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Class 3. Parallel to both planes, or to the axis.
The only case of this class is as figured ; and as the line meets no point in either plane, its traces cannot be marked. Its projecting planes are parallel also to the coordinate planes ; and since they form with the axis, the sections of a plane by parallel planes, these projections are also parallel to the axis. Their distances from the axis on each plane are the same as the distances of the line from the other plane.
CLASS 4. Perpendicular to one plane, and therefore parallel to the other.
In this case the projection on the plane to which the line is parallel will be perpendicular to the axis ; and as the projection on the other plane is indefinite in position (all planes through the line being perpendicular to the plane), any line whatever through the point of section will define the trace, when taken in conjunction with AC.
It is, however, not necessary to trace the line AD at all, if we terminate CA at A, the point in which it meets the plane. The two orthographs will hence be simply as marked in the margin. This, because the
Parallel to Elevation.
Parallel to Plan.
simplest method, is that most commonly adopted in practice.
THE ORTHOGRAPH REPRESENTATION OF A POINT. A point may be considered in three different ways: as the intersection of three planes ; as the intersection of a plane and line ; or as the intersection of two lines which are in the same plane. For special purposes, one of these views may be preferable to the others; but as appertaining to the general theory, they all coalesce under the second form in a very convenient state for practical use.
As in the line, so with regard to the point, we may define it by means of two of the intersecting figures that pass through the point; and we are at liberty to choose those positions which will most simply express those figures in reference to the coordinate planes. We shall, however, better perceive the relative values of the different special modes of representation by examining their results in succession. It may be advantageous to the student, however, to defer the consideration of the more complex expressions of a point in orthograph till he has performed a few of the preliminary problems; and we shall therefore merely lay before him in this place the mode of definition which has been found most convenient, and which is consequently most usually employed.
The method is this :
Take three planes, two of them parallel to the coordinate planes, and the third perpendicular to them both. The projections of the line formed by the first two are parallel to the axis, and the intersections of the third with both planes will be perpendicular to the axis and pass through the same point.
In the first of these figures suppose P to be the point; then abı, azbe are the projections of a line parallel to the axis, which passes through the point to be represented : and piQ, Qp, are the traces of a plane perpendicular to them both, which also passes through P. Then it is quite clear that p, is the projection of the point P of the line AB upon the plan ; and that p, is that of the same point upon the elevation. Moreover, since a B and pop, are planes, both perpendicular to the elevation, it follows that Ppa, their intersection, is perpendicular to the eleration. In like manner Pp, is perpendicular to the plan. Also, it is clear that pipe is a rectangle, or Pp: = p.Q and Ppı = p.Q; whilst Pp, Pp, are the distances of the point P from the plan and elevation respectively.
We pass from this to the first immediate orthograph; in which we see that the point is defined by two horizontal lines, one on each coordinate plane, and by a line perpendicular to them both. For in revolving to the plan the elevation p.Q is always at right angles to OX, and hence in orthograph continuous of p.Q.
But as these horizontal lines a,b, agbe serve no other purpose in definition than to mark the distance of the point from each plane, we may, knowing those distances, define the point as in the second orthograph by terminating the perpendicular through Q at those distances respectively. This is the most simple mode of all.
The points po, p, are called the projections of the point P; and the lines Pp, Pp, in the eidograph are the projecting lines. They are defined as the perpendiculars from the point to the coordinate planes in other works on this subject.
This subject is one of scarce less importance in Descriptive Geometry than it is in algebra and its applications; and much attention has been given to it by the French writers on the subject, and more especially with respect to graphic exhibition. We shall follow them strictly in this respect, but in the literal notation we shall employ a method which is more symmetrical at the same time that it is more discriminative than theirs.
1. Graphic Notation.
1. The data and quæsita of a problem are always marked in continued line, when they are upon the visible parts of the plan and elevation ; and when behind the elevation or below the plan, so as to be concealed by these planes, they are marked in round dots; thus,
and .... 2. Lines used in construction, as the intersections of the projecting planes with the coordinate planes, etc. (except they belong to the preceding class), are marked in traits, whether visible or hidden, by the coordinate planes ; thus
3. When of the constructive lines some are very important in relation to the problem, they are marked in a mixture of points and traits ; thus,
The number of interposed dots implying a greater degree of relative importance.
