TWISTED CUBICS
§ rot. If two pencils with centres Si and S2 are made projective, then to a ray in one corresponds a ray in the other, to a plane a plane, to a flat or axial pencil a projective flat or axial pencil, and so on.
There is a double infinite number of lines in a pencil. We shall see that a single infinite number of lines in one pencil meets its corresponding ray, and that the points of intersection form a curve in space.
Of the double infinite number of planes in the pencils each will meet its corresponding plane. This gives a system of a double infinite number of lines in space. We know (§ 5) that there is a quadruple infinite number of lines in space. From among these we may select those which satisfy one or more given conditions. The systems of lines thus obtained were first systematically investigated and classified by Plucker, in his Geometrie des Ra mes. He uses the following names:
A treble infinite number of lines, that is, all lines which satisfy one condition, are said to form a complex of lines; e.g. all lines cutting a given line, or. all lines touching a surface.
A double infinite number of lines, that is, all lines which satisfy two conditions, or which are common to two complexes, are said to form a congruence of lines; e.g. all lines in a plane, or all lines cutting two curves, or all lines cutting a given curve twice.
A single infinite number of lines, that is, all lines which satisfy three conditions, or which belong to three complexes, form a ruled surface; e.g. one set of lines on a ruled quadric surface, or developable surfaces which are formed by the tangents to a curve.
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.
§ Io2. Let li be the line S1S2 as a line in the pencil Si. To it corresponds a line 12 in S2. At each of the centres two corresponding lines meet. The two axial pencils with li and l2 as axes are projective, and, as their axes meet at S2, the intersections of corresponding planes form a cone of the second order (§ 58), with S2 as centre. If oir1 and 72 be corresponding planes, then their intersection will be a line p2 which passes through S2. Corresponding to it in Si will be a line pi which lies in the plane an, and which therefore meets p2 at some point P. Conversely, if p2 be any line in S2 which meets its corresponding line pi at a point P, then to the plane 12 P2 will correspond the plane lipl, that is, the plane S1S2P. These planes intersect in P2, so that p2 is a line on the quadric cone generated by the axial pencils h and 12. Hence:
All lines in one pencil which meet their corresponding lines in the other form a cone of the second order which has its centre at the centre of the first pencil, and passes through the centre of the second.
From this follows that the points in which corresponding rays meet lie on two cones of the second order which have the ray joining their centres in common, and form therefore, together with the line SIS2 or ll, the intersection of these cones. Any plane cuts each of the cones in a conic. These two conics have necessarily that point in common in which it cuts the line li, and therefore besides either one or three other points. It follows that the curve is of the third order as a plane may cut it in three, but not in more than three, points. Hence:
The locus of points in which corresponding lines on two projective pencils meet is a curve of the third order or a " twisted cubic " k, which passes through the centres of the pencils, and which appears as the intersection of two cones of the second order, which have one line in common.
A line belonging to the congruence determined by the pencils is a secant of the cubic; it has two, or one, or no points in common with this cubic, and is called accordingly a secant proper, a tangent, or a secant improper of the cubic. A secant improper may be considered,
to use the language of coordinate geometry, as a secant with imaginary points of intersection.
§ to3. If al and a2 be any two corresponding lines in the two pencils, then corresponding planes in the axial pencils having al and ¢2 as axes generate a ruled quadric surface. If P be any point on the cubic k, and if p2 be the corresponding rays in Si and S2 which meet at P, then to the plane a1 pi in SI corresponds ¢2 Pz in S2. These therefore meet in a line through P.
This may be stated thus:
Those secants of the cubic which cut a ray al, drawn through the centre Si of one pencil, form a ruled quadric surface which passes through both centres, and which contains the twisted cubic k. 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 meeting at A, then the ruled quadric surface becomes a cone of the second order, having
A as centre. Or all lines of the congruence which pass through a point on the twisted cubic k form a cone of the second order. In other words,
the projection of a twisted cubic from any point in the curve on to any plane is a conic.
If al is not a secant, but made to pass through any point Q in space, the ruled quadric surface determined by al will pass through
Q. There will therefore be one line of the congruence passing through
Q, and only one. For if two such lines pass through Q, then the lines S1Q and S2Q will be corresponding lines; hence Q will be a point on the cubic k, and an infinite number of secants will pass through it. Hence:
Through every point in space not on the twisted cubic one and only one secant to the cubic can be drawn.
§ 104. The fact that all the secants through a point on the cubic form a quadric cone shows that the centres of the projective pencils generating .the cubic are not distinguished from any other points on the cubic. If we take any two points S, S' on the cubic, and draw the secants through each of them, we obtain two quadric cones, which have the line SS' in common, and which intersect besides along the cubic. 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 correspondence is determined. These two pencils will generate a cubic, and the two cones of secants having S and S' as centres will be identical with the above cones, for each has five rays in common with one of the first, viz. the line SS' and the four lines determined for the correspondence; therefore these two cones .intersect in the original cubic. This gives the theorem
On a twisted cubic any two points may be taken as centres of projective pencils which generate the cubic, corresponding planes being those which meet on the same secant.
