Plane and solid analytic geometry; an elementary textbook . ft, and yr be values of a, ft, and 7, for whichthis expression vanishes. Then, as a, ft, and 7 approacha, ft, and 7, the value of p will increase indefinitely,and the line through the origin whose direction anglesare a, ft, and y may be said to meet the surface atinfinity. Such a line is called an asymptotic line. Since the equation —— 1 - v—?- = 0 is the a1 ¥ 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 t
Plane and solid analytic geometry; an elementary textbook . ft, and yr be values of a, ft, and 7, for whichthis expression vanishes. Then, as a, ft, and 7 approacha, ft, and 7, the value of p will increase indefinitely,and the line through the origin whose direction anglesare a, ft, and y may be said to meet the surface atinfinity. Such a line is called an asymptotic line. Since the equation —— 1 - v—?- = 0 is the a1 ¥ 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 254 ANALYTIC GEOMETRY OF SPACE [Ch. V, § 33 by p2 and transforming to rectangular coordinates, itbecomes *l+£. _ ?!=n a2 62 c2 Similarly it may be shown that the asymptotic coneof the biparted hyperboloid is x2 y1 a2 b2 <? 0. 33. The paraboloids. ^ ± ^ = 2 cz. — The surface—- + y~ — 2 cz passes through the origin, but does not cut (X 0 the axes at any other point. Sections made by planes. Fig. 20. Fig. 21. parallel to the X-F-plane are ellipses whose axes increaseas the section recedes from the origin. Sections made byplanes parallel to the other coordinate planes are parabo- Ch. V, § 33] QUADRIC SURFACES 255 las, which have their axes parallel to the Z-axis. Thissurface is shown in Fig. 20. It is called an elliptic parabo-
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