A first course in projective geometry . Fig. THE CONSTRUCTION OF A CONIC 249 Thus, for every straight line wedraw through O, we obtain a freslipoint on the curve. The principle of the constructionis that homographic pencils atvertices B and C are completelydeterminable when two triads ofcorresponding rajs BA, BD, BEand CA. CD, CE are known. The ranges determined by theseon X, y, having a common point,are in perspective. So we have B{ADEF} = {ADiEiFi} = 0{ADiEiFi} = {AD3E2F,l = C{ADEF}. .-. by §5, Chap. XIV., F is apoint on the curve. Thus, for every point we takeon o, we obtain a fresh
A first course in projective geometry . Fig. THE CONSTRUCTION OF A CONIC 249 Thus, for every straight line wedraw through O, we obtain a freslipoint on the curve. The principle of the constructionis that homographic pencils atvertices B and C are completelydeterminable when two triads ofcorresponding rajs BA, BD, BEand CA. CD, CE are known. The ranges determined by theseon X, y, having a common point,are in perspective. So we have B{ADEF} = {ADiEiFi} = 0{ADiEiFi} = {AD3E2F,l = C{ADEF}. .-. by §5, Chap. XIV., F is apoint on the curve. Thus, for every point we takeon o, we obtain a fresh tangent tothe curve. The principle of the constructionis that homographic ranges onStraight lines h, c are completelydeterminable when two triads ofcorresponding points (ha), {hd), {be)and (ca), {cd), {ct) are known. The pencils subtended by theseat X, Y, having a common ray, arein perspective. So we have = {ad^2f-2} = (^{^d^f} .: by § 5, Chap. XIVtangent to the curve. /is § 3. Pascals and Brianchons theorems may also be usedfor these problems
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