The

ordinates amongst their projections, that the projections of a plane curve of a given algebraical order are curves of the same algebraical order. The projections of a circle are ellipses; the projections of a parabola of a given order are parabolas of the same order. projections of a straight tangent to a plane curve are straight tangents to the projections of that curve. The projections of a point of contrary flexure in a plane curve are points of contrary flexure in its projections.

12. Drawings of a Machine.—A third plane of projection, perpendicular to the first two, is often employed, not as being mathematically necessary, but as being more convenient for the representation of certain lines. Thus, for example, the drawings of a machine usually consist of three projections on three planes at right angles to each other; one horizontal (the plan), and the other two vertical (the elevations). Any two of those projections are mathematically sufficient to show the whole dimensions and figure of the machine; and from any two the third can be constructed; but it is convenient, for purposes of measurement, calculation, and construction, to have the whole three projections.

In the application of the rules about to be stated in the sequel of this Section, the two planes of projection may be held to represent any two of the three views of a machine; and the axis of projection will then have the directions stated in the following table:

Views Represented by the Planes of

Projection.

Direction of the Axis
of Projection.

Longitudinal Elevation and Plan,............Longitudinal.
Longitudinal and Transverse Elevations,.... Vertical.
Plan and Transverse Elevation,..... ...Transverse.

Projections of figures upon planes oblique to the principal planes of projection may be used for special purposes.

SECTION II.-Traces of Lines and Surfaces.

13. By a Trace is meant the intersection of a line with a surface, or of one surface with another. The trace of a line upon a surface is a point; the trace of one surface upon another is a line.

In descriptive geometry the term traces is specially employed, when not otherwise specified, to denote the intersections of a line or surface with the planes of projection.

14. Traces of a Straight Linc.—The position of a straight line is completely determined when its traces are known. For example, the straight line A C, in fig. 2, has its position completely deter mined by its traces, A and C, being the points where it cuts the

two planes of projection. The rabatment of the trace C is represented by c.

A straight line parallel to one of the planes of projection has

known. For example, the plane A B C, in fig. 2, has its position completely determined by its traces, B A and B C.

A plane perpendicular to one of the planes of projection has its trace on the other plane of projection perpendicular to the axis of projection. A plane perpendicular to both planes of projection has for its traces two lines perpendicular to the axis. Thus, in fig. 1, page 3, the traces of the plane A B C D are D C and D B, both perpendicular to X X

A plane parallel to one of the planes of projection has a trace on the other plane of projection only, being a straight line parallel to X X.

If a plane traverses a straight line, the traces of the plane traverse the traces of the line.

SECTION III-Rules Relating to Straight Lines.

16. General Explanations.-In each of the figures illustrating the following rules the axis of projection is represented by X X; and in general the part of the figure above that line represents the rabatment of the vertical plane of projection, and the part below, the horizontal plane of projection. The projections of points on the horizontal plane are in general marked with capital letters, and the projections on the vertical plane with small letters.

17. Given (in fig. 3), the Traces, A, b, of a Straight Line, to Draw its Projections.-From A and 6 let fall A a and b B perpendicular to X X. Then a will be the vertical projection of the

trace A, and B the horizontal projection of the trace b. Join a b, A B; these will be the projections required.

(It may here be remarked, that a A and a b are the traces of a plane traversing the given line, and perpendicular to the vertical plane of projection, and that B A and B b are the traces of a plane traversing the given line, and perpendicular to the horizontal plane of projection.)

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18. Given (in fig. 3), the Projections, A B, a b, of a Straight Line, to Find its Traces.-From a and B, where the given projections meet the axis, draw a A and Bb perpendicular to X X, cutting the given projections in A and b respectively. These points will be the required traces.

projections (as b) draw d be parallel to X X; through the other end, a, of the same projection, draw a d perpendicular to XX, cutting dbe in d; make de the other projection, A B; join a e; this

=

will be the length required.

