The essentials of descriptive geometry . Likewisedraw through O a line parallel to V and making the given gradewith H, and make it equal in length to OQ. op, then, is thelength of the plan view reqxiired. Revolve these two lines untilP and Q coincide at N; on and on then will be the required planand elevation of the center Hne of the tunnel. Note. The sum of the angles given must not exceed 90degrees or the problem becomes impossible. 68. Proposition 16. Given two planes to find the angle be-tween them. Discussion. The angle between the planes is measured in aplane perpendicular to both planes
The essentials of descriptive geometry . Likewisedraw through O a line parallel to V and making the given gradewith H, and make it equal in length to OQ. op, then, is thelength of the plan view reqxiired. Revolve these two lines untilP and Q coincide at N; on and on then will be the required planand elevation of the center Hne of the tunnel. Note. The sum of the angles given must not exceed 90degrees or the problem becomes impossible. 68. Proposition 16. Given two planes to find the angle be-tween them. Discussion. The angle between the planes is measured in aplane perpendicular to both planes or to their Hne of , then, a plane be passed through any point of a Hne common 78 ESSENTIALS OF DESCRIPTIVE GEOMETRY to both planes and the intersection of this plane with both of theothers be found the angle thus formed will be the required anglebetween the given planes, and its size may be determined. Construction. In Fig. 62 is shown a plan and elevation of atrough. The problem is to find the angle between its Fig. 61. Assume the point 0 on the line common to the planes of the twosides, and through this point, by Article 48, pass a plane perpen-dicular to the line. This plane is rRr. Now by Article 54 findthe intersection of R and S, and R and T. This is OP in the firstcase and OQ in the second, and POQ is the required angle. Itstrue size, pOnq, may be found by Article 50. 69. Special Cases, i. Find the angle a plane makes with HandV. PROBLEMS ON POINTS, LINES, AND PLANES 79 2. Find the angle between two planes which are parallel tothe G. L. 70. Proposition 17. Given one trace of a plane and the anglethe plane makes with the corresponding plane of projection tofind the other trace.
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