. Differential and integral calculus, an introductory course for colleges and engineering schools. ipsoid is a sphere of radius The Hyperboloid of One Sheet, x _u y _ _ a2 + b2 ~ c2 274 GEOMETRY OF THREE DIMENSIONS §185 It is easily seen that the trace in the xy-plsme is an ellipse, andthat the traces in the zz-plane and the yz-plsme are z-axis is the conjugate axis of each of these hyperbolas, andconsequently the four foci lie in the xy-pl&ne. Setting z = k in the equation of the surface, we get, after easyreductions, X2 y2 _ ~> / 7 o\ — J-) which is the equation of the
. Differential and integral calculus, an introductory course for colleges and engineering schools. ipsoid is a sphere of radius The Hyperboloid of One Sheet, x _u y _ _ a2 + b2 ~ c2 274 GEOMETRY OF THREE DIMENSIONS §185 It is easily seen that the trace in the xy-plsme is an ellipse, andthat the traces in the zz-plane and the yz-plsme are z-axis is the conjugate axis of each of these hyperbolas, andconsequently the four foci lie in the xy-pl&ne. Setting z = k in the equation of the surface, we get, after easyreductions, X2 y2 _ ~> / 7 o\ — J-) which is the equation of the right cylinder which projects uponthe £?/-plane the curve of section of thegiven surface by the plane z = k, or is theequation of this projection, or the equa-tion of the curve of section itself. Henceevery section made by a plane z = k isan ellipse, and this ellipse becomes largeras | k | increases. The solid bounded bythe surface may be conceived as made up of a series of ellipsesstrung upon the z-axis. On writing y = k in the equation of the surface and transform-ing, there results. (-» (>-» The section is an hyperbola. When | k \ \b\, the transverse axis and foci are inthe 2/z-plane and the conjugate axis is in the :n/-plane. Whenk = b, the section is a pair of right lines intersecting in the remarks apply to the sections made by the planesx = k. Let the student give the complete discussion of thesesections. 4. The Hyperboloid of Two Sheets, x2 _ y2 _ z2 a2 b2 c2 §185 THE POINT, THE PLANE, AND THE SURFACE 275 Writing this in the form t. _l zl = *L. i -« -,2 1: b2 C* a1 it is seen that when | x \ < \a\,y and z cannot both be real, which means that the surface has no real points between the planes x = — a and x = +a. The traces in the xy-pl&ne and the zz-plane are hyperbolas with their foci in the o;-axis, while the trace in the yz- plane is of course imaginary. It can be shown without difficulty that the planes y = k,z = k, cut the surface
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1912
