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PROPOSITION X. To find the shade and shadow of a given cylinder. Let a, b, a, b, be the projections of the axis of the cylinder, a,, b, being its traces. Take any point c in ab and find the horizontal or vertical trace of the shadow cast by it, y, or Yz; then we get the plane P which contains the shadow of the axis.
The neutral planes are parallel to this, and their traces are tangents to the trace of the cylinder; and hence are found to be N, N and Q. Q: also the vertical traces are parallel to PP2, and N, Q are found; and hence NN, and QQ, are the boundaries of the shadow on the vertical plane.
When the cylinder cuts the vertical plane, the visible shadow terminates; and the curve which bounds the shadow coincides with the trace of the cylinder on the vertical plane. This is easily marked by finding the intersection of the shadow of an edge and the projection of the same edge on the vertical plane.
For the shade we have the part of the cylinder on the right of N, Q. (in this figure, the light coming in on the left), and its corresponding vertical projection
(bounded by double lines in the figure) to represent it.
To find the shadow and the shade of a given cone. Through v,v2, the vertex of the cone, draw a line parallel to the given ray: this will be the shadow line of the vertex of the cone. Let its traces be w1, Wą.
Now, the shadow planes pass through this line and touch the cone itself: whence their traces upon the horizontal plane will be tangents to the trace of the cone.
Whence drawing tangents from w, to the base of the cone, we have the traces of the two neutral edges, and hence can effect their constructions in the usual manner, as represented by the double lines in the figure.
Again, the horizontal shadows tend to wi, and from their sections with the axis, the vertical tend to wg. The shade and shadow are hence determined.
NOTE.-There is no practical simplification either in respect of the cone or cylinder, when they are vertical, except that the traces of the shadow on the vertical plane do not exist : the cylinder in this case not meeting that plane, and the cone not meeting it within the visible limits.
Given the azimuth and altitude of the sun to find the directions of
the ray on the coordinate planes.
Let XX' be the axis of the coordinate planes, AB the direction of the meridian with respect to it: make BAa, equal to the given azimuth, and make the angle a, Aa the altitude (on either side of Aa, as may be convenient), and draw from any point a, the line a, a perpendicular to Aaj: then lastly in the perpendicular a, ad, to the axis make aa, = a, a and join a A. Then will Aa, and Aa, be the parallels of the ray upon the coordinate planes.
For evidently Aa, is the horizontal projection by the nature of the conditions; and that aa, is the projection X
X on the vertical plane of the projecting line of a point situated in the ray; and therefore Aa, is the vertical projection of the ray itself.
Note.-This, in fact, is finding the projections of that particular ray which passes through the intersection of the meridian and elevation trace on the horizontal plane.
It is also obvious how to construct the converse problem, should it be required, viz. :
Given the projections of the ray to find the angle which it makes with either plane.
TOPOGRAPHIC PROJECTION,* OR THE METHOD
1. It has been established in our preliminary course, that every point, every line, and every curve becomes perfectly defined, when its horizontal and vertical projections are exhibited in orthograph. Planes, it has been seen, are otherwise represented, viz., by means of their traces on the planes of projection. It has also been seen that the simple surfaces upon which we have treated (the cylinder, the cone, and the sphere), are also represented upon a different principle, that of representing an arbitrarily selected specimen of the line or circle by which they are respectively generated. Let us recal these.
* The Topographic Projection differs from the old Orthographic Projection on a single plane, in giving the scale of the line or plane in addition to the projection. In the topographic the information is completely given; in the orthographic only partially. Or, at least, the complementary information is given in such a diversity of ways as to destroy all systematic representation. The consequential processes in the topographic method, hence, have a uniformity which cannot be given to those in the orthographic. Nicholson, indeed, attempted to remove this objection to the orthographic projection by means of his “ directing diagram" (uearly 40 years ago in Rees's Cyclopædia, and more recently in his work on Projection), but with limited success.
2. A cylinder is generated by a line, which is always parallel to a given line, and which has its horizontal or vertical trace in a given curve.
