TWISTED CUBICS § rot. If two pencils with centres Si and S2 are made projective, then to a See also:

• RAY (Lat. raia)
• RAY (or WRAY, as he wrote his name till 1670), JOHN (1628-1705)

It follows that all lines in which corresponding planes in twoprojective pencils meet form a congruence. We shall see this congruence consists of all lines which cut a twisted cubic twice, or of all secants to a twisted cubic. § See also:

• IO2

From this follows that the points in which corresponding rays meet See also:

• LIE, JONAS LAURITZ EDEMIL (1833—1908)
• LIE, MARIUS SOPHUS (1842–1899)

Of such surfaces an infinite number exists. Every ray through Si or S2 which is not a secant determines one of them. If, however, the rays a1 and a2 are secants See also:

• MEETING (from " to meet," to come together, assemble, 0. Eng. metals ; cf. Du. moeten, Swed. mota, Goth. gamotjan, &c., derivatives of the Teut. word for a meeting, seen in O. Eng. Wit, moot, an assembly of the people; cf. witanagemot)

If we make these two pencils having S and S' as centres projective by taking four rays on the one cone as corresponding to the four rays on the other which meet the first on the cubic, the See also:

• CORRESPONDENCE
• CORRESPONDENCE (from med. scholastic Lat. correspondentia, corrcspondere, compounded of Lat. cum, with, and respond ere, to answer; cf. Fr. correspondance)

It is therefore often called See also:

• DRAWING

If we remember that a line is perpendicular to a plane that perpendicular to every line in the plane if only it is perpendicular to any two intersecting lines in the plane, we see that the axis which is perpendicular both to AA, and to AA2 is also perpendicular to A,Ao and to A2Ao because these four lines are all in the same plane. Hence, if the plane 'r2 be turned about the axis till it coincides with the plane then A2Ao will be the continuation of A,Ao. This position of the planes is represented in fig. 38, in which the line A,A2 is perpendicular to the axis x. Conversely any two points A1, As in, a line perpendicular to the axis will be the projections of some point in space when the plane nr2 is turned about the axis till it is perpendicular to the plane ii,, because in this position the two perpendiculars to the planes 2r, and ire through the points A, and A, will be in a plane and therefore meet at some point A. See also:

• REPRESENTATION

We suppose that the plane a2 is turned as indicated in fig. 37, so that the point P comes to Q and ,$,.s R to S, then the quadrant in which the point A lies is called the first, and we say g_-B2 that in the first quadrant a point lies above As- -.,A the horizontal and in front of the vertical II t plane. Now we go See also:

• ROUND (O. Fr. rond, Lat. rotundus, the Fr. is the source also of Du. rond; Ger., Swed., Dan. and Nor. rond)

If a point lies in the vertical plane, its plan lies in the axis and the elevation coincides with the point itself. If a point lies in the axis, both its plan and elevation lie in the axis and coincide with it. Of each of these propositions, which will easily be seen to be true, the converse holds also. § 3. Representation of a Plane.—As we are thus enabled to represent points in a plane, we can represent any finite figure by representing its See also:

• SEPARATE

Thus a plane parallel to the horizontal plane of the plan has only one finite trace, viz. that with the plane of elevation. If the plane passes through the axis both its traces coincide with the axis. This is the only case in which the representation of the plane by its two traces fails. A third plane of projection is therefore introduced, which is best taken perpendicular to the other two. We call it simply the third plane and denote it by in. As it is perpendicular to in, it may be taken as the plane of elevation, its line of intersection y with a, being the axis, and be turned down to coincide with 2r,. This is represented in fig. 40. OC is the axis xy whilst OA and OB are the traces of the third plane. They lie in one line y. The plane x is rabatted about y to the horizontal plane. A plane a through the axis xy will then show in it a trace a3.

In fig. 4o the lines OC and OP will thus be the traces of a plane through the axis xy, which makes an See also:

• ANGLE (from the Lat. angulus, a corner, a diminutive, of which the primitive form, angus, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Gr. ayKOS, a bend; both connected with the Aryan root ank-, to

These may be any two lines, for, bringing the planes 7r1, Ti into their original position, the planes through these lines perpendicular to TI and Ti respectively will intersect in some line a which has a1, a2 as its projections. Secondly. —A line a is represented by its traces—that is, by the points in which it cuts the two planes T1, Ti. Any two points may be taken as the traces of a line in space, for it is determined when the planes are in their original position as the line joining the two traces. This representation becomes undetermined if the two traces coincide in the axis. In this case we again use a third plane, or else the pro- jections of the line. The fact that there are different methods of representing points and planes, and hence two methods of representing lines, suggests the principle of duality (See also:

