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Theses and Dissertations
1928
A study of the hyper-quadrics in Euclidean space of
four dimensions
Clarence Selmer Carlson
University of Iowa
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Carlson, Clarence Selmer. "A study of the hyper-quadrics in Euclidean space of four dimensions." MS (Master of Science) thesis, State
University of Iowa, 1928.
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Part of the Geometry and Topology Commons
A STUDY OP THE HYPER-QUADRICS
IN
EUCLIDEAN SPACE OF F OUR DIMENSIONS
by
Clarence Selmer Carlson
A thesis submitted in partial
fulfillment o f the requirements for the
degree of Master of Science, in the
Graduate College of the State University
of Iowa.
July 1928
I wish to express my appreciation to
Dr* Roscoe Woods for his liberal gifts of time
and energy to help, guide, and encourage me in the
preparation of this thesis.
Vvft.
Table of Contents
Section 1.
Introduction
Section 2.
Some Properties of Euclidean
Four-Space.
Section 3.
pp 1-3
pp.3-11
Systems of Hyperplanes
pp 11-14
Section 4.
Systems of Hyperspheres
pp 15-19
Section 5.
Classification of Hyperquadrics
pp 20-32
Section 6 .
Generation of Hyper-quadrics
in S/i
Section 7.
pp 33-38
Tangent Hyperplanes and
Polar Hyperplanes pp 39-45
Section 8.
Some Theorems,Proofs and
Humerical Examples.pp 45-56
Section 1
Introduction
The idea of a space of four-dimensions is quite
easily developed, in which many of the ideas and
methods of the usual three-dimensional geometry can
he readily extended to the new space♦Naturally it is
necessary to make a few fundamental definitions and
assumptions.
A point
(also referred to in this thesis as an
S 0) in a four-dimensional space
( also referred to as
an S/t) is defined as any set of values of the four
ratios Xi : x^ 5x 3 :x^ 5x 5
of the five variables.
In non-homogeneous form the point is a set of values
of the four variables
(x,y,z,w,)*
A straight line in 3^ is defined as a one-di­
mensional extent (Si) determined by the equations
pxj = Yf + lzi
(i«l, 2 ,3,4,5,).......(l)
where y^ and x^ are two fixed points and 1 is an
independent parameter.
A plane is defined as a two-dimensional extent
d termined by the equations
I.Woods Frederick S.
j
Higher Geometry Pg.362
-2-
pxi = y i + lzi + lzi
(i=l,2,3,4,5,)..... (2 )
where y*, zn*, and wj are three fixed points not on
the same straight line and 1 , m,
-1
are inde­
pendent parameters.
A hyperplane
(also referred to as a three-fold
or three-space) is defined as a three-dimensional
extent
(SS ) determined by the equations
px* = y-f +
+ m w 7* + nu^
(i=i,2,3,4,5,)..(3)
where yj , zn-, wj, and uj are fixed points not in the
same plane and 1 , m, n, are independent parameters.
On the basis of these definitions it is evident
that a straight line is completely arid uniquely de­
termined by two of its points, a plane by any three
of its points and a hyperplane by any four of its
points which are not coplanar.Also, if two points
lie in a plane, the line determined by them lies in
the plane, if three points lie in a hyperplane,
the
plane determined by them lies in the hyperplane.
Eliminating p, 1, m, n, from the equations of the
hyperplane we find that any hyperplane may be rep­
resented by a linear equation in the co-ordinates
Xj
(i=l,2 ,3,4,5,)
equation in x D* as
and conversely, any linear
ZTa 5x i = 0
(i=i,2 ,3,4,5,)
represents a hyperplane.
In like manner, any plane may be represented by
-3-
two linear equations in the co-ordinates Xj
(i=l,2,3,
4,5,) and conversely; any straight line may he rep­
resented hy three linear equations and conversely; and
finally, four linear equations of the type mentioned
in general represent a point, the common point of
intersection.
Section 2
Some Properties of Euclidean Pour-Space,
Article 1. Transition.
To arrive at some of the
properties of Euclidean four-space let us write
xi = xi / x 5
(i=lf2,3,4,)* If xg /
0 the co-or­
dinates are finite, and are said to represent a point
in finite space.
