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LINES OF MEASURES.

55. Any line which lies in the picture plane will be its own perspective, and show the true length of the line (24 it). Such a line is called a Line of Measures.

In the last problem,. the line ae, being in the picture plane, was a line of measures ; that is to say, its length could be laid off directly from the given data, and from this length the lengths of the remaining lines in the perspective drawing could be established. Fig. 20 shows a similar problem. The line ae lies in the picture plane, and are" is, therefore, a line of measures for the object.

56. Besides this principal line of measures, other lines of measures may easily be established by extending any vertical plane in the object until it intersects the piCture plane. This intersec tion, since it lies in the picture plane, will show in its size, and will be an auxiliary line of measures. All points in it will show at their true height above the plane of the ground. Thus, in Fig. 20, are" is the principal line of measures, and shows the true height of the block. If the rear vertical faces of the block are extended till they intersect the picture plane, these intersec tions (at'n" and olpP) will be auxiliary lines of measures, and will also show the true height of the block. It will be noticed in the figure that ?VW and o"p" are each equal to aPeP. Either one of these lines could have beeu used to determine the vertical height of the perspective of the block. For illustration, suppose It is de sired to find the height of the pe'rspective, using the line oPpP as the line of measures. Assume the vanishing points (ed and v") for the two systems'of horizontal edges in the block to have been established as in the previous case. Now extend the line be (in the diagram), which represents the horizontal projection of the face ebf, till it intersects 11PP. From this intersection drop a vertical line op which will represent the intersection of the vertical face elf with the picture plane, and will be a line of measures for the faCe. ..pP, where this line of measures intersects will be the point where the lower horizontal edge (produced) of the face cbf intersects the picture plane. MeasUre off the distance pPoP equal to the true height of the block, as given by the. elevation. Two lines drawn thiough oP and pP respectively, and vanishing at ed, will represent the perspectives of the upper and lower edges_ of the face ebf, produced. The perspective (bP), of the point (5, will be

found on the perspective of the upper edge of the face v1;1; verti cally below the intersection of I-1PP with the horizontal projection of a visual ray drawn through the point b in the diagram. A tical line through will intersect the lower horizontal edge of the face elf in the point fr. Lines drawn respectively through bP and fP, vanishing at eb, will establish the perspectives of the upper and lower horizontal edges of the face able The points aP and eP will he found vertically under the points a and e in the diagram. The remainder of the perspective projection may now he determined.

57. The! perspectives of any points on the faces of the block may be found by means of the diagram and one of the lines of measures.

Let the points g°, h°, k°, and /°, in the given elevation, de termine a square on the face able of the block. Let the points g,h, k, and 1, represent the position of the square in the diagram. Extend the upper and lower horizontal edges of the square, as shown in elevation, until they intersect the vertical edge ave° in the points t° and vv. To determine the perspective of the square, lay off on areP, which is a line of measures for the face alfe, the divisions tr and vr taken directly from the elevation. Two lines drawn through tP and vP respectively, vanishing at eb, will represent the perspectives of the upper and lower edges (pro duced) of the square. will be found on the perspective of the upper edge, vertically under the intersection of ItYP with the horizontal projection of a visual ray drawn through the point g in the diagram. The position of kP may be established in a similar manner. Vertical lines drawn through gP and kP respectively, will complete the perspective of the square. • 58. The auxiliary line of measures might have been used instead of are. In this case, oPpP should be divided by the points u/P and yP, in the same way that ae; in elevation, is divided by the points t and v. Through teP and draw horizontal lines lying in the plane cbf, for which o' pr is a line of measures.. These lines will vanish at v", and intersect the vertical edge VP of the block. From these intersections draw horizontal lines lying in the plane abef, vanishing at vab, and representing the upper and lower edges of the square. The remainder of the square may be determined as in the previous case.