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USE OF AUXILIARY PLANES.

72. In finding shadows on some of the double-curved sur faces 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 east is parallel to one of the co-ordinate planes.

73. Problem XIII: To find the shadow in a spherical hol-. low. • Fig. 20 shows in plan and elevation a spherical hollow whose plane has been assumed parallel to V. .

Applying to the elevation, the projections of the ray It, • 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 6v are the lim its of this shade line avcvbv. The remaining portion of the line avdviv 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 DOW east the shadow of the line avevbv on the spher ical surface of the hollow, and having no ground line, (since neither Lie V nor the II projection of the spherical hollow is a line,) we use auxiliary planes.

If we pass through hollow, parallel to the plane of the line aeb (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 whose diameter is ob tained from the line whyh in the plan. This line of intersection will show in plan as a straight line, xhyh.

Cast the shadow of the line aeb on this auxiliary P. This is not difficult because the plane P was assumed parallel to aeb, 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 aPsoPsbPs. This is the shad ow of the shade line of the object on the auxiliary plane P.. It will be noted

that this shadow aPsoPsbPs crossed the line of inter section, made by P with the spherical surface, at the two points no and nP. In plan these points would be soll and nh which are two points in the required on the spherical surface for they are the shadows of two points in the shade line aeb and they are also on the surface of the spherical hollow since they are on the line of intersection xy which lies in that spherical • surface. • With one auxil iary plane we thus obtafn two poilits 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 war rant its outline being in elevation and plan with accuracy. The shadow in plan is determined by projection from the shallow in elevation, which is found first.

74. 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.

5. 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.

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 of a spherical hollow. Neither the Hor V pro jection 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.

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