The reader of the following treatise is supposed to know the first six books and the 11th book of Euclid, or the arti cle GEOMETRY in this Dictionary.
On the determination of the perspective representations of given objects on a plane surface.
If straight lines be drawn from the point of sight to the several points of any object, the intersections of these with the perspective plane will mark out the perspective repre sentation of the object.
Let 0 (Fig. 2.) be the point of sight, HK the perspec tive plane, and ABC any figure. If straight lines be drawn from 0 to every point of ABC, the figure a b c, formed by their intersection with the perspective plane HK, will be the perspective representation of ABC.
For as the rays of light proceed from the objects to the eye in straight lines, it is clear that no ray can come from the point A to an eye placed at 0, except by passing through the point a. If, therefore, the point a could be coloured so as to convey to the eye at 0 the same impres sion with the point A, the absence of A would not be per ceived. And as the same- is true of all the other points, it is evident that the figure a b c, when properly coloured, would have to an eye placed at 0, exactly the same ap pearance as the original object ABC; therefore, the inter section a b c of straight lines, drawn from 0, the point of sight, to the several parts of the original object ABC, with the perspective plane, is the perspective representation of that object.
COR. 1. if a straight line be drawn from the point of sight to any point, its intersection with the perspective plane is the perspective representation of that point.
Con. 2. The perspective representation of a straight line not passing through the point of sight is a straight line. For if straight lines be drawn from the point of sight to every point in the given straight line, they will form a Mane triangle ; whose intersection with the per spective plane is a straight line. Thus a b is the perspec tive representation of AB.
Con. 3. The perspective representation of any straight line which passes through the point of sight is a point.
Con. 4. The perspective representation of a straight line is the straight line which joins the perspective repre sentations of its extremities.