Descriptive geometry . ent to a cone contains the vertex. Problem 31. To pass a plane tangent to a cone at a given pointin the surface. Analysis. The plane is determined by the element whichpasses through the given point, and a line tangent to the baseat the point where this element intersects the base (§§ 155,162, &). Construction. Case I. 77/<= base of the cone lien in H or V. Example 1 (Fig. 274). The base of this cone lies in a (oh. a), lying in the element E (Eh. E*) be the given pointin the surface. Since the required tangent plane contains theelement E, find the traces sx and t
Descriptive geometry . ent to a cone contains the vertex. Problem 31. To pass a plane tangent to a cone at a given pointin the surface. Analysis. The plane is determined by the element whichpasses through the given point, and a line tangent to the baseat the point where this element intersects the base (§§ 155,162, &). Construction. Case I. 77/<= base of the cone lien in H or V. Example 1 (Fig. 274). The base of this cone lies in a (oh. a), lying in the element E (Eh. E*) be the given pointin the surface. Since the required tangent plane contains theelement E, find the traces sx and t-, of E. Then the of the tangent plane passes through slf and is determined XIX, § 162] TANGENT PLANES 185 by the fact that HQ must be tangent to the base of the cone(§ 162, c). The trace VQ is now determined, since it mustpass through tx and the point in which HQ intersects GL. Let it also be required to find the tangent plane at the pointb, lying in the element F. The Zf-trace, HT, of this plane,. Fig. 274. passes through the //-trace, s2, of F, and is tangent to the baseof the cone. The F-trace, VT, passes through the V-trace of F,but this point is inaccessible. Recourse must be had to someauxiliary line lying in the plane T, and so situated that itsF-trace is accessible. (See § 108.) As HT is known, a hori-zontal principal line of the plane may be drawn through anypoint of the line F (§ 108, Ex. 3). A convenient point is thegiven point b. Through 6* draw U parallel to HT; throughbv draw Lv parallel to GL. Then the line L lies in the planeT, and the F-trace, t2, of L is a point in VT. Example 2 (Fig. 275). The base of this cone lies in planes are passed at the points a and b. For the 186 DESCRIPTIVE GEOMETRY [XIX, § 162 plane Q, tangent at a, the trace VQ is drawn tangent to thebase of the cone through the F-trace of the element E. Sincethe i7-trace of the element E is not available, a point on HQ isfound by means of the auxiliary line L, a vert
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