72. In finding shadows on some of the double-curved surfaces of revolution, such as the surface of the spherical hollow, the scotia and the torus, we can make use of auxiliary planes to advantage, when the plane of the line whose shadow is to be cast is parallel to one of the co-ordinate planes.

73. Problem XIII. To find the shadow in a spherical hoI= low.

Fig. 29 shows in plan and elevation a spherical hollow whose plane has been assumed parallel to Y.

Applying to the elevation, the projections of the ray R, we determine the amount of the edge of the hollow which will cast a shadow on the spherical surface inside. The points of tangency av and bv are the limits of this shade line avcvbv. The remaining portion of the line avdvbv is not a shade line since the light would reach the spherical surface adjacent to it and also reach the plane surface on the other side of avdvbv outside the spherical hollow.

We must now cast the shadow of the line avcvbv on the spherical surface of the hollow, and having no ground line, (since neither the V nor the H projection of the spherical hollow is a line,) we use auxiliary planes.

If we pass through the spherical hollow, parallel to the plane of the line acb (in this case parallel to V) an auxiliary plane P, it will cut on the spherical surface a line of intersection xy; in elevation this will show as a circle xvyv, whose diameter is obtained from the line xh yh in the plan. This line of intersection will show in plan as a straight line, xhyh.

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Cast the shadow of the line acb on this auxiliary plane P. This is not difficult because the plane P was assumed parallel to acb, and in this particular case, avcvbv is the arc of a circle. To east its shadow on P it is only necessary to cast the shadow of its center Ov, using the line P as a ground line and to draw an arc of same length and radius. We thus obtain the arc a,PsCPsbPs This is the shadow of the shade line of the object on the auxiliary plane P. It will be noted that this shadow aPscPsbPs crossed the line of inter-section, made by P with the spherical surface, at the two points mp and np. In plan these points would be mh and nh which are two points in the required shadow on the spherical surface for they are the shadows of two points in the shade line acb and they are also on the surface of the spherical hollow since they are on the line of intersection my which lies in that spherical surface. With one auxiliary plane we thus obtain two points in the shadow of the hollow. In Fig. 30 a number of auxiliary planes have been used to obtain a sufficient number of points, 1, 2, 3, 4, etc., of the shadow, to warrant its outline being in elevation and plan with accuracy. The shadow in plan is determined by projection from the shadow in elevation, which is found first.

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7 4 The separate and successive steps in this method of determining the shadow of an object by the use of auxiliary planes are as follows:

1. Determine the shade line by applying to the object the projections of the ray of light.

2. Pass the auxiliary planes through the object parallel to the plane of the shade line.

3. Find the line of intersection which each auxiliary plane makes with the object.

4. Cast the shadow of the shade line on each of the auxiliary planes.

5. Determine the point or points where the shadow on each auxiliary plane crosses the line of intersection made by that plane with the object.

6. Draw a line through these points to obtain the required shadow,

75. Problem XIV. To find the shadow on the surface of a scotia.

This problem is similar in method and principle to that for finding the shadow on a spherical hollow. Neither the H or V projection of the surface of the scotia is a line, and we therefore must resort to some method other than that generally used. The follow. ing is the most accurate and convenient although the shadow can be found by a method to be explained in the next problem.

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 on the scotia is formed by the shadow of the upper fillet or right cylinder upon the surface of the scotia. "We determine the shade lines of the cylinder, Problem XI, by applying to the plan the projections of the ray, Fig. 31. These determine the shade elements at xh and yh and also the portion of the 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 elements of the cylinder falls not on the scotia itself, but beyond on the II or V plane, or some other object, hence we can neglect them for the present.

Having determined the shade line, there is another preliminary step to be taken before finding its shadow. That is, to determine the highest point in the shadow avs. 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 avs and the shade line xvavyv 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 shadow avs is determined, therefore, as follows:

The point ah lying on the diagonal Poh is evidently the point in the shade line which will cast the highest point in the shadow, for, considering points in the shade line on either side of ah, it will become evident that the rays through them must in tersect the scotia surface at points lower down than the point avs.

The point a lies in a plane of light P, which passes through the axis oh of the scotia. This plane, therefore, cuts out of the scotia surface a line of intersection exactly like the profile avcv. If we revolve the plane P and its line of intersection about the axis oh until it is parallel to V, the line of intersection will then coincide with this profile avcv, the point av having moved to the point a'v.

If, before revolving, we had drawn the projections of the ray of light, Rv, through the point av, they would be the lines avbv and ahbh After the revolution of the plane P these projections of the ray are the lines av bv and ahbh. The point b, being in the axis, does not move in the revolution of the plane P. The point avs, the intersection of the projection of the ray Rv with the profile avcc, indicates that the ray Rv has pierced the scotia surface. If now the plane P is revolved back to its original position, this point a,vs will move in a horizontal line in elevation to the point avs, and the point avs thus obtained is the shadow of the point av on the surface of the scotia and is also the highest point of the shadow.

FIG.31.

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FIG.32.

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T7. 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 II) 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 xhahyh on each of these auxiliary planes. This is done by casting the shadow of its center O and drawing ares equal to xhahyh.

The points of intersection, 2h, 3h, 4h, 5h, 6h, etc., are pointa 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 taking one of the auxiliary planes at the line MN. The points lv, 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 surfaces, 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 arc xhahyh.