Problem XVII
To Find the Dividing-Points of a Given Inclined Line
Let P, Fig. 46., be the vanishing-point of the inclined line, and V the vanishing-point of the relative horizontal.
Find the dividing-points of the relative horizontal, D and D′.
Through P draw the horizontal line X Y.
With center P and distance D P describe the two arcs D X and D′ Y, cutting the line X Y in X and Y.
Then X and Y are the dividing-points of the inclined line.27
Obs. The dividing-points found by the above rule, used with the ordinary measuring-line, will lay off distances on the retiring inclined line, as the ordinary dividing-points lay them off on the retiring horizontal line.
Another dividing-point, peculiar in its application, is sometimes useful, and is to be found as follows:—
Let A B, Fig. 47., be the given inclined line drawn in perspective, and A c the relative horizontal.
Find the vanishing-points, V and E, of A c and A B; D, the dividing-point of A c; and the sight-magnitude of A c on the measuring-line, or A C.
From D erect the perpendicular D F.
Join C B, and produce it to cut D E in F. Join E F.
Then, by similar triangles, D F is equal to E V, and E F is parallel to D V.
Hence it follows that if from D, the dividing-point of A c, we raise a perpendicular and make D F equal to E V, a line C F, drawn from any point C on the measuring-line to F, will mark the distance A B on the inclined line, A B being the portion of the given inclined line which forms the diagonal of the vertical rectangle of which A C is the base.