Chapter XXXII: Book XIII (4)

This proves also that a plane cuts the surface in a curve of the
second order, as no line can have more than two points in common with
it. To show that this is a curve of the same kind as those considered
before, we have to show that it can be generated by projective flat
pencils. We prove first that this is true for any plane through the
centre of one of the pencils, and afterwards that every point on the
surface may be taken as the centre of such pencil. Let then [alpha]1
be a plane through S1. To the flat pencil in S1 which it contains
corresponds in S2 a projective axial pencil with axis a2 and this cuts
[alpha]1 in a second flat pencil. These two flat pencils in [alpha]1
are projective, and, in general, neither concentric nor perspective.
They generate therefore a conic. But if the line a2 passes through S1
the pencils will have S1 as common centre, and may therefore have two,
or one, or no lines united with their corresponding lines. The section
of the surface by the plane [alpha]1 will be accordingly a line-pair
or a single line, or else the plane [alpha]1 will have only the point
S1 in common with the surface.

Every line l1 through S1 cuts the surface in two points, viz. first in
S1 and then at the point where it cuts its corresponding plane. If now
the corresponding plane passes through S1, as in the case just
considered, then the two points where l1 cuts the surface coincide at
S1, and the line is called a tangent to the surface with S1 as point
of contact. Hence if l1 be a tangent, it lies in that plane [tau]1
which corresponds to the line S2S1 as a line in the pencil S2. The
section of this plane has just been considered. It follows that--

_All tangents to quadric surface at the centre of one of the
reciprocal pencils lie in a plane which is called the tangent plane to
the surface at that point as point of contact._

_To the line joining the centres of the two pencils as a line in one
corresponds in the other the tangent plane at its centre._

_The tangent plane to a quadric surface either cuts the surface in two
lines, or it has only a single line, or else only a single point in
common with the surface._

_In the first case the point of contact is said to be hyperbolic, in
the second parabolic, in the third elliptic._

S 95. It remains to be proved that every point S on the surface may be
taken as centre of one of the pencils which generate the surface. Let
S be any point on the surface [Phi]' generated by the reciprocal
pencils S1 and S2. We have to establish a reciprocal correspondence
between the pencils S and S1, so that the surface generated by them is
identical with [Phi]. To do this we draw two planes [alpha]1 and
[beta]1 through S1, cutting the surface [Phi] in two conics which we
also denote by [alpha]1 and [beta]1. These conics meet at S1, and at
some other point T where the line of intersection of [alpha]1 and
[beta]1 cuts the surface.

In the pencil S we draw some plane [sigma] which passes through T, but
not through S1 or S2. It will cut the two conics first at T, and
therefore each at some other point which we call A and B respectively.
These we join to S by lines a and b, and now establish the required
correspondence between the pencils S1 and S as follows:--To S1T shall
correspond the plane [sigma], to the plane [alpha]1 the line a, and to
[beta]1 the line b, hence to the flat pencil in [alpha]1 the axial
pencil a. These pencils are made projective by aid of the conic in
[alpha]1.

In the same manner the flat pencil in [beta]1 is made projective to
the axial pencil b by aid of the conic in [beta]1, corresponding
elements being those which meet on the conic. This determines the
correspondence, for we know for more than four rays in S1 the
corresponding planes in S. The two pencils S and S1 thus made
reciprocal generate a quadric surface [Phi]', which passes through the
point S and through the two conics [alpha]1 and [beta]1.

The two surfaces [Phi] and [Phi]' have therefore the points S and S1
and the conics [alpha]1 and [beta]1 in common. To show that they are
identical, we draw a plane through S and S2, cutting each of the
conics [alpha]1 and [beta]1 in two points, which will always be
possible. This plane cuts [Phi] and [Phi]' in two conics which have
the point S and the points where it cuts [alpha]1 and [beta]1 in
common, that is five points in all. The conics therefore coincide.

