Descriptive geometry . ^cth with a hori-zontal plane, and revolving about a verticalaxis through o. Point e will describe the circleof the upper base cgl, point d will describe thecircle of the lower base dmf^ and the point e,the nearest to the axis, will describe the circleelan, which is called the circle of the other points of the generatrix will simi-larly describe circles, and by drawing theseparallels of the surface, the meridian line willbe determined, and is an hyperbola. Thuspoint s, in the generatrix cd^ will be in theposition t when it lies in the principal me-ridian plane,
Descriptive geometry . ^cth with a hori-zontal plane, and revolving about a verticalaxis through o. Point e will describe the circleof the upper base cgl, point d will describe thecircle of the lower base dmf^ and the point e,the nearest to the axis, will describe the circleelan, which is called the circle of the other points of the generatrix will simi-larly describe circles, and by drawing theseparallels of the surface, the meridian line willbe determined, and is an hyperbola. Thuspoint s, in the generatrix cd^ will be in theposition t when it lies in the principal me-ridian plane, and f will be a point in the ver-tical contour, which is an hyperbola. 156. Through any point of the surface todraw an element. If one projection of thepoint be given, draw a parallel of the surfacethrough this point to determine the other pro-jection. Through the horizontal projectionof this point draw a tangent to the circle ofthe gorge, and it will be the horizontal pro-jection of the required element, the extremi-. Fig. 194, HYPERBOLOID OF REVOLUTION OF ONE NAPPE 121 ties of which lie in the horizontal projectionsof the upper and lower bases of the either extremity may be regarded aslying in the upper base of the surface, thereare two tangents which may be drawn throughthe given point. They will make equalangles with the horizontal coordinate plane,and intersect at the circle of the gorge. 157. The Generatrix may be governed byThree Rectilinear Directrices. Fig. 194. If twoelements, ah and cd^ be drawn through point eof the circle of the gorge, either may be takenas the generatrix of the surface. One isknown as an element of the first generation,and the other as an element of the secondgeneration. Conceive ah as fixed and cd as the genera-trix. In the revolution about the axis, cdwill at all times intersect ah^ if these lines beextended indefinitely. This may be provedas follows : If cd be in the position indicatedby gf, then the horizontal projections of aha
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