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
_AYC_ be drawn in perspective, and the vanishing-point of _AY_
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 _YY′_ to the vanishing-point of _AA′_ and _A′Y′_ to
the vanishing-point of _AY_, 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.,[34] it is not necessary to draw each separately, but merely
to determine their breadths on the line _AV_, 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 _YV_ and _AV_.
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 _ABCD_, Fig. 73., be a rectangular inclined plane, and let
it be required to find the vanishing-point of its diagonal _BD_.
Find _V_, the vanishing-point of _AD_ and _BC_.
Draw _AE_ to the opposite vanishing-point, so that _DAE_ may represent
a right angle.
Let fall from _B_ the vertical _BE_, cutting _AE_ in _E_.
Join _ED_, and produce it to cut the sight-line in _V′_.
[Illustration: Fig. 72.]
Then, since the point _E_ is vertically under the point _B_, the
horizontal line _ED_ is vertically under the inclined line _BD_.
[Illustration: Fig. 73.]
So that if we now let fall the vertical _V′P_ from _V′_, and produce
_BD_ to cut _V′P_ in _P_, the point _P_ will be the vanishing-point of
_BD_, and of all lines parallel to it.[35]
[34] The diagram is inaccurately cut. _YV_ should be a right line.
[35] The student may perhaps understand this construction better
by completing the rectangle _ADFE_, drawing _DF_ to the
vanishing-point of _AE_, and _EF_ to _V_. The whole figure,
_BF_, may then be conceived as representing half the gable roof
of a house, _AF_ the rectangle of its base, and _AC_ the
rectangle of its sloping side.
In nearly all picturesque buildings, especially on the
Continent, the slopes of gables are much varied (frequently
unequal on the two sides), and the vanishing-points of their
inclined lines become very important, if accuracy is required
in the intersections of tiling, sides of dormer windows, etc.
Obviously, also, irregular triangles and polygons in vertical
planes may be more easily constructed by finding the
vanishing-points of their sides, than by the construction given
in the corollary to Problem IX.; and if such triangles or
polygons have others concentrically inscribed within them, as
often in Byzantine mosaics, etc., the use of the
vanishing-points will become essential.