This proves that all those points P on [Phi]' lie on [Phi] which have
the property that the plane SS2P cuts the conics [alpha]1, [beta]1 in
two points each. If the plane SS2P has not this property, then we draw
a plane SS1P. This cuts each surface in a conic, and these conics have
in common the points S, S1, one point on each of the conics [alpha]1,
[beta]1, and one point on one of the conics through S and S2 which lie
on both surfaces, hence five points. They are therefore coincident,
and our theorem is proved.

S 96. The following propositions follow:--

_A quadric surface has at every point a tangent plane._

_Every plane section of a quadric surface is a conic or a line-pair._

_Every line which has three points in common with a quadric surface
lies on the surface._

_Every conic which has five points in common with a quadric surface
lies on the surface._

_Through two conics which lie in different planes, but have two points
in common, and through one external point always one quadric surface
may be drawn._

S 97. _Every plane which cuts a quadric surface in a line-pair is a
tangent plane._ For every line in this plane through the centre of the
line-pair (the point of intersection of the two lines) cuts the
surface in two coincident points and is therefore a tangent to the
surface, _the centre of the line-pair being the point of contact_.

_If a quadric surface contains a line, then every plane through this
line cuts the surface in a line-pair (or in two coincident lines)._
For this plane cannot cut the surface in a conic. Hence:--

_If a quadric surface contains one line p then it contains an infinite
number of lines, and through every point Q on the surface, one line q
can be drawn which cuts p._ For the plane through the point Q and the
line p cuts the surface in a line-pair which must pass through Q and
of which p is one line.

_No two such lines q on the surface can meet_. For as both meet p
their plane would contain p and therefore cut the surface in a
triangle.

_Every line which cuts three lines q will be on the surface_; for it
has three points in common with it.

_Hence the quadric surfaces which contain lines are the same as the
ruled quadric surfaces considered in_ SS 89-93, but with one important
exception. In the last investigation we have left out of consideration
the possibility of a plane having only one line (two coincident lines)
in common with a quadric surface.

S 98. To investigate this case we suppose first that there is one
point A on the surface through which two different lines a, b can be
drawn, which lie altogether on the surface.

If P is any other point on the surface which lies neither on a nor b,
then the plane through P and a will cut the surface in a second line
a' which passes through P and which cuts a. Similarly there is a line
b' through P which cuts b. These two lines a' and b' _may_ coincide,
but then they must coincide with PA.

If this happens for one point P, it happens for every other point Q.
For if two different lines could be drawn through Q, then by the same
reasoning the line PQ would be altogether on the surface, hence two
lines would be drawn through P against the assumption. From this
follows:--

_If there is one point on a quadric surface through which one, but
only one, line can be drawn on the surface, then through every point
one line can be drawn, and all these lines meet in a point. The
surface is a cone of the second order_.

_If through one point on a quadric surface, two, and only two, lines
can be drawn on the surface, then through every point two lines may be
drawn, and the surface is ruled quadric surface._

_If through one point on a quadric surface no line on the surface can
be drawn, then the surface contains no lines._

Using the definitions at the end of S 95, we may also say:--

_On a quadric surface the points are all hyperbolic, or all parabolic,
or all elliptic._

As an example of a quadric surface with elliptical points, we mention
the sphere which may be generated by two reciprocal pencils, where to
each line in one corresponds the plane perpendicular to it in the
other.

S 99. _Poles and Polar Planes._--The theory of poles and polars with
regard to a conic is easily extended to quadric surfaces.

Let P be a point in space not on the surface, which we suppose not to
be a cone. On every line through P which cuts the surface in two
points we determine the harmonic conjugate Q of P with regard to the
points of intersection. Through one of these lines we draw two planes
[alpha] and [beta]. The locus of the points Q in [alpha] is a line a,
the polar of P with regard to the conic in which [alpha] cuts the
surface. Similarly the locus of points Q in [beta] is a line b. This
cuts a, because the line of intersection of [alpha] and [beta]
contains but one point Q. The locus of all points Q therefore is a
plane. _This plane is called the polar plane of the point P, with
regard to the quadric surface. If P lies on the surface we take the
tangent plane of P as its polar._

The following propositions hold:--

1. _Every point has a polar plane_, which is constructed by drawing
the polars of the point with regard to the conics in which two planes
through the point cut the surface.

