SCHOLIUM. It follows from the above proposition, that, if the system of straight lines, or the cone of rays drawn from the point of sight to the objects be the same, their perspective representations will be the same. Hence ob jects of very different figure and appearance may have the same or similar perspective representations.
The straight line which joins the vanishing point with the intersection of any straight line, is the perspective re presentation of that line extended ad infinitum ; and the representation of any part of that original line is some portion of the former.
Let IB, (Fig. 3.) any line indefinitely extended towards B, intersect the perspective plane in I, and let V be its vanishing point ; then VI is the perspective representa tion of the line IB, indefinitely extended towards B, and the representation of AB any portion of IB, is a portion of VI. For, let 0 be the point of sight, and join VO. By Def. 8. OV is parallel toj IB. Therefore OV, VI, and IB, are all in one plane. Hence every straight line drawn from 0 to the line IB must intersect VI somewhere between V and I. If the lines be drawn from 0 towards the remote extremity of IB, they will intersect VI towards the point V ; and though the intersections may be as near as possible to the point V, they cannot be above it, OV being parallel to IB. It is clear, then, that if straight lines be drawn from 0 to every point in IB indefinitely extend ed towards B, they will form a plane whose intersection with the perspective plane is VI ; VI is therefore the per spective representation of IB indefinitely extended to wards B. If straight lines be drawn from 0 to any two points A, B, in the line AB, they will intersect VI some where between V and 1; and a b, the perspective repre sentation of AB, is a portion of VI.
Cox. 1. The perspective representations of parallel lines converge to a point in the picture. This is clear, for it has just been proved, that the perspective represen tation of any line passes through its vanishing point ; but all parallels have the same vanishing point ; their repre sentations, therefore, pass through the same point in the picture. Thus VAL VI are the representations of the
parallels IB. AIN.
Cox. 2. The perspective representations of all lines perpendicular to the perspective plane, pass through the centre of the picture; for the straight line drawn from the point of sight to the centre of the picture is perpendicu lar to the perspective plane, and is therefore parallel to the other perpendiculars. Hence the centre of the pic ture is the vanishing point of all the perpendiculars; but every representation passes through the vanishing point of its original line ; therefore all the representations of the perpendiculars pass through the centre of the pic ture.
The perspective representation of a straight line paral lel to the picture, is parallel to its original ; and has to it the same ratio which the distance of the point of sight from the perspective plane has to its distance from a plane drawn through the original straight line parallel to the perspective plane.
Let MN (Fig. 4.) be the perspective plane, 0 the point of sight, and AB a straight line parallel to MN. Through AB draw the plane PQ, parallel to MN, and let OHG be drawn at right angles to the planes MN, PQ, intersecting them in the points H, G; the perspective representation of AB shall be parallel to A B, and shall have to it the same ratio which OH has to OG. For, if straight lines be drawn from 0 to every point in AB, they will form a plane triangle OAB, whose intersection a b with the perspective plane will be (Prop. 1.) the representation of AB. But the intersections of parallel planes by a third plane are pa rallel (E. 12 of 11.) Therefore a b is parallel to AB. Again, by similar triangles, a b : AB = Ob : OB. But parallel planes cut straight lines in the same ratio (E. 1 ith Book). Hence Ob : 013 = OH : OC; and a b : AB = OH : OC.