76. As in any problem in shades and shadows, the first step is to determine the shade line.
The scotia is bounded above and below by fillets which are portions of right cylinders. The shadow of the scotia is formed by the shadow of the upper fillet or right cylinder upon the sur face of the scotia. We determine the shade lines of the cylinder, Problem XI, by applying to the plan the projections of the lay, Fig. 31. These determine the shade elements at xh and yt and also the portion of tho perimeter of the fillet, xhahyh, which is to cast the. shadow on the scotia.
In this case, as in most scotias, the shadow of the shade ele ments of the cylinder falls not on the scotia itself, but beyond on the H or V plane, or some other object, hence we can neglect them for the present.
Having determined the shade line, there is another prelim inary step to be taken before finding its shadow. That is, to deter mine the highest point in the shadow os. We do this to know where it is useless to pass auxiliary planes through the scotia. Such planes would evidently be useless between the point ays and the shade line eavyv in elevation. Also because we could not be sure that in passing the auxiliary planes we were passing a plane which would determine this highest point.
The highest point of shad ow ays is determined, there fore, as follows: The point at', lying on the diagonal Pob is evidently the point in the shade line which will cast the highest point in the shadow; for, con- • sidering points in the shade line on either side of ah, it 'will become evident that the rays through them must in• tersest the scotia surface at points lower down than the point ays.
The point a lies in a plane of light P, which passes through the axis ob of the scotia. This plane, therefore, cuts out of the scotia surface a line of intersection exactly like the profile ave. If we revolve the plane P and its line of intersection about the axis ob until it is parallel to V, the line of intersection will then coincide with this profile avev, the point av having moved to the point a'v.
If, before revolving, we had drawn the projections of the ray of light, It'', through the'point a", they would be the lines avb° and ahbh. After the revolution of the plane P these projections of the ray are the lines dvbv and a'hbh. The point b, being ix the axis, does not move in the revolution of the plane P. TIB point dvs, the intersection of the projection of the. ray RI with the profile aver, indicates that the ray n'y has pierced the •scotia surface. If now the plane P is revohjed back to its original posi tion, this point a'" will move in a horizontal line in elevation to a", and the point a" thus obtained is the shadow of the point ay on the surface of the scotia and is also the highest point of the shadow.
77. The remainder of the process is, from now on, similar to the method just explained in the previous problem. See Fig. 32.
We pass auxiliary planes, A, B, C, etc., (in this case parallel to H) through the scotia.
We determine in plan their respective lines of intersection with the scotia: they will be circles.
Cast the shadow of the arc xhaliyh on each of these auxiliary planes. This is done by casting the shadow of its center 0 and drawing arcs equal to xhahyh.
The points of intersection, 2h, 3h, 4h, 5h, 6h, etc., are points in the required shadow in plan. The points lh and 10h are the ends, where the shadow leaves the scotia, and these are determined by tak ing one of the auxiliary planes at the line MN. The points 1v, 2v, 3v, etc., are obtained in the elevation by projection from the plan.
The shade of the lower fillet is determined by Problem XI.
78. In case the fillets are conical instead of cylindrical sur. faces, as is sometimes the case in the bases of columns where the scotia moulding is most commonly found, care must be taken to first determine the shade elements of the conical surface. This supposition of conical surfaces would mean a larger arc for the shade line than the arcrehahyh.