2. _If Q is a point in the polar of P, then P is a point in the polar
of Q_, because this is true with regard to the conic in which a plane
through PQ cuts the surface.

3. _Every plane is the polar plane of one point, which is called the
Pole of the plane._

The pole to a plane is found by constructing the polar planes of three
points in the plane. Their intersection will be the pole.

4. _The points in which the polar plane of P cuts the surface are
points of contact of tangents drawn from P to the surface_, as is
easily seen. Hence:--

5. _The tangents drawn from a point P to a quadric surface form a cone
of the second order_, for the polar plane of P cuts it in a conic.

6. _If the pole describes a line a, its polar plane will turn about
another line a'_, as follows from 2. _These lines a and a' are said to
be conjugate with regard to the surface._

S 100. The pole of the line at infinity is called the _centre_ of the
surface. If it lies at the infinity, the plane at infinity is a
tangent plane, and the surface is called a _paraboloid_.

_The polar plane to any point at infinity passes through the centre,
and is called a diametrical plane._

_A line through the centre is called a diameter. It is bisected at the
centre. The line conjugate to it lies at infinity._

_If a point moves along a diameter its polar plane turns about the
conjugate line at infinity_; that is, _it moves parallel to itself,
its centre moving on the first line._

_The middle points of parallel chords lie in a plane_, viz. in the
polar plane of the point at infinity through which the chords are
drawn.

_The centres of parallel sections lie in a diameter which is a line
conjugate to the line at infinity in which the planes meet._

TWISTED CUBICS

S 101. If two pencils with centres S1 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 (S 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 Raumes_. 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 two
projective 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.

S 102. Let l1 be the line S1S2 as a line in the pencil S1. To it
corresponds a line l2 in S2. _At each of the centres two corresponding
lines meet._ The two axial pencils with l1 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 (S 58), with S2
as centre. If [pi]1 and [pi]2 be corresponding planes, then their
intersection will be a line p2 which passes through S2. Corresponding
to it in S1 will be a line p1 which lies in the plane [pi]1, and which
therefore meets p2 at some point P. Conversely, if p2 be any line in
S2 which meets its corresponding line p1 at a point P, then to the
plane l2p2 will correspond the plane l1p1, 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 l1 and l2. 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 S1S2 or
l1, 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 l1, 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.

S 103. If a1 and a2 be any two corresponding lines in the two pencils,
then corresponding planes in the axial pencils having a1 and a2 as
axes generate a ruled quadric surface. If P be any point on the cubic
k, and if p1, p2 be the corresponding rays in S1 and S2 which meet at
P, then to the plane a1p1 in S1 corresponds a2p2 in S2. These
therefore meet in a line through P.

This may be stated thus:--

_Those secants of the cubic which cut a ray a1, drawn through the
centre S1 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 S1 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 a1 is not a secant, but made to pass through any point Q in space,
the ruled quadric surface determined by a1 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._

S 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 [alpha] in S will be cut by its
corresponding plane [alpha]' always in the same secant a. Whilst S'
moves along the curve the plane [alpha]' will turn about a, describing
an axial pencil.

