Descriptive geometry for students in engineering science and architecture; a carefully graded course of instruction . MORE DIFFICULT CASES OF INTERPENETRATION OF SOLIDS, AND THEPROJECTION OF THE INTERSECTIONS OF THEIR SURFACES Section 29. When the intersecting surfaces are those of the cone and thesphere, then all that is necessary, in order to obtain the projection of pointscommon to both surfaces, is to make use of planes which give circular sections ofthe cone, and pass through the sphere. If the circular sections of the sphere, MORE DIFFICULT CAbES OF INTEKPEXETRATIOX OF SOLIDS lu5 by thes
Descriptive geometry for students in engineering science and architecture; a carefully graded course of instruction . MORE DIFFICULT CASES OF INTERPENETRATION OF SOLIDS, AND THEPROJECTION OF THE INTERSECTIONS OF THEIR SURFACES Section 29. When the intersecting surfaces are those of the cone and thesphere, then all that is necessary, in order to obtain the projection of pointscommon to both surfaces, is to make use of planes which give circular sections ofthe cone, and pass through the sphere. If the circular sections of the sphere, MORE DIFFICULT CAbES OF INTEKPEXETRATIOX OF SOLIDS lu5 by these planes, intersect the circular sections of the cone by the same planes,then the points of intersection are points on both surfaces. Another method of determining the intersection of the surfaces of these by the use of intersecting spherical surfaces instead of intersecting planes, anddepends on the following facts:— (i) That a sphere whose centre is in the axis of a right circular cone willintersect the surface of that cone, if it intersects at all. in a circle perpendicularto the axis of the cone;. Fig. io: Fig. 103. (2) That if a sphere intersects another sphere the intersection of their sur-faces is the circumference of a circle, the plane of which is perpendicular to the linejoining their centres; and (3) That if a sphere, with its centre in the axis of a right circular cone inter-penetrates at the same time another sphere, then the intersection of the circularsections it makes with both solids will be points common to the surfaces of bothsolids. Illustrations of the use of both methods referred to are given in Figs. 102and 103. In Fig. 102, a right circular cone with its axis vertical, and a sphere 106 DESCRIPTIVE GEOMETRY whose centre is cc, are arranged so that their surfaces intersect. The centreof the sphere is not necessarily at the same distance from the as the axis ofthe cone. By the use of horizontal section planes a number of points will befound
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