Elements of descriptive geometry, with applications to isometric projection and othe forms of one-plane projection; a text-book for colleges and ingineering schools by O. E. Randall.

CHAPTER XV

INTERSECTION OF SURFACES BY LINES
346. General Instructions. If the given line is a straight line or
a curved line of single curvature, pass an auxiliary plane through
the line and determine its intersection with the given surface.
The point or points in which the given line intersects the lines
cut from the surface by the auxiliary plane will be the points
sought.
Since an infinite number of planes may be drawn through a
straight line, it will be wise, in connection with straight lines, to
use those auxiliary planes which will cut from the surface the
simplest lines. If the given line is a curved line of single curvature the plane of the curve must be used as the auxiliary plane.
347. On the Character of Lines in which Planes intersect Surfaces.
Since prisms and pyramids have plane surface faces, the intersections of these faces by planes will always be straight lines whatever the position of the cutting plane.
Since the elements of cylindrical surfaces are parallel straight
lines, the intersections of such surfaces by planes which are parallel to the elements are parallel straight lines.
Since the elements of conical surfaces all pass through the vertex, the intersections of such surfaces by planes which contain the
vertex will be straight lines passing through the vertex.
The intersection of a sphere by a plane is always a circle. If
the cutting plane contains the center of the sphere, the circle is a
great circle.
348. Problem 221. To find the point in which a given straight line
intersects a given plane (see Section 151).
349. Problem 222. To find the points in which a given straight
line intersects the surface of a square prism.
Analysis. Pass the auxiliary plane through the line and parallel
to the edges of the prism.
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