Descriptive geometry . 160. Through a right line to pass a plane tan-gent to any double-curved surface of revolution. By the use of the hyperboloid of revolutionof one nappe as an auxiliary surface, it is pos-sible to make a general solution of problemsrequiring the determination of tangent planesto double curved surfaces of revolution, asfollows: Principle. If the given right line be re-volved about the axis of the double-curvedsurface of revolution, it will generate an hy-perboloid of revolution. A plane tangent toboth surfaces and containing the given line,which is an element of ooe of them
Descriptive geometry . 160. Through a right line to pass a plane tan-gent to any double-curved surface of revolution. By the use of the hyperboloid of revolutionof one nappe as an auxiliary surface, it is pos-sible to make a general solution of problemsrequiring the determination of tangent planesto double curved surfaces of revolution, asfollows: Principle. If the given right line be re-volved about the axis of the double-curvedsurface of revolution, it will generate an hy-perboloid of revolution. A plane tangent toboth surfaces and containing the given line,which is an element of ooe of them, will bethe required plane. Since one, and only onemeridian plane at a point of tangency will beperpendicular to the tangent plane, and as thesurfaces of revolution have a common axis, itfollows that one meridian plane will cut aline from the tangent plane which will passthrough the points of tangency on both sur-faces and be tangent to both meridian line and the given line will determinethe tangent HYPERBOLOID OF REVOLUTION OF ONE NAPPE 123 Method. 1. Draw the principal meridiansection of the hyperboloid of revolutionwhich has the given line for its Draw a tangent to the principal meridiansections of both surfaces. 3. Revolve thisline about the axis of the surfaces until itintersects the given line, observing that itspoint of tangency with the hyperbola is apoint of the given line. 4. Determine theplane of this tangent and the given line. Construction. Fig. 195. Having de-scribed the hyperboloid of revolution withthe given line A as its generatrix, draw c|5j tangent to the meridian curves. It will bethe vertical projection of the revolved positionof a line tangent to both surfaces. In counter-revolution this line will intersect the givenline A at <?, which is the counter-revolvedposition of the point of tangency Cy Thismust be so, since line A is an element of thehyperboloid of revolution and must be in con-tact with the parallel through c
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