. Plane and solid analytic geometry . represented by the equation The (1) a2 52 ^2 is called a hyperholoid of one sheet or an unparted hyperholoid. If a = b, it is in particular a hyper-holoid of revolution of one sheet(Ch. XXII, § 8). In the general case, a =^h, thesurface can be constructed by themethod of § 1. The sections bythe vertical coordinate planes,X = 0 and y = 0, are the hyper-bolas ^^M ^^H^^^ K / ^^A^m^^^M ^^^^H ^^^^^^^^^^^K^^/^rx ^ VxV^o!^^^^^^| HHMHMIP ^ ^^^^^H y\/\r4^^J Fig. 3 (2) 62 C2 X^ a2 --=!. The sections by the planes z = k are ellipses. The smallestone is th
. Plane and solid analytic geometry . represented by the equation The (1) a2 52 ^2 is called a hyperholoid of one sheet or an unparted hyperholoid. If a = b, it is in particular a hyper-holoid of revolution of one sheet(Ch. XXII, § 8). In the general case, a =^h, thesurface can be constructed by themethod of § 1. The sections bythe vertical coordinate planes,X = 0 and y = 0, are the hyper-bolas ^^M ^^H^^^ K / ^^A^m^^^M ^^^^H ^^^^^^^^^^^K^^/^rx ^ VxV^o!^^^^^^| HHMHMIP ^ ^^^^^H y\/\r4^^J Fig. 3 (2) 62 C2 X^ a2 --=!. The sections by the planes z = k are ellipses. The smallestone is the section by 2; = 0, the (x, 2/)-plane; it is known asthe minimaiii ellipse. The general one increases in size as itsdistance from the (.t, 2/)-plane increases. The surface can bethought of as generated by it; cf. Fig. 8, Ch. XXII. QUADRIC SURFACES 551 The Hyperholoid of Two Sheets. This surface, also knownas the biparted hyperholoid, is defined by the equation (3) t-t=-l or x^ ip- j^ a2 62 c2 = A particular case, when a = 6, is the hyperholoid of revolutionof two sheets. In the general case, a ^^ 5, the sections by the vertical coor-dinate planes are the hyperbolas conjugate to the hyperbolas(2). The (ic, 2/)-plane, 2 = 0, does notintersect the surface. This is trueof all the planes 2; = A:, for whichfc2 c^ meetit in ellipses, which increase in ^sizeas A: increases in numerical value;cf. Fig. 9, Ch. XXII. Center, Axes, Principal hyperboloid is symmetric inthe origin 0 and in the coordinate axes and coordinate planes; 0 is the center, the coordinateaxes, the axes, and the coordinate planes, the principal planesfor each surface. The sections by the principal planes are theprincipal sections. The Asymptotic Cone. The hyperboloids (1) and (3) arecalled conjugate hyperboloids. We have seen that each verticalcoordinate plane intersects them in conjugate hyperbola
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