Descriptive geometry . of the lines will be lines of the required tan-gent planes. 5. Counter-revolve the auxil-iary plane containing the lines of the tangentplanes. 6. Determine the planes defined bythe given line and each of the tangent linesobtained by 4. These will be the requiredtangent planes. Construction. Fig. 157. The given lineis A, and the center of the sphere is e. HXand VX are the traces of the auxiliary planeperpendicular to A and passing through e PLANE TANGENT TO SPHERE 81 (Art. 72, page 50). The point/ is that of theintersection of line A and plane X. Revolveplane
Descriptive geometry . of the lines will be lines of the required tan-gent planes. 5. Counter-revolve the auxil-iary plane containing the lines of the tangentplanes. 6. Determine the planes defined bythe given line and each of the tangent linesobtained by 4. These will be the requiredtangent planes. Construction. Fig. 157. The given lineis A, and the center of the sphere is e. HXand VX are the traces of the auxiliary planeperpendicular to A and passing through e PLANE TANGENT TO SPHERE 81 (Art. 72, page 50). The point/ is that of theintersection of line A and plane X. Revolveplane X into the vertical coordinate plane ande will be the revolved position of the centerof the sphere, and f the revolved position ofthe point /. Draw the great circle of the sphere and the tangents C and B. In counter-revolution these lines will be atC and B, intersecting A -dt f. C and A willbe two intersecting lines of one of the requiredtangent planes, S, and the second plane, iV, willbe determined by lines O and CHAPTER V INTERSECTION OF PLANES WITH SURFACES,AND THE DEVELOPMENT OF SURFACES of any 115. To determine the intersectionsurface with any secant plane. General Method. 1. Pass a series ofauxiliary cutting planes which will cut lines,straight or curved, from the surface, and rightlines from the secant plane. 2. The inter-sections of these lines are points in the re-quired curve of intersection. This method is applicable alike to prisms,pyramids, cylinders, cones, or double-curved surfaces of revolution. The auxiliary cuttingplanes may be used in any position, but forconvenience they should be chosen so as^ tocut the simplest curves from the surface, thatis, straight lines or circles. With solids such as prisms, pyramids,single-curved, or other ruled surfaces, theabove method consists in finding the intersec-tion of each element with the oblique plane byArt. 61, page 44. 82 INTERSECTION OF PLANE WITH PYRAMID 83 116. A tangent to the curve of intersection of a plane
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