27. A sphere 1.5 inches in diameter rests on the horizontal plane and its centre is 1.5 inches in front of the vertical plane of projection. A point on the horizontal plane, 3.75 inches in front of the vertical plane of projection and 2.25 inches from the plan of the centre of the sphere is the vertex of a cone which envelops the sphere. Draw the vertical trace of the cone, also the plan and elevation of the circle of contact between the cone and sphere.

28. A circular section of an oblique cylinder is 2 inches in diameter and the axis of the cylinder is inclined at 40° to this section. Draw the true form of the section of the cylinder by a plane perpendicular to its axis.

29. An oblique cylinder is given in Fig. 534. Draw the vertical trace of the

cylinder, that is, determine the section of the cylinder by the vertical plane of projection. Draw also the plan and elevation of a sub-contrary section.

30. An oblique cone is given in Fig. 535. Draw the plan and also the true form of the parabolic section of this cone by a plane whose vertical trace is PQ and which is perpendicular to the vertical plane of projection. Draw also the elevation of the hyperbolic section by the vertical plane whose horizontal trace

is RS.

CHAPTER XIX

SPECIAL PROJECTIONS OF PLANE FIGURES AND SOLIDS

232. Projections of a Figure whose Plane and a Line in it have Given Inclinations.-First determine the traces of a plane containing the figure. The working of the problem is much simplified by first assuming this plane perpendicular to the vertical plane of projection. In this position the elevation of the figure will be a straight line coinciding with the vertical trace of the plane. Afterwards any other elevation may be obtained by Art. 157, p. 191.

Draw L'M (Fig. 537) making with XY an angle equal to the given inclination of the plane of the figure. Draw MN the horizontal trace at right angles to XY. In the

plane L'MN place a line RS having the inclination a of the line of the figure given (Art. 174, p. 214).

Now imagine the plane L'MN, with the line RS upon it, to rotate about its horizontal trace MN until it comes into the horizontal plane. The point S being in MN will remain stationary while the point R will describe an arc of a circle in the vertical plane of projection with M as centre. Hence, if with M as centre and Mr' as radius the arc r'R, be described, meeting XY at R1, R18 will be the position of the line RS when that line is brought into the horizontal plane.

FIG. 537.

Mark off on R1s a length A,B, equal to the length of that line of the figure whose inclination a is given, and on A,B, construct the given figure. Note that the line AB of the figure is not necessarily one of its sides, but may be any line whatever in the plane of the figure and occupying a definite position in relation to the figure. Next imagine the figure thus drawn on the horizontal plane to rotate about MN until its inclination is 0. All points in the figure will describe arcs of circles whose plans will be straight lines perpendicular to MN and whose elevations will be arcs of circles having their centres at M and having radii equal to the distances of the points

from MN. Thus the point C, will describe an arc of a circle whose plan is C,c perpendicular to MN and whose elevation is the arc c'e' of a circle whose centre is M. c is of course in a straight line through dat right angles to XY.

As a verification of the construction it should be noticed that if any line, say C,B1, of the figure constructed on the horizontal plane be produced, if necessary, to meet MN it will do so at the point t where the plan be of the same line in its inclined position meets it.

It must be borne in mind that no line of the figure can have a greater inclination than that of its plane, that is, a must not be greater than 0.

233. Projections of a Plane Figure, the Inclinations of two Intersecting Lines in it being given.-Determine by Art. 187, p. 219, the traces of the plane containing the lines whose inclinations are given, taking the horizontal trace at right angles to the ground line. Rotate this plane, with the lines upon it, about its horizontal trace, as in the preceding Article, until it comes into the horizontal plane. On the lines thus brought into the horizontal plane construct the given figure and proceed to determine the plan and elevation of the figure in its inclined position exactly as in the preceding Article.

234. Projections of a Plane Figure having given the Heights of three Points in it.-Note that the difference between the heights of any two points must not exceed the true distance between the points.

Let A, B, and C be the points whose heights are given.

Construct on the horizontal plane a triangle A,B,C, (Fig. 538) equal to the triangle ABC. With centre A, and radius equal to the height of the point A describe the circle EFH. With centre B, and radius equal to the height of the point B describe the circle KLM. With centre C, and radius equal to the height of the point C describe the circle NPQ. Draw HL to touch the circles EFH and KLM, and produce it to meet A,B, produced at R. Also draw EP to touch the circles EFH and NPQ, and produce it to meet AC, produced at S. RS is the horizontal trace of the plane which will contain the points A, B, and C.