4. To distinguish the traces of a plane from the projections of a line, in all the cases, a slight cross trait is placed at or near the end of a trace ; thus,
-.-.|, or _..-..]
In actual drawings much simplicity would be effected by denoting the traces, projections, and auxiliary lines by different colours. No fixed rule for this purpose has, however, been adopted.
2. Literal Notation. 1. In the eidograph, a point is represented by a Roman capital, and its projection on the plan and elevation respectively by the small letter with a subscribed numeral 1, 2. Thus a point A in space
referred by projection to the plan and elevation will be denoted by a,, Az. If we represent it on the profile, we shall denote it by az.
2. A plane, being defined by its traces, will be denoted by Roman capitals, where it cuts the axis, the other literal denomination on the plan being marked with the same letter having 1 subscribed, and on the elevation by 2. Thus a plane whose traces meet the axis will be expressed by S,SS,, SiS being on the plan and SS, on the elevation.
3. The traces of a line on the coordinate planes is represented by the Greek letters a, B, Y, d, etc., with subscribed numerals. Thus the line aß will signify generally a line which meets the planes in a and B; and as a is on the plan, it will be marked an; and ß on the elevation, it will be marked Bc. The projections of these points in the axis will therefore, in conformity with the principle, bed, and B.
4. The projecting planes of a line will bear that name which has two of its points of the saine name with numerals subscribed. Thus Baba, is called the plane B, and «,««ß, is called the plane a.
Or, again, by the letter at which the projecting plane cuts the axis, which is more simple, and is that to be used in the work.
PRELIMINARY PROPOSITION S. The following are the problems referred to at p. 140, as being required in Descriptive Geometry.
The postulate assumed here, besides the postulates and propositions of plane geometry, is :
That through three points not in one line, or through one point and a line, or through two lines which are parallel, a plane may be drawn or constituted.
In any plane so drawn, of course all the operations of plane geometry may be performed.
PROPOSITION I. To draw a perpendicular to a given plane from a given point. 1. Let the given point A be without the given plane MN.
Through A draw any plane QB cutting MN in BC; and in the plane QB draw the line AD perpendicular to BC; in the plane MN, the line DE also perpendicular to BC; and in the plane ADE the line AE perpendicular to DE. This is the perpendicular required.*
For since BC is perpendicular both to AD and DE, it is perpendicular to the plane ADE (Pls. II. 2); and hence the plane MN through BC is perpendicular to the plane ADE (Pls. 11. 16).
Wherefore AE is drawn in one of two perpendicular planes ADE and MN, at right angles to the common section DE; and is, therefore, perpendicular to MN (Pís. Def. 4.).
2. Let the given point A be in the given plane MN.
Through A draw any line BC in MN, and through BC any two planes PC, QC; in these planes draw the lines AD, AE perpendicular to BC; P through AD, AE draw a plane cutting MN in GH; and in this plane M draw AK perpendicular to GH. This will be the perpendicular required.
For, as before, AD, AE being perpendicular to BC, the planes DAE, MN are perpendicular; and AK in one of them DAE being perpendicular to the common section GH, is perpendicular to the other.
SCHOLIUM. The mode usually directed for the construction of this case is :-take any other point without MN, and by the preceding case draw a perpendicular to MN; and then through Å draw AK parallel to this perpendicular.
This seems shorter: its brevity, however, is only in the statement, not in the work. Either method may be used as shall be most convenient in each particular case of its application.
PROPOSITION II. Through a given point to draw a plane parallel to a given plane. Let A be the given point, and MN
F the given plane.
Draw any line AB to meet MN in B; and in MN any two lines BC, BD; in the plane ABC draw AE parallel to BC, and in ABD draw AF parallel to BD. Then the plane through AE, AF is that required.
For the two lines CB, BD which meet in B being parallel to the two EA, AF which meet in A, they are in parallel planes (Pls. 1. 7).
* If it should happen that ADE is a right angle, the construction will terminate with drawing AD; but this can only happen accidentally.