The same operation may be performed on the other plane of projection.

20. Given (in fig. 4), the Projections, A, ɑ, of a Point, and the Projections, A B, a b, of a Straight

Line through that Point, to Lay off a given Distance from the Point along the Line. In any convenient position, draw a straight line, B 6, perpendicular to XX, meeting the projections of the given straight line in two points, B, b, which are the projections of one point; then perform the construction described in Article 19, so as to find a e. From the point a, in the line a e, lay off the given distance, a f. Through ƒ draw fh parallel to X X, cutting ab in g; ag will be one of the projections of the given distance. Then draw g G perpen

b

dicular to X X, cutting A B in G; A G will be the other projection of the given distance.

Another method of finding G is to lay off A G

=

hf.

21. Given (in fig. 4), the Projections, a b, A B, of a Straight Line, to Find the Angle which it makes with One of the Planes of Pro

jection (for example, the horizontal plane).-Perform the construction described in Article 19; then de a is the angle made by the given line with the horizontal plane. The same construction performed in the horizontal plane of projection will give the angle made by the given line with the vertical plane of projection.

22. Given (in fig. 5), the Projections, a b and A B, a c and A C, of a Pair of Straight Lines which Intersect each other in the Point whose Projections are ɑ, A, to find the Angle between those Lines.---In either of the planes of projection (for example, the vertical

plane) find the points, d, e, where the projections of the given line cut the axis X X; these will be also the vertical projections of the horizontal traces of the lines. Through e and d draw e E, d D, perpendicular to X X, cutting A C and A B in E and D respectively; these points will be the horizontal traces of the lines. Join DE (which will be the horizontal trace of the plane containing the lines), and on it let fall the perpendicular F A. Join A a (which of course is perpendicular to X X); let it cut X X in G. Make Gf=A F, and join af. In FA produced, take FH = a f; join H E, H D; EHD will be the angle required.

REMARK. The triangle E H D is the rabatment upon the horizontal plane of the triangle whose projections are E AD and e a d.

22 A. Given (in fig. 5), the Projections, a b and A B, of a Straight Line, and One Trace (say D E) of a Plane Traversing that Line, to Find the Projections of a Straight Line which shall, at a given Point, ɑ, Å, make a given Angle in the given Plane with the given Straight Line.-Join A a, which will be perpendicular to X X. On DE let fall the perpendicular A F. In X X take Gƒ= A F; join aj. In F A produced take F H = af. Join HD; and draw HE, making DHE the given angle, and cutting D E in E. From E let fall Ee perpendicular to X X; join A E, a e; these will be the projections of the line required.

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SECTION IV.-Rules Relating to Planes.

23. Given, the Projections of Three Points, to draw the Traces of a Plane Passing through them.—-Draw straight lines from one of the points to the two others; find, by Article 18, the traces of those straight lines; through those traces, on the two planes of projection respectively, draw two straight lines; these will be the traces required.

23 A. Given, the Projections of Two Points and of a Straight Line, 10 Draw the Traces of a Plane Traversing the Points and Parallel to the Line.-Through the projections of either of the given points draw straight lines parallel respectively to the corresponding projections of the given line; these will be the projections of a straight line through the given point, parallel to the given straight line; then, by Article 23, find the traces of a plane traversing the new straight line and the other given point.

24 Given (in fig. 6), the Traces of a Plane, B A, BC, to Find the Angle which it makes with one of the Planes of Projection (for example, the vertical plane). -From any convenient point, 4, in the horizontal trace let fall A D perpendicular to X X. From D let fall De perpendicular to B C. In D B lay off Dj De. Join ƒ A (this will represent the perpendicular distance from BC of the point x whose projections are D and A). AfD will be the angle required.

=

25. Given (in fig. 7), the Traces of a Plane, B A, B C, to Find the Angle which it makes with the Axis of Projection, X X.In either of the two traces (for

B

Fig. 6.

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