AUTHORITIES.--In this article we have given a purely geometrical
theory of conics, cones of the second order, quadric surfaces, &c. In
doing so we have followed, to a great extent, Reye's _Geometrie der
Lage_, and to this excellent work those readers are referred who wish
for a more exhaustive treatment of the subject. Other works especially
valuable as showing the development of the subject are: Monge,
_Geometrie descriptive_: Carnot, _Geometrie de position_ (1803),
containing a theory of transversals; Poncelet's great work _Traite des
proprietes projectives des figures_ (1822); Mobins, _Barycentrischer
Calcul_ (1826); Steiner, _Abhangigkeit geometrischer Gestalten_
(1832), containing the first full discussion of the projective
relations between rows, pencils, &c.; Von Staudt, _Geometrie der Lage_
(1847) and _Beitrage zur Geometrie der Lage_ (1856-1860), in which a
system of geometry is built up from the beginning without any
reference to number, so that ultimately a number itself gets a
geometrical definition, and in which imaginary elements are
systematically introduced into pure geometry; Chasles, _Apercu
historique_ (1837), in which the author gives a brilliant account of
the progress of modern geometrical methods, pointing out the
advantages of the different purely geometrical methods as compared
with the analytical ones, but without taking as much account of the
German as of the French authors; Id., _Rapport sur les progres de la
geometrie_ (1870), a continuation of the _Apercu_; Id., _Traite de
geometrie superieure_ (1852); Cremona, _Introduzione ad una teoria
geometrica delle curve piane_ (1862) and its continuation _Preliminari
di una teoria geometrica delle superficie_ (German translations by
Curtze). As more elementary books, we mention: Cremona, _Elements of
Projective Geometry_, translated from the Italian by C. Leudesdorf
(2nd ed., 1894); J.W. Russell, _Pure Geometry_ (2nd ed., 1905).
(O. H.)

III. DESCRIPTIVE GEOMETRY

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, S 18). 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.

S 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 [pi]1 and [pi]2. 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 AA1 and
AA2 to the planes [pi]1 and [pi]2 so that A1 and A2 are the horizontal
and vertical projections of A.

If we remember that a line is perpendicular to a plane that is
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 AA1 and to AA2 is also perpendicular to A1A0
and to A2A0 because these four lines are all in the same plane. Hence,
if the plane [pi]2 be turned about the axis till it coincides with the
plane [pi]1, then A2A0 will be the continuation of A1A0. This position
of the planes is represented in fig. 38, in which the line A1A2 is
perpendicular to the axis x.

Conversely any two points A1, A2 in a line perpendicular to the axis
will be the projections of some point in space when the plane [pi]2 is
turned about the axis till it is perpendicular to the plane [pi]1,
because in this position the two perpendiculars to the planes [pi]1
and [pi]2 through the points A1 and A2 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
A1, A2 in the plane on a line perpendicular to the axis represent a
point A in space_. If the line A1A2 cuts the axis at A0, and if at A1
a perpendicular be erected to the plane, then the point A will be in
it at a height A1A = A0A2 above the plane. This gives the position of
the point A relative to the plane [pi]1. In the same way, if in a
perpendicular to [pi]2 through A2 a point A be taken such that A2A =
A0A1, then this will give the point A relative to the plane [pi]2.

S 2. The two planes [pi]1, [pi]2 in their original position divide
space into four parts. These are called the four quadrants. We suppose
that the plane [pi]2 is turned as indicated in fig. 37, so that the
point P comes to Q and R to S, then the quadrant in which the point A
lies is called the first, and we say that in the first quadrant a
point lies above the horizontal and in front of the vertical plane.
Now we go round the axis in the sense in which the plane [pi]2 is
turned and come in succession to the second, third and fourth
quadrant. In the second a point lies above the plane of the plan and
behind the plane of elevation, and so on. In fig. 39, which represents
a side view of the planes in fig. 37 the quadrants are marked, and in
each a point with its projection is taken. Fig. 38 shows how these are
represented when the plane [pi]2 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.

S 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 [pi]1 and [pi]2, and no plane would appear different from any
other. But any plane [alpha] cuts each of the planes [pi]1, [pi]2 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 [alpha].