Of the two projective pencils at S and S' we may keep the first fixed, and move the centre of the other along the curve. The pencils will hereby remain projective, and a plane a in S will be cut by its corresponding plane a always in the same secant a. Whilst S' moves along the curve the plane a' will turn about a, describing an axial pencil.
This branch of geometry is concerned with the methods for representing solids and other figures in three dimensions by drawings in one plane. The most important method is that which was invented by Monge towards the end of the 18th century. It is based on parallel projections to a plane by rays perpendicular to the plane. Such a projection is called orthographic (see PROJECTION, § r8). If the plane is horizontal the projection is called the plan of the figure, and if the plane is vertical the elevation. In Monge's method a figure is represented by its plan and elevation. It is therefore often called drawing in plan and elevation, and sometimes simply orthographic projection.
§ 1. We suppose then that we have two planes, one horizontal, the other vertical, and these we call the planes of plan and of elevation respectively, or the horizontal and the vertical plane, and denote them by the letters and in. Their line of intersection is called the axis, and will be denoted by xy.
If the surface of the drawing paper is taken as the plane of the plan, then the vertical plane will be the plane perpendicular to it through the axis xy. To bring this also into the plane of the drawing paper we turn it about the axis till it coincides with the horizontal plane. This process of turning one plane down till it coincides with another is called rabatting one to the other. Of course there is no necessity to have one of the two planes horizontal, but even when this is not the case it is convenient to retain the above names.
The whole arrangement will be better understood by referring to fig. 37. A point A in space is there projected by the perpendicular
D. R 'D,
AA, and AA2 to the planes 7r, and 'r2 so that and A2 are the horizontal and vertical projections of A.
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.
Representation of Points.—We have thus the following method of representing in a single plane the position of points in space: we take in the plane a line xy as the axis, and then any pair of points Ai, A2 in the plane on a line perpendicular to the axis represent a point A in space. If the line A,A2 cuts the axis at Ao, and if at Ai a perpendicular be erected to the plane, then the point A will be in it at a height A,A=AoA2 above the plane. This gives the position of the point A relative to the plane a,. In the same way, if in a perpendicular to x2 through A2 a point A be taken such that A2A
AoA,, then this will give the point A relative to the plane 7I"2.
§ 2. The two planes in, ir2 in their original position divide space into four parts. These are called the four quadrants. 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 round the axis in the c, 8, ;A, S
sense in which the plane ire is turned and Q b
come in succession to the second, third III Iv
and fourth quadrant. In the second a  C CQ
point lies above the plane of the plan and  D
behind the plane of elevation, and so on. R D, In fig. 39, which represents a side view of
the planes in fig. 37 the quadrants are FIG. 39, marked, and in each a point with its pro
jection is taken. Fig. 38 shows how these are represented when the plane in is turned down. We see that
A point lies in the first quadrant if the plan lies below, the elevation above the axis; in the second if plan and elevation both lie above; in the third if the plan lies above, the elevation below; in the fourth if plan and elevation both lie below the axis.
If a point lies in the horizontal plane, its elevation lies in the axis and the plan coincides with the point itself. 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 separate points. It is, however, not possible to represent a plane in this way, for the projections of its points completely cover the planes a, and a2, and no plane would appear different from any other. But any plane a cuts each of the planes 2r,, ,r2 in a line. These are called the traces of the plane. They cut each other in the axis at the point where the latter cuts the plane a.
A plane is determined by its two traces, which are two lines that meet on the axis, and, conversely, any two lines which meet on the axis determine a plane.
If the plane is parallel to the axis its traces are parallel to the axis. Of these one may be at infinity; then the plane will cut one of the planes of projection at infinity and will be parallel to it. 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 angle POQ with the horizontal plane.
We can also find the trace which any other plane makes with as. In rabatting the plane
irs its trace OB with the plane 'e2 will come to the position OD. Hence a plane 13 having the traces CA and CB will have with the third plane the trace #s, or AD if OD =OB.
B ,'C' B,
Bo Co D„ y
IC2
7o8
It also follows immediately that
If a plane a is perpendicular to the horizontal plane, then every point in it has its horizontal projection in the horizontal trace of the plane,
as all the rays projecting these points lie in the plane itself.
Any plane which is perpendicular to the horizontal plane has its vertical trace perpendicular to the axis.
Any plane which is perpendicular to the vertical plane has its horizontal trace perpendicular to the axis and the vertical projections of all points in the plane lie in this trace.
§ 4. Representation of a Line.—A line is determined either by two points in it or by two planes through it. We get accordingly two representations of it either by projections or by traces.
First.—A line a is represented by its projections al and a2 on the two planes irl and Ti. 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 (section ii., Projective Geometry, § 45). It is worth while to keep this in mind. It is also worth remembering that traces of planes or lines always lie in the planes or lines which they represent. Projections do not as a rule do this excepting when the point or line projected lies in one of the planes of projection.