In this thesis it has been found to be very ad­
vantageous for the sake of brevity and efficiency to
make frequent shifts from homogeneous to non-homogeneous co-ordinates.7i/here ambiguity would not result
the transition has been made by making xg = 1 without
mention of the fact and vice versa.
Article 2.
The Line.
Prom the equations (i)
defining a line we have the symmetric form
where P(y^) and Q,(z^) are two points on the line
(i=l,2,3,4,).This equation may also be written
xi - yi)/Ai = (xg - y*)/A^ = (x^ - vg)/Ag = (x4 * y4)
/ A/\_ .•••••( 5)
where the numbers A 4 (i=l,2,3,4,) are determined from
the two points used to determine the ^line and are
defined as the directions of the line.
-5-
Two lines with directions A i , and B i
(i=L ,2,3,4,)
respectively are said to make an angle 0 with each
2
other
defined by the equation
the equation of the straight line takes the form
Clearly,with this form of the straight line equation,
the angle between the lines is defined by the equation
Prom the values of lj and
above it is clear that3 *
ht
= i ......................... (9)
i
where 1^ is the cosine of the angle which the line
makes with the axis Xj_, Likewise the sine of the angle
which the line makes with this axis becomes
Sin20 =
Article 3,
/Ml-f
- lijaf)2 ................ (10)
The Hyperplane,
It was seen that the
2.Woods,Frederick S. Higher Geometry pg.369
2#Schoute,Dr.P*E*
Mehrdimensionale Geometrie.pg.133
3,Ibid page 128
-
6-
hyperplane could be represented by a single linear
equation in five homogeneous variables.
The normal
form of the hyperplane is shown to be
where p is the normal.
If (ll) is equated to d,
d will represent the distance from a point xj to the
S 3 given by (ll).
Article 4.
Distance Relations.
The distance
between two points P and Q is defined for a rectang.
5
ular syatem of co-ordinates * as
where x^ and y^ are the co-ordinates of the points
P and Q, respectively.
To find the distance from a point to a line, let
the line be given in the form (7) and the point
E^tb-i ,bn,b^,b 4 ,) not lying on the line, where the
lj = cos a j b e i n g
the angle with the axis of x^.
how if P^(a^fapfa 3 *a4 f) be a point on the line, and
if we let P be the foot of the perpendicular from Pp
to'the line, and l e t .9 be the angle between the given
line and the line PqPp> while s,t the same time d is
4. .Sciioute,D r .P.K.
5. Ibid pg 127
Mehrdimensionale Geomwtrie pg!168
-7-
is the length of the segment P1P2, then
(P2 P) = (P1P2)sin 9
(P2P)2 = (P1P2) sin20 = d 2- d2
c o s29
The direction cosines of the lineP1P2 are m i =
(b i -ai)/d,
from
(10),
(i=l, 2,3,4,) and thus Cose**^ (b^-a-il/d
so that we have
f.5
Schoute *
has a beautiful generalization for n-
dimensional geometry
which should he mentioned here.
He says that if in Sm
a space of
and a point
are given, the locus of the line through P and per­
pendicular to
^ is an
In order to find the system of equations rep­
resenting the S^,let the co-ordinates of the point he
(a-j ,ap,.. .a^) and let the equations of the Sn-(* he
given hy the independent equations
(k=l,2,3,4,..d) when referred to a rectangular system
of co-ordinates.
By the familiar properties of the normal form
(Hesse) the equations of the directed perpendicular from
0 to these d spaces are
6.Schoute,Dr.H.f. Mehrdimensionale Geometrie pg 168
-
8-
Then, the determinant
when
is infinite, becomes the equation of the
through P and the infinite point of the d perpendic­
ular, for in that case it simplifies to the form
where each determinant of the highest rank which can be
taken from this form constitutes one equation of the
desired system.
Article 5.
Parallelism.
Any two of the configurations^
line, plane or hyperplane are said to be parallel if
their complete intersection is infinitely distant.
