[Footnote 32: ] This point coincides in the figure with the extremity of the horizontal diameter, but only accidentally.] [Return to text]
[Footnote 33: ] The engraving is a little inaccurate; the inclosing line should touch the dotted semicircles at A and B. The student should draw it on a large scale.] [Return to text]
[p94]
]PROBLEM XVI.
It is often possible to shorten other perspective operations considerably, by finding the vanishing-points of the inclined lines of the object. Thus, in drawing the gabled roof in Fig. 43., if the gable A Y C be drawn in perspective, and the vanishing-point of A Y determined, it is not necessary to draw the two sides of the rectangle, A′ D′ and D′ B′, in order to determine the point Y′; but merely to draw Y Y′ to the vanishing-point of A A′ and A′ Y′ to the vanishing-point of A Y, meeting in Y′, the point required.
Again, if there be a series of gables, or other figures produced by parallel inclined lines, and retiring to the point V, as in [Fig. 72.],[Footnote 34] ] it is not necessary to draw each separately, but merely to determine their breadths on the line A V, and draw the slopes of each to their vanishing-points, as shown in [Fig. 72.] Or if the gables are equal in height, and a line be drawn from Y to V, the construction resolves itself into a zigzag drawn alternately to P and Q, between the lines Y V and A V.
The student must be very cautious, in finding the vanishing-points of inclined lines, to notice their relations to the horizontals beneath them, else he may easily mistake the horizontal to which they belong.
Thus, let A B C D, [Fig. 73.], be a rectangular inclined plane, and let it be required to find the vanishing-point of its diagonal B D.
Find V, the vanishing-point of A D and B C.
Draw A E to the opposite vanishing-point, so that D A E may represent a right angle.
Let fall from B the vertical B E, cutting A E in E.