We have hitherto been examining the conditions of horizontal and
vertical lines only, or of curves inclosed in rectangles.
[Illustration: Fig. 39.] [Illustration: Fig. 40.]
We must, in conclusion, investigate the perspective of inclined lines,
beginning with a single one given in position. For the sake of
completeness of system, I give in Appendix II. Article III. the
development of this problem from the second. But, in practice, the
position of an inclined line may be most conveniently defined by
considering it as the diagonal of a rectangle, as _AB_ in Fig. 39.,
and I shall therefore, though at some sacrifice of system, examine it
here under that condition.
If the sides of the rectangle _AC_ and _AD_ are given, the slope of
the line _AB_ is determined; and then its position will depend on that
of the rectangle. If, as in Fig. 39., the rectangle is parallel to the
picture plane, the line _AB_ must be so also. If, as in Fig. 40., the
rectangle is inclined to the picture plane, the line _AB_ will be so
also. So that, to fix the position of _AB_, the line _AC_ must be
given in position and magnitude, and the height _AD_.
[Illustration: Fig. 41.]
If these are given, and it is only required to draw the single line
_AB_ in perspective, the construction is entirely simple; thus:—
Draw the line _AC_ by Problem I.
Let _AC_, Fig. 41., be the line so drawn. From _a_ and _c_ raise the
vertical lines _ad_, _cb_. Make _ad_ equal to the sight-magnitude of
_AD_. From _d_ draw _db_ to the vanishing-point of _ac_, cutting _bc_
in _b_.
Join _ab_. Then _ab_ is the inclined line required.
[Illustration: Fig. 42.]
If the line is inclined in the opposite direction, as _DC_ in
Fig. 42., we have only to join _dc_ instead of _ab_ in Fig. 41., and
_dc_ will be the line required.
I shall hereafter call the line _AC_, when used to define the position
of an inclined line _AB_ (Fig. 40.), the “relative horizontal” of the
line _AB_.