_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 [pi]3. As it
is perpendicular to [pi]1, it may be taken as the plane of elevation,
its line of intersection [gamma] with [pi]1 being the axis, and be
turned down to coincide with [pi]1. 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 [gamma]. The plane is rabatted about [gamma] to
the horizontal plane. A plane [alpha] through the axis xy will then
show in it a trace [alpha]3. In fig. 40 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 [pi]3. In
rabatting the plane [pi]3 its trace OB with the plane [pi]2 will come
to the position OD. Hence a plane [beta] having the traces CA and CB
will have with the third plane the trace [beta]3, or AD if OD = OB.

It also follows immediately that--

_If a plane [alpha] 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._

S 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 a1 and a2 on the
two planes [pi]1 and [pi]2._ These may be any two lines, for, bringing
the planes [pi]1, [pi]2 into their original position, the planes
through these lines perpendicular to [pi]1 and [pi]2 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 [pi]1, [pi]2._ 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 projections 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_, S 41). 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 A, B, C; planes in space by Greek letters
[alpha], [beta], [gamma]; lines in space by small letters a, b, c;
horizontal projections by suffixes 1, like A1, a1; vertical
projections by suffixes 2, like A2, a2; traces by single and double
dashes [alpha]' [alpha]", a', a". Hence P1 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 [alpha] has the traces
[alpha]' and [alpha]".

S 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 planes 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.

S 6. _To find the projections of a line which joins two points A, B
given by their projections A1, A2 and B1, B2_, we join A1, 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
perpendiculars 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
a1 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 lie 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.

S 7. _To decide whether a point A, given by its projections, lies in a
plane [alpha], given by its traces_, we draw a line p by joining A to
some point in the plane [alpha] and determine its traces. If these lie
in the traces of the plane, then the line, and therefore the point A,
lies in the plane; otherwise not. This is conveniently done by joining
A1 to some point p' in the trace [alpha]'; this gives p1; 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 [alpha].

S 8. _Parallel planes have parallel traces_, because parallel planes
are cut by any plane, hence also by [pi]1 and by [pi]2, 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.

S 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.

S 10. 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 case 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.

S 11. _To draw a plane through a given point parallel to a given plane
[alpha]_, we draw through the point two lines which are parallel to
the plane [alpha], 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 (S 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
[alpha] will be the vertical trace [beta]" of the required plane
[beta], and a line parallel to the horizontal trace of [alpha] meeting
[beta]" on the axis will be the horizontal trace [beta]'.

Let A1 A2 (fig. 42) be the given point, [alpha]' [alpha]" the given
plane, a line l1 through A1, parallel to [alpha]' and a horizontal
line l2 through A2 will be the projections of a line l through A
parallel to the plane, because the horizontal plane through this line
will cut the plane [alpha] in a line c which has its horizontal
projection c1 parallel to [alpha]'.

S 12. We now come to the metrical properties of figures.

_A line is perpendicular to a plane if the projections of the line are
perpendicular to the traces of the plane._ We prove it for the
horizontal projection. If a line p is perpendicular to a plane
[alpha], every plane through p is perpendicular to [alpha]; hence also
the vertical plane which projects the line p to p1. As this plane is
perpendicular both to the horizontal plane and to the plane [alpha],
it is also perpendicular to their intersection--that is, to the
horizontal trace of [alpha]. It follows that every line in this
projecting plane, therefore also p1, the plan of p, is perpendicular
to the horizontal trace of [alpha].

_To draw a plane through a given point A perpendicular to a given line
p_, we first draw through some point O in the axis lines [gamma]',
[gamma]" perpendicular respectively to the projections p1 and p2 of
the given line. These will be the traces of a plane [gamma] which is
perpendicular to the given line. We next draw through the given point
A a plane parallel to the plane [gamma]; this will be the plane
required.

Other metrical properties depend on the determination of the real size
or shape of a figure.

