. The principles of projective geometry applied to the straight line and conic . in perspective and consequently KL (s) is parallel to w. The triangles A^BiC^ and ABC are similar by construction. Also since AC_ _SC _b AC~SC ^ the sides AC and AiCi, and therefore the triangles A^B^Ci and ABC, are equal. (6) To construct the perspective of a given quadrangle A BCD so that it may be(i) a, square. Let Fi and Fg be a pair of diagonal points of the quadrangle. Take Fj V^ asvanishing line and let the third pair of sides of the quadrangle meet F, Fg in F3and F4. On V^V^ and on F3F4, as diameters, desc
. The principles of projective geometry applied to the straight line and conic . in perspective and consequently KL (s) is parallel to w. The triangles A^BiC^ and ABC are similar by construction. Also since AC_ _SC _b AC~SC ^ the sides AC and AiCi, and therefore the triangles A^B^Ci and ABC, are equal. (6) To construct the perspective of a given quadrangle A BCD so that it may be(i) a, square. Let Fi and Fg be a pair of diagonal points of the quadrangle. Take Fj V^ asvanishing line and let the third pair of sides of the quadrangle meet F, Fg in F3and F4. On V^V^ and on F3F4, as diameters, describe two circles which willintersect at some point ^S. Take *S as centre of perspective and some line s parallelto w as axis. The perspective of the quadrangle will then be a square. A2)2)lications of Perspective 53 For since Fj and Vo are on the vanishing line, the resulting figure is a parallelo-gram. Since SVi and SV2 are at right angles and are parallel to the sides, it isa rectangle. Since S\\ and .ST^ are at right angles and are parallel to the diagonals, it isa (ii) a square of given dimensions of the square in (i) vary directly as the distance between w ands (Art. 24). Let d be the length of the side of the given square. On V^Qo takea point Q2 such that \\ ^ dVoQ-i BC- If a line parallel to w through Q/ be taken as axis of perspective, the resultingfigure will be a square of side d. (iii) an equilateral triangle and its A BCD be the quadrangle and let AD, ED, CD meet BC, CA, ABrespectively in AiBiCi. Let BC meet BiCi in TFj and AB meet AiBi in the ranges ABC^W^ and CBAi\\\ are harmonic (Art. 31 (2)). Take W1W3as vanishing line (w). Let AD and CD meet w in TF^ and Wq respectively. OnWi \Vi and on TF3 Wq as diameters describe circles intersecting in S. Take S ascentre of perspective and any line parallel to w as axis of perspective. 54 Principles of Projective Geometry In the perspective figure the points corresponding t
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