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4. Given the projections and projectors of three angular points of a parallelogram, to find those of the fourth.
5. A circle whose radius is y is projected on a plane whose trace is given, and this projection again projected on a second plane, which is to be determined, when it again becomes a circle whose diameter is y; find the inclination of the line in which the third plane intersects the first, to the second plane.
6. Given three points, D, E, F, in reference to the plane of projection, through which three sides of a triangle ABC are to pass, and given, likewise, the inclinations of the three sides of the triangle to the plane of projection ; to construct the triangle, its projection, and the inclination of its plane to the plane of projection.
DESCRIPTIVE GEOMETRY. In the investigations respecting figures in space, the main, or indeed the only, real object kept in view has been to deduce the geometrical properties of the lines and planes formed according to the hypotheses of the propositions. For this purpose the method of proceeding which we have employed is not only the most obvious and natural, but the most simple in conception and the most easy of development; and were the deduction of speculative truths the only purpose of our researches, there would be no necessity for the adoption of any other method, at least as far as this class of figures was concerned. However, as in plane geometry, so also in that of space, the ultimate object at which we aim, in reference to social life, is of a practical character; and it hence becomes necessary to accommodate our researches to meet the social wants created by the arts, whether of the painter, the architect, the engineer, or the artisan.
It requires but little consideration to convince us that beyond the deduction of speculative truths the method employed by Euclid with respect to the geometry of space, is unsuited to this purpose: for it implies the construction of lines and planes, and the determination of points which cannot be practically fulfilled. Neither, indeed, is it proposed to actually fulfil these conditions by the following methods; but to substitute for the proposed operations certain others, by which the same constructive results are obtained as would have been obtained by the theoretical constructions of the old geometry could they have been actually performed. Moreover, to render this object really attainable, it is quite clear that the entire series of operations to be actually performed inust be performable upon a single plane, represented by the paper upon which the drawing is made. That this can be accomplished will presently be seen.
One mode of accomplishing this has, indeed, been already shown, under the head of Orthographic Projection. This is, in reality, the foundation of all methods, however different in appearance they may be in their ultimate development and practice. Besides the perspective projection, there are two methods of applying the orthographic projection effectively in practice: the Descriptive Geometry and the Method of Contours. A still further subordinate mode of application has of late years been brought into use in this country (and in this country only) under the title of Isometric Projection, which is found to be very serviceable in architectural and engineering drawings. Each of these subjects will be here taken in succession, so as to render each auxiliary to the study of the succeeding ones.
DEFINITIONS. Let us conceive any two planes which intersect each other to be given in a fixed position, and making any dihedral angle with each other; then all other figures in space are said to be given when the positions of their defining parts with respect to these two planes are given.
Dec. 1. These planes are called coordinate planes, and their intersection the axis.
Der. 2. If a plane be given with respect to the coordinate planes, it will cut them in two lines (generally) AS, SB, which are called the truces of the plane, upon the planes ZOX, YOX.
Now when the traces are given, the plane itself becomes fixed; for no other plane than that one can pass at the same time through both the traces; and the two traces are essential to the definition of the plane since one alone might be common to innumerable planes, and could not therefore express one individual of the series.
DEF. 3. Two planes intersect in a straight line; and hence if the two planes be given, the straight line will also be given. Let, then, the two planes be given by their traces BS, SA and B'S', S'A': then the intersections of the traces D and C will give the points in which the coordinate planes are cut by the line itself. These points are called the traces of the line; and they define its position completely, since no other line could pass through those two points.
DEF. 4. A point not in either plane cannot be directly exhibited: but we may conceive it given as the common section of three planes, whose traces are given; or as the intersection of two lines whose traces are given. This latter form for the point is analogous to the definition of a line by means of two planes. We shall presently consider a modification of this.
DEF. 5. Instead of taking the angle of ordination (the dihedral angle of the coordinate planes) as any whatever, most of the constructions and the whole of the reasonings are much simplified by taking it
rectangular : and this plan is universally adopted in the processes of Descriptive Geometry.
The plane XY is taken horizontal, and XZ vertical; and are respectively called the horizontal plane, and the vertical plane; and the section OX is called the axis, or the ground line.
