PERSPECTIVE OF A POINT PLACED IN ANY POSITION WITHIN THE SQUARE
Let the point _P_ be the one we wish to put into perspective. We have but to repeat the process of the previous problem, making use of our measurements on the base, the diagonals, &c.
Indeed these figures are so plain and evident that further description of them is hardly necessary, so I will here give two drawings of triangles which explain themselves. To put a triangle into perspective we have but to find three points, such as _fEP_, Fig. 148 A, and then transfer these points to the perspective square 148 B, as there shown, and form the perspective triangle; but these figures explain themselves. Any other triangle or rectilineal figure can be worked out in the same way, which is not only the simplest method, but it carries its mathematical proof with it.
PERSPECTIVE OF A SQUARE PLACED AT AN ANGLE NEW METHOD
As we have drawn a triangle in a square so can we draw an oblique square in a parallel square. In Figure 150 A we have drawn the oblique square _GEPn_. We find the points on the base _Am_, as in the previous figures, which enable us to construct the oblique perspective square _n·G·E·P·_ in the parallel perspective square Fig. 150 B. But it is not necessary to construct the geometrical figure, as I will show presently. It is here introduced to explain the method.
Fig. 150 B. To test the accuracy of the above, produce sides _G·E·_ and _n·P·_ of perspective square till they touch the horizon, where they will meet at _V_, their vanishing point, and again produce the other sides _n·G·_ and _P·E·_ till they meet on the horizon at the other vanishing point, which they must do if the figure is correctly drawn.
In any parallel square construct an oblique square from a given point--given the parallel square at Fig. 150 B, and given point _n·_ on base. Make _A·f·_ equal to _n·m·_, draw _f·S_ and _n·S_ to point of sight. Where these lines cut the diagonal _AC_ draw horizontals to _P·_ and _G·_, and so find the four points _G·E·P·n·_ through which to draw the square.
ON A GIVEN LINE PLACED AT AN ANGLE TO THE BASE DRAW A SQUARE IN ANGULAR PERSPECTIVE, THE POINT OF SIGHT, AND DISTANCE, BEING GIVEN.
Let _AB_ be the given line, _S_ the point of sight, and _D_ the distance (Fig. 151, 1). Through _A_ draw _SC_ from point of sight to base (Fig. 151, 2 and 3). From _C_ draw _CD_ to point of distance. Draw _Ao_ parallel to base till it cuts _CD_ at _o_, through _O_ draw _SP_, from _B_ mark off _BE_ equal to _CP_. From _E_ draw _ES_ intersecting _CD_ at _K_, from _K_ draw _KM_, thus completing the outer parallel square. Through _F_, where _PS_ intersects _MK_, draw _AV_ till it cuts the horizon in _V_, its vanishing point. From _V_ draw _VB_ cutting side _KE_ of outer square in _G_, and we have the four points _AFGB_, which are the four angles of the square required. Join _FG_, and the figure is complete.