Elementi di prospettiva

Le prime righe

[p13] PROBLEM II. TO DRAW A RIGHT LINE BETWEEN TWO GIVEN POINTS. [Geometric diagram] Fig. 6. Let A B, Fig. 6., be the given right line, joining the given points A and B. Let the direct, lateral, and vertical distances of the point A be T D, D C, and C A. Let the direct, lateral, and vertical distances of the point B be T D′, D C′, and C′ B. Then, by Problem I., the position of the point A on the plane of the picture is a. And similarly, the position of the point B on the plane of the picture is b. Join a b. Then a b is the line required. [p14] COROLLARY I. If the line A B is in a plane parallel to that of the picture, one end of the line A B must be at the same direct distance from the eye of the observer as the other. Therefore, in that case, D T is equal to D′ T. Then the construction will be as in Fig. 7.; and the student will find experimentally that a b is now parallel to A B.[Footnote 11] [Geometric diagram] Fig. 7. And that a b is to A B as T S is to T D. Therefore, to draw any line in a plane parallel to that of the picture, we have only to fix the position of one of its extremities, a or b, and then to draw from a or b a line parallel to the given line, bearing the proportion to it that T S bears to T D. [p15] COROLLARY II. If the line A B is in a horizontal plane, the vertical distance of one of its extremities must be the same as that of the other. Therefore, in that case, A C equals B C′ (Fig. 6.). And the construction is as in Fig. 8. [Geometric diagram] Fig. 8. In Fig. 8. produce a b to the sight-line, cutting the sight-line in V; the point V, thus determined, is called the Vanishing-Point of the line A B. Join T V. Then the student will find experimentally that T V is parallel to A B.[Footnote 12] [p16] COROLLARY III. If the line A B produced would pass