• SECTION
• SECTION (Lat. sectio, cutting, secare, to cut)

Before we do this, however, we shall explain the notation used; for it is of See also:

• GREAT

For example, the traces of a line are two points in the line whose projections are known or at all events easily found. They are the traces themselves and the feet of the perpendiculars from them to the axis. Hence if a' a' (fig. 41) are the traces of a line a, and if the per- pendiculars from them cut the axis in P and Q respectively, then the line a'Q will be the horizontal and a'P the vertical projection of the line. Conversely, if the projections a1, a2 of a line are given, and if these cut the axis in Q and P respectively, then the perpendiculars Pa' and Qa' to the axis ` drawn through these points cut the projections al and a2 in the traces a' and a'. To find the line of intersection of two planes, we observe that this line lies in both planes; its traces must therefore he in the traces of both. Hence the points where the horizontal traces of the given planes meet will be the horizontal, and the point where the vertical traces meet the vertical trace of the line required. § 7. To decide whether a point A, given by its projections, lies in a plane a, given by its traces, we draw a line p by oining A to some point in the plane a and determine its traces. If these lie in the [DESCRIPTIVE traces of the plane, then the line, and therefore the point A, lies in the plane; otherwise not. This is conveniently done by joining AI to some point p' in the trace a'; this gives pi; and the point where the perpendicular from p' to the axis cuts the latter we join to A2; this gives p2. If the vertical trace of this line lies in the vertical trace of the plane, then, and then only, does the line p, and with it the point A, lie in the plane a.

§ 8. Parallel planes have parallel traces, because parallel planes are cut by any plane, hence also by arj and by Rr2, in parallel lines. Parallel lines have parallel projections, because points at infinity are projected to infinity. If a line is parallel to a plane, then lines through the traces of the line and parallel to the traces of the plane must meet on the axis, because these lines are the traces of a plane parallel to the given plane. § 9. To draw a plane through two intersecting lines or through two parallel lines, we determine the traces of the lines; the lines joining their horizontal and vertical traces respectively will be the horizontal and vertical traces of the plane. They will meet, at a finite point or at infinity, on the axis if the lines do intersect. To draw a plane through a line and a point without the line, we join the given point to any point in the line and determine the plane through this and the given line. To draw a plane through three points which are not in a line, we draw two of the lines which each join two of the given points and draw the plane through them. If the traces of all three lines AB, BC, CA be found, these must lie in two lines which meet on the axis. § to. We have in the last example got more points, or can easily get more points, than are necessary for the determination of the figure required—in this case the traces of the plane.

This will happen in a great many constructions and is of considerable importance. It may happen that some of the points or lines obtained are not convenient in the actual construction. The horizontal traces of the lines AB and AC may, for instance, fall very near together, in which case the line joining them is not well defined. Or, one or both of them may fall beyond the drawing paper, so that they are practically non-existent for the construction. In this ease the traces of the line BC may be used. Or, if the vertical traces of AB and AC are both in convenient position, so that the vertical trace of the required plane is found and one of the horizontal traces is got, then we may join the latter to the point where the vertical trace cuts the axis. The draughtsman must remember that the lines which he draws are not mathematical lines without thickness, and therefore every drawing is affected by some errors. It is therefore very desirable to be able constantly to check the latter. Such checks always See also:

• PRESENT

Let AI A2 (fig. 42) be the given point, a' a' the given plane, a line 11 through Al, parallel to a' and a horizontal line l2 through A2 will be the projections of a line 1 through A parallel /R" to the plane, because the horizontal plane through this line will cut the plane a in a line c which has its horizontal projection ci parallel to a'. § 12. We now come to the metrical properties of figures. A line is perpendicular to a plane if the projections of the line are per- pendicular to the traces of the plane. We prove it for the horizontal projection. If a line p is perpendicular to a plane a, every plane through p is perpendicular to a; hence also the vertical plane which projects the line p to PI. As this plane is perpendicular both to the horizontal plane and to the plane a, it is also perpendicular to their intersection—that is, to the horizontal trace of a. It follows that every line in this projecting plane, therefore also pi, the plan of p, is perpendicular to the horizontal trace of a. To draw a plane through a given point A perpendicular to a given line p, we first draw through some point 0 in the axis lines y , y' perpendicular respectively to the projections p1 and ¢2 of the given line. These will be the traces of a plane y which is perpendicular to the given line. We next draw through the given point A a plane parallel to the plane y; this will be the plane required.

709 Other metrical properties depend on the determination of the real point two lines perpendicular to the two planes and determine the See also:

• SIZE

Points treatment of the two cases, we shall consider only the case of rabatt- in this line coincide with their elevations. Hence it is given in See also:

• ING

43) and See also:

• A1A