3. A cone is generated by a line which always passes through a given point, and has its horizontal or vertical trace in a given curve.
4. Å sphere is generated by the revolution of a given semicircle about a diameter which is given in position.
5. Although we have employed the generating line and the generating circle respectively for the ordinary constructions of Descriptive Geometry, in representing these surfaces, these are not the only views under which we may consider these surfaces generated. In fact, when we come to consider surfaces generally in reference to the practical applications of Descriptive Geometry to the arts of construction, we shall seldom find these definitions to be the best adapted to our purpose, and hence I have dwelt upon them under this view but very slightly. Even this slight view of them under such an aspect has been adopted more for the sake of an easy transition from one view to the other than for any other reason ; though it must he remarked, that even these afford facilities in certain special cases for the investigation of particular problems that are greatly simpler than the general mode of construction that is usually employed; this is, however, only casual, and is no argument in favour of its supplanting generally the more effective method which will be hereafter developed.
6. The most simple of general methods of representation of surfaces for the purposes of plan-drawing is that of supposing the surface, however generated, to be cut by a series of planes parallel to one of the planes of projection; and the curve of section to be represented by its projections on the two planes. The sections are in practice made, most commonly, by planes parallel to the horizontal plane of projection : though, as in Descriptive Geometry, there is no preference of one over the other in theory, and the horizontal is chosen from its being more in accordance with our earlier practice in this class of drawings.
7. For the purpose of showing this more clearly, I shall resume the representation of a cone as already given, and as given by the means here spoken of.
Let v, v, be the vertex, and a, a, a point in the surface : then, if the point b, be in the trace of the cone on the horizontal plane of projection, any point in the surface is represented if we join the point b, in the trace, and either of the projections a, or a, of the point through which the generatrix passes. The construction, and therefore exhibition of the defining elements, of any point in the conical surface, can be effected, provided we have the requisite data for fixing that point.
8. If, instead of the preceding representation, we employ the method of horizontal sections, we shall have a series of curves 0, 1, 2, etc., upon the horizontal plane of projection similar to one another; whilst
the vertical projections are horizontal lines bounded by the lines which are the projections of the extreme generatrices, and which are also marked in corresponding numbers 0, 1, 2, etc.
It will be obvious that any point b on the surface of the cone will be as distinctly defined, by its projection on one of these planes, as by the other method. It will be seen that whether b, or bą be given, the other can, by means of known processes, be found. A very rational inquiry in this case will be, whether, since the vertical plane contains only straight lines, the constructions effected by means of them might not, possibly, be more simply performed without their aid by the substitution of other operations whose results shall be coeffective with them. In many cases it will be found to be possible, but not in all cases. The principal practical advantage is in those cases where (o the projections on the vertical plane are upon a small scale in comparison with the extent of those on the horizontal plane, or vice versa. To the engineer it occurs in military drawings, where the relief is small in comparison with the ground fortified; and to the civil engineer, in the delineation of ground which varies inconsiderably from the horizontal level. To him it is of importance for the purposes of railroad surveys, those of canals and turnpikes, the disposition of buildings, sewerage, supply of water, and parochial maps.
9. To show the general principle, let us suppose these horizontal sections to be made at distances from each other equal to any given unit of altitude. Then the horizontal projections, being drawn and numbered 0, 1, 2, etc., will suffice for the adequate representation of the sections themselves. The vertical projections may be omitted from the drawing, so long as our object is merely to enable the mind to form a general conception of the form of the surface; as it is easy to connect together the ideas of altitude and form-it being, in fact, nothing more than conceiving the several projections of the section (which are equal to them in all respects) to be raised up to the heights 1, 2, 3, etc., designated by the number of units affixed to the curve of projection on the borizontal plane.
10. In actual surveys, however, our object is an inverted one. The form of the surface is not given, but is required to be found by observation in each case. It cannot in general be a regular surface, that is, one whose geometrical genesis can be assigned. The surveys for this purpose will be explained elsewhere: and I shall assume that they can be made in what I am now discussing. The problem, as one of surveying, is :