Descriptive geometry . his trace, PQ, is an edge view of the plane(S 60), and gives the required angles, a with //and (3 with V,directly. 134 DESCRIPTIVE GEOMETRY [XIV, § 137 137. Planes making Given Angles with H and V. The pre-ceding problem gives rise to two converse problems, in whichit is required to find planes which make given angles with thecoordinate planes. Problem 19. Given one trace of a plane, and the angle which theplane makes with either H or V, to find the other trace of the are two cases, which are stated below. Analysis. The required result may be obtained by reve
Descriptive geometry . his trace, PQ, is an edge view of the plane(S 60), and gives the required angles, a with //and (3 with V,directly. 134 DESCRIPTIVE GEOMETRY [XIV, § 137 137. Planes making Given Angles with H and V. The pre-ceding problem gives rise to two converse problems, in whichit is required to find planes which make given angles with thecoordinate planes. Problem 19. Given one trace of a plane, and the angle which theplane makes with either H or V, to find the other trace of the are two cases, which are stated below. Analysis. The required result may be obtained by revers-ing the construction of Problem 18. Case I. Given one trace and the angle which the plane makesivith the same coordinate plane. Construction (Fig. 212). Let HQ and the angle «, which Qmakes with H, be given. Assume the axis A, lying in P~andperpendicular to II. Revolve the given trace HQ about thisaxis to the position IIR, perpendicular to GL. Complete theplane in this position by drawing VR at the given angle, a,. Fig. 212. Fig. 213. with H. Then R intersects the axis A at the point c (C, c*).Let R now be supposed to be revolved back about the axis Auntil IIR coincides with HQ ; the point c remains fixed. Sincec is in V, VQ must pass through c, and is determined by thispoint and the point in which HQ intersects GL. The given trace HQ may also be revolved to the positionHRU giving a second point <y\ and a second result, VQV Thedistances of cv and Cj from GL are obviously the same, henceVQ and VQt make equal angles with GL. XIV, § 137] THE REVOLUTION OF PLANES 135 It is evident in Fig. 212 that the construction is possible forany given angle a between 0° and 90°. If a = 90°, but oneresult is possible, and no construction is necessary. Why ? Case II. Given one trace and the angle which the planemakes with the other coordinate plane. Construction (Fig. 213). Let VQ and the angle «, which theplane makes with H, be given. Assume an axis, A, lying inV and perpendicular t
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