Problem VIII
To Draw a Square, Given in Position and Magnitude, in a Horizontal Plane
Let A B C D, Fig. 20., be the square.
As it is given in position and magnitude, the position and magnitude of all its sides are given.
Fix the position of the point A in a.
Find V, the vanishing-point of A B; and M, the dividing-point of A B, nearest S.
Find V′, the vanishing-point of A C; and N, the dividing-point of A C, nearest S.
(For since A B C D is a square, A C is equal to A B.)
Draw a V′ and c′ N, cutting each other in c.
Draw a V, and b′ M, cutting each other in b.
Then a c, a b, are the two nearest sides of the square.
Now, clearing the figure of superfluous lines, we have a b, a c, drawn in position, as in Fig. 21.
And because A B C D is a square, C D (Fig. 20.) is parallel to A B.
And all parallel lines have the same vanishing-point. (Note to Problem III.)
Therefore, V is the vanishing-point of C D.
Similarly, V′ is the vanishing-point of B D.
Therefore, from b and c (Fig. 22.) draw b V′, c V, cutting each other in d.
Then a b c d is the square required.
Notes
Draw the measuring-line through a, and make a b′, a c′, each equal to the sight-magnitude of A B.