Descriptive geometry . Fig. 276 (repeated). XIX, § 162] TANGENT PLANES 189 in the tangent plane at b (§ 162, b). But in attempting to passa plane through F and K, only the traces, s3 and t3, of Kcan be found; neither trace of F is available. Therefore,choose any point in F; a convenient point is the vertex, Fig. 277. Through o draw the auxiliary line L, parallel to K\ then Llies in the plane of F and K (§ 108, Ex. 1). Line L suppliesthe traces s4 and tt. These, taken in connection with the tracesof K, are sufficient to locate the traces, HT and VT, of therequired tangent plane. Problem 32.
Descriptive geometry . Fig. 276 (repeated). XIX, § 162] TANGENT PLANES 189 in the tangent plane at b (§ 162, b). But in attempting to passa plane through F and K, only the traces, s3 and t3, of Kcan be found; neither trace of F is available. Therefore,choose any point in F; a convenient point is the vertex, Fig. 277. Through o draw the auxiliary line L, parallel to K\ then Llies in the plane of F and K (§ 108, Ex. 1). Line L suppliesthe traces s4 and tt. These, taken in connection with the tracesof K, are sufficient to locate the traces, HT and VT, of therequired tangent plane. Problem 32. To pass a plane tangent to a cone through a givenpoint without the surface. (Two results.) Analysis. Pass a line through the given point and thevertex of the cone. Find the point in which this line piercesthe plane of the base of the cone. From this piercing point,draw a line tangent to the base of the cone. The requiredtangent plane is determined by the tangent line and the linefirst drawn. In general, two tangents may be drawn to thebase, giving two possible tangent planes. To prove this analysis, note that the vertex of the cone liesin every tangent plane (§ 162, d). Hence the Line firsl drawnmust lie in the required tangent plane. A tangent to the base 190 DESCRIPTIVE GEOMETRY [XIX, § 162 which
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