. Plane and solid analytic geometry; an elementary textbook. s2 6 _ cos2 7 b2 c2 Let a, yS, and 7 be values of «, /3, and 7, for whichthis expression vanishes. Then, as «, /3, and 7 approacha, /3, and 7, the value of p will increase indefinitely,and the line through the origin whose direction anglesare a\ /3, and 7 may be said to meet the surface atinfinity. Such a line is called an asymptotic line. Since the equation 1 —— ——— = 0 is the a2 o2 c2 only condition which must be satisfied by the polar coordi-nates of the points on all the asymptotic lines, it mustbe the equation of the asympt
. Plane and solid analytic geometry; an elementary textbook. s2 6 _ cos2 7 b2 c2 Let a, yS, and 7 be values of «, /3, and 7, for whichthis expression vanishes. Then, as «, /3, and 7 approacha, /3, and 7, the value of p will increase indefinitely,and the line through the origin whose direction anglesare a\ /3, and 7 may be said to meet the surface atinfinity. Such a line is called an asymptotic line. Since the equation 1 —— ——— = 0 is the a2 o2 c2 only condition which must be satisfied by the polar coordi-nates of the points on all the asymptotic lines, it mustbe the equation of the asymptotic cone, which containsall these asymptotic lines of the surface. Multiplying 242 ANALYTIC GEOMETRY OF SPACE [Ch. V, § { by p2 and transforming to rectangular coordinates, itbecomes ^ + T _ t. = 0 a2 b2 c2 Similarly it may be shown that the asymptotic coneof the biparted hyperboloid is :=0. 33. The paraboloids. x2 b2 = 2 ez. — The surface = 2 cz passes through the origin, but does not cutthe axes at any other point. Sections made by planes.
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