A first course in projective geometry . Fio. 94. Then PQ, P^ measure the deflections of the curves from thetangent line (Fig. 94). M 178 PROJECTIVE GEOMETRY PQConsider the ratio —-^. If it were unity for all positions of P near A, the curves would coincide in that neighbourhood, and their curvatures would be the same. We shall therefore not be inconsistent if we define the ratio of the curvatures of the curves at A in the general case PQ,as the limiting value of the ratio when P moves up to A. Pq Consider the circle. If any two points A, a (Fig. 95a) betaken on it and AP, aj) be the tange
A first course in projective geometry . Fio. 94. Then PQ, P^ measure the deflections of the curves from thetangent line (Fig. 94). M 178 PROJECTIVE GEOMETRY PQConsider the ratio —-^. If it were unity for all positions of P near A, the curves would coincide in that neighbourhood, and their curvatures would be the same. We shall therefore not be inconsistent if we define the ratio of the curvatures of the curves at A in the general case PQ,as the limiting value of the ratio when P moves up to A. Pq Consider the circle. If any two points A, a (Fig. 95a) betaken on it and AP, aj) be the tangents, PQ, pq at right anglesto these to meet the circle in Q, q ; then if AP = ap, it is easilyproved that PQ =^:> (a)
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