In general the projection of a figure differs both in size and shape
from the figure itself. But figures in a plane parallel to a plane of
projection will be identical with their projections, and will thus be
given in their true dimensions. In other cases there is the problem,
constantly recurring, either to find the true shape and size of a
plane figure when plan and elevation are given, or, conversely, to
find the latter from the known true shape of the figure itself. To do
this, the plane is turned about one of its traces till it is laid down
into that plane of projection to which the trace belongs. This is
technically called rabatting the plane respectively into the plane of
the plan or the elevation. As there is no difference in the treatment
of the two cases, we shall consider only the case of rabatting a plane
[alpha] into the plane of the plan. The plan of the figure is a
parallel (orthographic) projection of the figure itself. The results
of parallel projection (see PROJECTION, SS 17 and 18) may therefore
now be used. The trace [alpha]' will hereby take the place of what
formerly was called the axis of projection. Hence we see that
corresponding points in the plan and in the rabatted plane are joined
by lines which are perpendicular to the trace [alpha]' and that
corresponding lines meet on this trace. We also see that the
correspondence is completely determined if we know for one point or
one line in the plan the corresponding point or line in the rabatted
plane.

Before, however, we treat of this we consider some special cases.

S 13. _To determine the distance between two points A, B given by
their projections A1, B1 and A2, B2, or, in other words, to determine
the true length of a line the plan and elevation of which are given._

_Solution._--The two points A, B in space lie vertically above their
plans A1, B1 (fig. 43) and A1A = A0A2, B1B = B0B2. The four points A,
B, A1, B1 therefore form a plane quadrilateral on the base A1B1 and
having right angles at the base. This plane we rabatt about A1B1 by
drawing A1A and B1B perpendicular to A1B1 and making A1A = A0A2, B1B =
B0B2. Then AB will give the length required.

The construction might have been performed in the elevation by making
A2A = A0A1 and B2B = B0B1 on lines perpendicular to A2B2. Of course AB
must have the same length in both cases.

This figure may be turned into a model. Cut the paper along A1A, AB
and BB1, and fold the piece A1ABB1 over along A1B1 till it stands
upright at right angles to the horizontal plane. The points A, B will
then be in their true position in space relative to [pi]1. Similarly
if B2BAA2 be cut out and turned along A2B2 through a right angle we
shall get AB in its true position relative to the plane [pi]2. Lastly
we fold the whole plane of the paper along the axis x till the plane
[pi]2 is at right angles to [pi]1. In this position the two sets of
points AB will coincide if the drawing has been accurate.

Models of this kind can be made in many cases and their construction
cannot be too highly recommended in order to realize orthographic
projection.

S 14. _To find the angle between two given lines a, b of which the
projections a1, b1 and a2, b2 are given._

_Solution._--Let a1, b1 (fig. 44) meet in P1, a2, b2 in T, then if the
line P1T is not perpendicular to the axis the two lines will not meet.
In this case we draw a line parallel to b to meet the line a. This is
easiest done by drawing first the line P1P2 perpendicular to the axis
to meet a2 in P2, and then drawing through P2 a line c2 parallel to
b2; then b1, c2 will be the projections of a line c which is parallel
to b and meets a in P. The plane [alpha] which these two lines
determine we rabatt to the plan. We determine the traces a' and c' of
the lines a and c; then a'c' is the trace [alpha]' of their plane. On
rabatting the point P comes to a point S on the line P1Q perpendicular
to a'c', so that QS = QP. But QP is the hypotenuse of a triangle PP1Q
with a right angle P1. This we construct by making QR = P0P2; then P1R
= PQ. The lines a'S and c'S will therefore include angles equal to
those made by the given lines. It is to be remembered that two lines
include two angles which are supplementary. Which of these is to be
taken in any special case depends upon the circumstances.

_To determine the angle between a line and a plane_, we draw through
any point in the line a perpendicular to the plane (S 12) and
determine the angle between it and the given line. The complement of
this angle is the required one.

_To determine the angle between two planes_, we draw through any point
two lines perpendicular to the two planes and determine the angle
between the latter as above.

In special cases it is simpler to determine at once the angle between
the two planes by taking a plane section perpendicular to the
intersection of the two planes and rabatt this. This is especially the
case if one of the planes is the horizontal or vertical plane of
projection.