Draw XY perpendicular to RS meeting it at O. Draw A,a,' perpendicular to XY to meet it at a'. With centre O and radius Oa, describe an arc of a circle to meet at a' a parallel to XY which is at a distance from XY equal to the height of the point A. Oa' is the vertical trace of the plane which will contain the points A, B, and C.

X

R

E

N

a

A,

H

FIG. 538.

The theory of the above construction for finding the traces of the

T

plane containing the points A, B, and C is similar to that of the construction given in Art. 187, p. 219, which should be again referred to. It is evident that the angles A,RH and A,SE are the inclinations of the straight lines AB and AC respectively.

The plan abc and the elevation a'b'c' of the triangle ABC are next determined as shown, the construction being the same as in Art. 232.

If ABC is not the complete figure given, a figure equal to it must be built up on the triangle A,B,C,. Then the projections of the remainder of the figure are obtained in the same way as the projections of the part ABC.

235. Projections of a Plane Figure when it has been turned about a Horizontal Line in it till the Plan of an Opposite Angle is equal to a given Angle.-Let ABCD denote the given figure, AC the horizontal line about which it is turned, and B the angle whose plan is to be equal to a given angle.

Construct the figure aB,cD, (Fig. 539) equal to the given figure ABCD. On ac describe a segment of a circle (Art. 14, p. 18) to contain an angle equal to that into which

the angle B is to be projected.

As the figure revolves about AC the point B will describe an arc of a circle whose plan is a straight line through B1 perpendicular to ac. Let this perpendicular meet the arc of the segment of a circle which has been described on ac at the point b. Join ab and be. Then abc is the plan of the angle B when the figure has been turned as required.

Draw XY at right angles to ac, meeting ac produced at a'. Draw Bb,' at right angles to XY to meet it at b'. With centre a' and radius a'b' describe the arc b'b', and through b draw a perpendicular to XY to meet this arc at b. a'b' is the vertical trace and aa' is the horizontal trace of the plane containing the figure ABCD when it occupies the position required.

FIG. 539.

B,

From D, draw D,d,' perpendicular to XY to meet it at d''. With centre a' and radius a'd' describe the arc d'd' to meet b'a' produced at d'. Draw d'd perpendicular to XY to meet a line through D, parallel to XY at d. Join ad and cd. abcd is the plan and d'b is an elevation of the figure ABCD as required.

236. Projections of a Solid when a Plane Figure on it and a Line in that Figure have given Inclinations.-The plane figure may be a face of the solid or it may be a section of it. Considering the most general case, the first step is to determine the projections of the plane figure as in Art. 232, p. 272. On the rabatment of the plane figure on the horizontal plane the feet of the

perpendiculars from the angular points of the solid on the plane of the figure must next be located. The projections of the feet of these perpendiculars are next found, and the projections of the perpendiculars can then be drawn. The projections of the angular points of the solid are therefore determined.

Two examples are illustrated by Figs. 540 and 541.

In the first example (Fig. 540) a right square prism, side of base 1.5 inches, altitude 1.8 inches, is shown when its base is inclined at 50° and one side of that base is inclined at 20°. The plan abcd and elevation a'b'c'd' of the square base are first determined as in Art. 232, p. 272. In this case the feet of the perpendiculars from the angular points of the solid on the plane of the base are at the angular points of that base. Hence it is now only necessary to draw from a', b', c', and d' perpendiculars to the vertical trace of the plane of the base and make them 1.8 inches long in order to obtain the elevations e', f', g', and h' of the other angular points of the solid. The plans of these perpendiculars are at right angles to the horizontal trace of the plane of the base and are therefore parallel to XY.

In the second example (Fig. 541) a cube of 1.5 inches edge is shown when the plane of two diagonals of the solid is inclined at 60° and one of these diagonals is inclined at 40°. The plane of the two diagonals divides the cube into two right triangular prisms, the triangular ends of which are each equal to one half of a face of the cube. The common face of these two triangular prisms is a rectangle of which two opposite sides are edges of the cube while the other two