Having now shown how to represent 'points, planes and lines, we have to state the conditions which must hold in order that these elements may lie one in the other, or else that the figure formed by them may possess certain metrical properties. It will be found that the former are very much simpler than the latter.
Before we do this, however, we shall explain the notation used; for it is of great importance to have a systematic notation. We shall denote points .in space by capitals Pi, B, C; planes in space by Greek letters a, t3, y; lines in space by small letters a, b, c; horizontal projections by suffixes I, like AI, a1; vertical projections by suffixes 2, like A2, a2; traces by single and double dashes a' a', a', a'. Hence PI will be the horizontal projection of a point P in space; a line a will.have the projections a1, a2 and the traces a' and a.; a plane a has the traces a' and a'.
§ 5. If a point lies in a line, the projections of the point lie in the projections of the line.
If a line lies in a plane, the traces of the line lie in the traces of the plane.
These propositions follow at once from the definitions of the projections and of the traces.
If a point lies in two lines its projections must lie in the projections of both. Hence
If two lines, given by their projections, intersect, the intersection of their plans and the intersection of their elevations must lie in a line perpendicular to the axis, because they must be the projections of the point common to the two lines.
Similarly—If two lines given by their traces lie in the same plane or intersect, then the lines joining their horizontal and vertical traces respectively must meet on the axis, because they must be the traces of the plane through them.
§ 6. To find the projections of a line which joins two points A, B given by their projections Al, A2 and B1, B2, we join Ai, B1 and A2, B2; these will be the projections required. 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 nonexistent 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 present themselves when the same result can be obtained by different constructions, or when, as in the above case, some lines must meet on the axis, or if three points must lie in a line. A careful draughtsman will always avail himself of these checks.
§ 11. To draw a plane through a given point parallel to a given
plane a, we draw through the point two lines which are parallel to the plane a, and determine the plane through them; or, as we know that the traces of the required plane are parallel to those of the given one (§ 8), we need only draw one line l through the point parallel to the plane and find one of its traces, say the vertical trace l"; a line through this parallel to the vertical trace of a will be the vertical trace of the required plane 13, and a line parallel to the horizontal trace of a meeting 13" on the axis will be the horizontal
trace 13'.
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 size or shape of a figure. angle between the latter as above.
In general the projection of a figure differs both in size and shape In special cases it is simpler to determine at once the angle between from the figure itself. But figures in a plane parallel to a plane the two planes by taking a plane section perpendicular to the interof projection will be identical with their projections, and will thus section of the two planes and rabatt this. This is especially the be given in their true dimensions. In other cases there is the case if one of the planes is the horizontal or vertical plane of proproblem, constantly recurring, either to find the true shape and jection.
size of a plane figure when plan and elevation are given, or, con Thus in fig. 45 the angle PIQR is the angle which the plane a
versely, to find the latter from the known true shape of the figure makes with the horizontal plane.
itself. To do this, the plane is turned about one of its traces till it § 15. We return to the general case of rabatting a plane a of
is laid down into that plane of projection to which the trace belongs. which the traces a' a" are given.
This is technically called rabatting the plane respectively into the Here it will be convenient to determine first the position which plane of the plan or the elevation. As there is no difference in the the trace a"—which is a line in a—assumes when rabatted. 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 ing a plane a into the plane of the plan. The plan of the figure is its true dimension, and we can measure off along it the true distance a parallel (orthographic) projection of the figure itself. The results between two points in it. If therefore (fig. 45) P is any point in a" of parallel projection (see PROJECTION, §§ 17 and 18) may there originally coincident with
fore now be used. The trace a' will hereby take the place of what its elevation P2, and if 0 p
formerly was called the axis of projection. Hence we see that corre is the point where a" cuts sponding points in the plan and in the rabatted plane are joined by the axis xy, so that 0 is lines which.:are perpendicular to the trace a' and that corresponding also in a', then the point P lines meet on this trace. We also see that the correspondence is will after rabatting the o completely determined if we know for one point or one line in the plane assume such a posi
plan the corresponding point or line in the rabatted plane. tion that OP=OP2. At
Before, however, we treat of this we consider some special cases. the same time the plan is
§ 13. To determine the distance between two points A, B given by their an orthographic projection iQ
projections A1, Bl and A2, B2, or, in other words, to determine the true of the plane a. Hence the a'
length of a line the plan and elevation of which are given. line joining P to the plan
Solution.—The two points A, B in space lie vertically above their PI will after rabatting be
plans Al, Bl (fig. 43) and A1A=AoA2, B1B=BoB2. The four points perpendicular to a'. But
A, B, Al, Bl therefore form a plane Pi is known; it is the foot p
quadrilateral on the base A1B1 and of the perpendicular from having right angles at the base. P2 to the axis xy. We
A / This plane we rabatt about A1B1 draw therefore, to find P,
by drawing A1A and B1B per from P1 a perpendicular P1Q to a' and find on it a point P such that pendicular to A1B1 and making OP=OP2. Then the line OP will be the position of a" when
End of Article: TWISTED 

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