DEF. 6. In some cases it is advantageous to take a plane perpendicular to the axis, viz., YZ instead of one or other of the planes XY, XZ. This, however, will be noticed hereafter; and it is only necessary to remark, that no additional information is afforded of the position of the plane by the introduction of the plane ZY into the figure; since it is already given by means of the two traces AS, BS; and it is clear that two traces upon any pair of planes gives the trace upon the third.
The three planes are very often called by a technical names taken from the arts : XY, V the plan; XZ, the elevation ; and ZY the Y section (or profile): and as they are of perpetual occurrence in the application of Descriptive Geometry, we shall employ them from the outset; and when it is not otherwise expressed, we shall consider the two planes we use as the horizontal and vertical planes, that is, the plan and elevation.
The primary principle in Descriptive Geometry is, to perform all the operations of solution upon these two planes, whilst every operation upon each shall be successive to an operation upon one or other plane, and in no case require simultaneous use of both. At the same time, for the mere purpose of demonstration, we may make any conceivable hypothetical construction though not upon these planes, as in the geometry of Euclid's 11th Book. That the data, as far as planes, lines, and points are concerned, may be exhibited in this way we have already seen; and upon these planes also must the quæsita be finally exhibited as well as found. When other figures than these are concerned, the principle is the same; but this will be better understood when we have completed the present section of our course.
If these conditions can be fulfilled, it can evidently make no real difference in the two parts of the drawing (the plan and elevation) in what position they stand with respect to each other; and hence that one part may be removed to any other position that may be convenient in a practical point of view. One of the most obvious changes of place, and that which is found to be practically most convenient, is to cause one of the planes to revolve about the axis or ground line till it coincides with the extension of the plane of the plan beyond the ground line. This is the method universally adopted.
We might, however, have revolved the plan round the ground line till it coincided with the downward extension of the elevation ; and no difference of general appearance or relation would exist in the figure in one method from that in the other. In one case we should have both upon the same horizontal plane, and in the other both upon the same vertical plane.
This is best illustrated by a model where the relative positions of
the parts are better seen than in a mere pictorial representation.
In this motion every point in XZ describes the quadrant of a circle ; and all the figures traced upon that plane retain their original magnitudes and relation of their several parts to one another. In the figure, SB' represents SB the vertical trace; and it is easy to see that however it may change its position with respect to the other trace SA, it does not change with respect to the axis OX; and hence, so long as the relation of SB to SA is independent of the angle formed by them in space, no constructive change results from this supposed revolution.
DEF. 7. The following terms are used in reference to this revolution of the coordinate planes :- .
(1) To plan the elevation, is to bring the elevation into continuity with the plan;
(2) To elevate the plan, is to bring the plan into continuity with the elevation;
(3) To plan the section, is to bring the section by revolution round OY, into continuity with the plan. Def. 8. We shall designate the geometry of Euclid by the term Theoretical Geometry, to distinguish it from the Descriptive Geometry of which we here treat.
Def. 9. The model, or the representation of the figure as it exists, and as we contemplate it in Theoretical Geometry, we shall call the eidograph, or likeness; and the drawing which we execute as its substitute in Descriptive Geometry, from the one part being placed at right angles to its natural position, the orthograph.
DEF. 10. By the regions of space, we mean those portions of space which are separated by the coordinate planes from each other, and of which these planes are the limitations; these planes being indefinitely continued in their respective directions.
The first region is that above the H and before the V (H and V denoting the horizontal and vertical planes); the second, above H and behind V; the third, below H and behind V; y and the fourth, below H and before V. The regions being in succession from the spectator in revolving round the axis.
One or two other definitions will be given further on, when they will be better comprehended than at present.
In Descriptive, as in Theoretical Geometry, the properties of the eidographic figure, as well as of the orthographic, must be investigated ; for the eidographic properties are always the authorities for orthographic practice. This, of course, will often require us to give the figures of the problem in both kinds; and it will, for the most part, be found that those properties are not only most directly but most easily obtained from the eidograph.
THE TRACES OF PLANES. It is important to get a clear notion of the positions of the traces for particular positions of the planes; and conversely, of the planes themselves from the positions of the traces. For this purpose we give, attached to a verbal description, the figures both in eidograph and orthograph, of each peculiarity of case.
We shall minutely describe them in reference to the first region of space; and then only give the general representation of the cases for the other three regions. VERBAL DESCRIPTION.