Descriptive geometry . on ac (<<hch, avcv), Fig. 100, which is parallel toV, shows in the ^projection the angle, a, which the line itselfmakes with //. The line ac is the revolved position of the lineab. In the preceding construction we saw that during the revo-lution the angle which the line made with H did not a is the angle which the given line ab makes with is, in addition to finding the true length of the line ab inFig. 100, we have incidentally found the angle which this linemakes with //. Similarly, in Fig. Kll. we found, ;is an incidental part of theconstruct
Descriptive geometry . on ac (<<hch, avcv), Fig. 100, which is parallel toV, shows in the ^projection the angle, a, which the line itselfmakes with //. The line ac is the revolved position of the lineab. In the preceding construction we saw that during the revo-lution the angle which the line made with H did not a is the angle which the given line ab makes with is, in addition to finding the true length of the line ab inFig. 100, we have incidentally found the angle which this linemakes with //. Similarly, in Fig. Kll. we found, ;is an incidental part of theconstruction, the angle (3 which the line ab makes with 1. 56 DESCRIPTIVE GEOMETRY [VIII, § 80 80. Second Method for Finding True Length. The true lengthof a straight line may also be found by revolving the line aboutone of its own projections as an axis, until the line lies in acoordinate plane. Problem 3 (bis). To find the true length of a straight line. Second Method. By revolution about one of the projec-tions of the Fig. 102. Analysis (Fig. 102). Let ab be the given line, projected onany plane Q at aqbq. Revolve the plane abbqa? about aqbq asaxis into Q. Then a falls at ar, where ara equals aa, and isperpendicular to aqb (§ 76); b falls at br, with brbq equal tobbq and perpendicular to aqbq; hence arbr equals ab, and showsthe true length of the line. Construction. The projection ahbh, Fig. 103, corresponds toaqbq, Fig. 102, and is taken as the axis of revolution. Since a*is the projection of the point a in space, the distance, aah, ofpoint a from the axis is equal to the distance of the point afrom H, which shows in the ^projection as the distance from avto GL. Hence to find the revolved position ar, make ahar per-pendicular to ahbh, and the distance ahar equal to ave. Simi-larly, bhbr is perpendicular to ahbh, and is equal to bvf. Thenarbr shows the true length of the line ab. In Fig. 104 the F-projection is taken as the axis, and theline is revolved into V. The distances
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