Descriptive geometry . thisplane about the axis of the torus until it isparallel to V. In this position the verticalprojections of the elements will be Al and B^and their intersections with the circle cut fromthe torus will be at cl and dl. In counter-revolution these points will fall at <? and c?,thus determining the lowest points of thecurve on the inner and outer surface of thetorus. The meridian planes iV and M will cutcontour elements from the cylinder, and thevertical projections of their points of intersec-tion with the torus at e and/ will be deter-mined as in the previous case. In
Descriptive geometry . thisplane about the axis of the torus until it isparallel to V. In this position the verticalprojections of the elements will be Al and B^and their intersections with the circle cut fromthe torus will be at cl and dl. In counter-revolution these points will fall at <? and c?,thus determining the lowest points of thecurve on the inner and outer surface of thetorus. The meridian planes iV and M will cutcontour elements from the cylinder, and thevertical projections of their points of intersec-tion with the torus at e and/ will be deter-mined as in the previous case. In this mannerall the points of intersection may be found ; or,planes cutting parallels from the torus may beused, as H and S^ the former of which is tan-gent to the upper surface of the torus cuttingit and the cylinder in circles which intersectat points k and I. The remaining pointsnecessary to the determination of the curvemay be similarly found as shown by the auxil-iary plane S. INTERSECTION OF TORUS AND CYLINDER 107. Fig- 179. 108 DESCRIPTIVE GEOMETRY 143. To determine the curve of intersectionbetween an ellipsoid and a paraboloid, the axesof which intersect and are parallel to the ver-tical coordinate planes. Pkinciple. Auxiliary spheres having theircenters at the intersection of the axes of thesurfaces of revolution will cut the intersectingsurfaces in circles, one projection of which willbe right lines. Construction. Fig. 180 illustrates thiscase. It will be observed that the horizontalprojection of the parabola is omitted since thecurve of intersection is completely determinedby the vertical projections of the two surfacesand the horizontal projections of the parallelsof the ellipsoid.
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