homographic. But they have a common ray OK. 244 PROJECTIVE GEOMETRY .. P, Q, R are collinear, Q and R being the points of intersec-tion of OA and OB, with the tangents at X and Y respectively(Chap. XIIL, §2 (/>)). P. Fig. 124. But since K{PXOY} is harmonic, by the property of thecomplete quadrilateral, KO passes through the intersection ofXR and YQ. It follows, by the converse of Brianchons Theorem, that aconic can be drawn touching KX and KY at X and Y, and also AAand BB (ROQXKY is the Brianchon hexagon). Considering theconic as determined by the first five of these tangents, viz. twocoincident in KX, two in KY and AA, it may similarly be shewnthat it touches CC, and so on. Which proves the theorem. A corresponding theorem holds for the homographic sets oftangents referred to in § 3. EXAMPLES 245 Examples on Chapter XVI. DRAWING EXERCISES. 1. A, A; B, B; C, C are three pairs of corresponding points oftwo homographic ranges on the same straight line. Measuring from A always in the same sense, AA=1, AB = 1*5,AB=2-5, AC = 4, AC = 6. Construct the self-corresponding points of the two ranges and thepoint of each range which corresponds to the point at infinity on theother range. 2. A, B, 0, D are fou
Size: 1478px × 1690px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