Thus in fig. 45 the angle P1QR is the angle which the plane [alpha]
makes with the horizontal plane.

S 15. We return to the general case of rabatting a plane [alpha] of
which the traces [alpha]' [alpha]" are given.

Here it will be convenient to determine first the position which the
trace [alpha]"--which is a line in [alpha]--assumes when rabatted.
Points in this line coincide with their elevations. Hence it is given
in its true dimension, and we can measure off along it the true
distance between two points in it. If therefore (fig. 45) P is any
point in [alpha]" originally coincident with its elevation P2, and if
O is the point where [alpha]" cuts the axis xy, so that O is also in
[alpha]', then the point P will after rabatting the plane assume such
a position that OP = OP2. At the same time the plan is an orthographic
projection of the plane [alpha]. Hence the line joining P to the plan
P1 will after rabatting be perpendicular to [alpha]'. But P1 is known;
it is the foot of the perpendicular from P2 to the axis xy. We draw
therefore, to find P, from P1 a perpendicular P1Q to [alpha]' and find
on it a point P such that OP = OP2. Then the line OP will be the
position of [alpha]" when rabatted. This line corresponds therefore to
the plan of [alpha]"--that is, to the axis xy, corresponding points on
these lines being those which lie on a perpendicular to [alpha]'.

We have thus one pair of corresponding lines and can now find for any
point B1 in the plan the corresponding point B in the rabatted plane.
We draw a line through B1, say B1P1, cutting [alpha]' in C. To it
corresponds the line CP, and the point where this is cut by the
projecting ray through B1, perpendicular to [alpha]', is the required
point B.

Similarly any figure in the rabatted plane can be found when the plan
is known; but this is usually found in a different manner without any
reference to the general theory of parallel projection. As this method
and the reasoning employed for it have their peculiar advantages, we
give it also.

Supposing the planes [pi]1 and [pi]2 to be in their positions in space
perpendicular to each other, we take a section of the whole figure by
a plane perpendicular to the trace [alpha]' about which we are going
to rabatt the plane [alpha]. Let this section pass through the point Q
in [alpha]'. Its traces will then be the lines QP1 and P1P2 (fig. 9).
These will be at right angles, and will therefore, together with the
section QP2 of the plane [alpha], form a right-angled triangle QP1P2
with the right angle at P1, and having the sides P1Q and P1P2 which
both are given in their true lengths. This triangle we rabatt about
its base P1Q, making P1R = P1P2. The line QR will then give the true
length of the line QP in space. If now the plane [alpha] be turned
about [alpha]' the point P will describe a circle about Q as centre
with radius QP = QR, in a plane perpendicular to the trace [alpha]'.
Hence when the plane [alpha] has been rabatted into the horizontal
plane the point P will lie in the perpendicular P1Q to [alpha]', so
that QP = QR.

If A1 is the plan of a point A in the plane [alpha], and if A1 lies in
QP1, then the point A will lie vertically above A1 in the line QP. On
turning down the triangle QP1P2, the point A will come to A0, the line
A1A0 being perpendicular to QP1. Hence A will be a point in QP such
that QA = QA0.

If B1 is the plan of another point, but such that A1B1 is parallel to
[alpha]', then the corresponding line AB will also be parallel to
[alpha]'. Hence, if through A a line AB be drawn parallel to [alpha]',
and B1B perpendicular to [alpha]', then their intersection gives the
point B. Thus of any point given in plan the real position in the
plane [alpha], when rabatted, can be found by this second method. This
is the one most generally given in books on geometrical drawing. The
first method explained is, however, in most cases preferable as it
gives the draughtsman a greater variety of constructions. It requires
a somewhat greater amount of theoretical knowledge.

If instead of our knowing the plan of a figure the latter is itself
given, then the process of finding the plan is the reverse of the
above and needs little explanation. We give an example.

S 16. _It is required to draw the plan and elevation of a polygon of
which the real shape and position in a given plane [alpha] are known._