26. Problem II. To find the shadow of a given line on a given plane.

A straight line is made up of a series of points, Rays of light passing through all of these points would form a plane of light. The intersection of this plane of light with either of the co-ordinate planes would be the shadow of the given line on that plane. This shadow would be a straight line because two planes always intersect in a straight line. This fact, and the fact that a straight line is determined by two points, enables us to cast the shadow of a given line by simply casting the shadows of any two points in the line and drawing a straight line between these points of shadow.

FIG-IO.

FIG-11l.

In Fig. 10, avbv and ahbh are the elevation and plan respectively of a given line ab in space. Casting the shadow of the ends of the line a and b by the method illustrated in Problem 1 and drawing the line avsbvs, we obtain the shadow of the given line ab on V.

27. Fig. 11 shows the construction for finding the shadow of the line ab when the shadow falls upon H.

28, Fig. 12 shows the construction for finding the shadow of a line so situated that part of the shadow falls upon V and the remainder on H. To obtain the shadow in such a case, it must be found wholly on either one of the co-ordinate planes. In Fig. 12, it has been found wholly on V, avS being the actual shadow of that end of the line, and bvs being the imaginary shadow of the end b on Y. Of the line avsbvs we use only the part avscvs, that being the shadow which actually falls upon Y.

FIG-12.

FIG'13.

The point where the shadow leaves V and the point where it begins on H are identical, so that the beginning of the shadow on H will be on the lower ground line directly below the point cvs; chs will then be one point in the shadow of the line on II, and casting the shadow of the end b we obtain bhs. The line chsbhs, drawn between these points, is evidently the required shadow on H.

29. Another method of casting the shadow of such a line as ah is to determine the entire shadow on each plane independently. This will cause the two shadows to cross the ground lines at the same point c, and of these two lines of shadows we take only the actual shadows as the required result. This method involves unnecessary construction, but should be understood.

30. Fig. 13 shows the construction of the shadow of a given line on a plane to which it is parallel. It should be noted that the shadow in this case is parallel and equal in length to ihe given line.

31. Fig. 14 shows the construction of the shadow of a given line on a plane to which the given line is perpendicular. It is to be noted that the shadow coincides in direction with the projection of the ray of light on that plane, and is equal in length to the diagonal of a square of which the given line is one side.

FIG'14.

32. Fig. 15 shows the construction for finding the shadow of a curved line on a given plane. Under these conditions we find. by Problem 1, the shadows of a number of points in the line-the greater the number of points taken the more accurate the resulting shadow. The curve drawn through these points of shadow is the required shadow.

33. In Fig. 16 the given line ah is in space and the problem is to find its shadow on two rectangular planes mnop and nrso, both perpendicular to II.

Consider first the shadow of ah on the plane mnop. The edge no is the limit of this plane on the right. Therefore from the point nh draw back to the given line the projection of a ray of light. This 45° line intersects the given line at ch. It is evident that of the given line ah, the part ac falls on the plane mnop and the remainder, eh, on the plane nrso.

FIG.16.

FIG.15.

To find the shadow of ac on the left-hand plane we must first determine our ground line. The ground line will he that pro-ji ction of the plane receiving the shadow which is a line. In this example the vertical projection of the plane mnop is the rectangle mvnvovpv This projection cannot, therefore, be used as a GL, The plan, or II projection, of this plane is, however, a line mhnh This line, therefore, will be used as the ground line for finding the shadow of ac on mnop.

We find the shadow of a to be at as and the shadow of c at cs, Problem I. The line ascs is, therefore, a part of the required shadow. The remaining part, csbs is found in a similar manner.

34. The above illustrates the method of determining the GL when the shadow falls upon some plane other than a co-ordinate plane. In case neither projection of the given plane is a line, the shadow must be determined by methods which will be explained later.