PROBLEM XVI.

TO FIND THE VANISHING-POINT OF A GIVEN INCLINED

LINE.

LF, in Fig. 43. or Fig. 44., the lines A Y and ▲'y' be produced, the student will find that they meet.

Let P, Fig. 45., be the point at which they meet.

From P let fall the vertical P V on the sight-line, cutting the sight-line in v.

Then the student will find experimentally that v is the vanishing-point of the line a c.*

Complete the rectangle of the base A c', by drawing A'c' to v, and c c' to the vanishing-point of a a'.

Join y' c'.

Now if y c and 'c' be produced downwards, the

student will find that they meet.

Let them be produced, and meet in P'.

Produce P V, and it will be found to pass through

the point P'.

The demonstration is in Appendix II. Article III

Therefore if a Y (or c y), Fig. 45., be any inclined

line drawn in perspective by Problem XV., and

AC the relative horizontal (A c in Figs. 39, 40.), also

drawn in perspective.

Through v, the vanishing-point of a c, draw the

vertical P P' upwards and downwards.

Р

Produce A Y (or c Y), cutting P P' in P (or P').

Then P is the vanishing-point of a Y (or P' of c v). The student will observe that, in order to find the point P by this method, it is necessary first to draw a portion of the given inclined line by Problem XV. Practically, it is always necessary to do so, and, therefore, I give the problem in this form.

Theoretically, as will be shown in the analysis of the problem, the point P should be found by draw ing a line from the station-point parallel to the given inclined line; but there is no practical means of drawing such a line; so that in whatever terms the problem may be given, a portion of the inclined line (A Y or C Y) must always be drawn in perspective before P can be found.

PROBLEM XVII.

TO FIND THE DIVIDING-POINTS OF A GIVEN INCLINED

LET P, Fig. 46., be the vanishing-point of the in

clined line, and v the vanishing-point of the relative horizontal.

Find the dividing-points of the relative borizontal, D and D'.

Through P draw the horizontal line x Y.

With centre 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.*

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.

The demonstration is in Appendix II., p. 138.