A
few theorems on parallelism may be useful in the main
}
problem of this
thesis*
These theorems have been taken
from Woods Higher Geometry page 371.
l.If a line is parallel to a plane the two lie in
the same hyperplane and determine that hyperplane.
Through any point in space
there passes a pencil of
lines parallel to a fixed plane.
2.Two planes are said to be simply parallel if they
intersect in a single infinite point and completely
-9-
parallel if they intersect in a line at infinity,thus
two completely parallel planes lie in the same hyper­
plane
(S3 ).
In the case of completely parallel planes,
any line of one is parallel to a pencil of lines of the
other,while if the planes are simply parallel,
there is
a unique direction in each plane such that lines with
that direction in either plane are parallel to lines
with the same direction in the other, hut lines with
any other direction in one plane are parrallel to no
lines of the other*
3.
If a plane and a hyperplane are parallel, any
line in the plane is parallel to each line of a bundle
in the hyperplane, and any plane in the hyperplane is
at least simply parallel to the given plane.
4.
If two hyperplanes are parallel, any plane of
one is completely parallel to some plane and hence toy4
pencil of planes of the other, and any plane of one is
t
simply parallel to any plane of the other to which it
is not completely parallel.
The condition that two lin s he parallel is that the
direction cosines he alike or their direction numbers he
proportional.
Two hyperplanes are parallel if their
coefficients in their equations he proportional*
If we have two planes given by the equations
-10-
the necessary and sufficient condition that they
intersect in a^Jfinite point i,e., "be simply parallel
i s that
shall be of rank fmir,while if the determina&ts of
the matrix
IA t ...... A/iU
have a constant ratio to
hi
......
the corresponding determinants of
* * * * *£|||
the planes (14) and (15) are completely parallel.
Article 5*
Perpendicularity.
We have already
discussed the angle between two lines*
Other perpen­
dicular properties may be given in theorem form:
1.
A line, perpendicular to three lines of a hyper­
plane, which are non-copla,nar and no two of which are
parallel, is perpendicular to the hyperplane, while a
line perpendicular to every plane in a hyperplane is
perpendicular to the hyperplane.
2.
Two planes such that every line of one is perpen­
dicular to every line of the other is said to be com­
pletely perpendicular, while two planes such that each
-11-
contains a line perpendicular to the other are nailed
semi-perpendicul3.r planes
3,
Two semi-perpendicular planes may he simply
parallel.
The direction of the parallel
lines of the
two planes is then orthogonal to the directions of the
perpendicular lines®
4.
If a plane is perpendicular to a hyperplane,
it
is completely perpendicular to each plane of a pencil
of parallel planes of the hyperplane and semi-perpen­
dicular to every other plane of the hyperplane.
5.
Finally, in four-dimensional Euclidean space the
geometry in any hyperplane,
for which A 2 + B 2 + C 2 + D 2/0
is that of the usual Euclidean three-dimensional geometry.
11a
Section 3.
Systems of Hyperplanes.
Article 7.
System of Hyperplanes Through a Plane.If
are equations of two intersecting S g ’s, the equation
k^P^ + k<?Pp = 0 , is for all real values of k^ and kp9
the equation of a hyperplane passing through hhe plane
(S^) determined hy (16).
The totality of hyperplanes
passing through a plane is called a pencil of hyperplanes,
and any two of these hyperplanes may he used to define
the plane.
The two equations P-j_ = 0, Pp = 0, will represent
the same hyperplane when the coefficients are pro­
portional i.e., every second order determinant of the
matrix shall vanish
llAq....... A 5
IIbi .... b5
In this case the multipliers k-j_ and kp can he found such
that k-^P^ +
= 0 is satisfied identically and
conversely.
Article 8 .
Bundles of Hyperplanes.
Three hyper­
planes not in the same pencil intersect is a straight
line.
All hyperplanes through the same line form a
-12-
a bundle, and any three of them not in the same
pencil determine the line.
Let
be the equations of three hyperplanes belonging to a
bundle.
Then kqPj + kp Pp +
= 0 ......... . (18)
is the equation of the bundle.
Article 9.
A Web of Hyperplanes*
Four linearly
independent hyperplanes intersect in a point.
All
hyperplanes through the same point form a threedimensional extent*.On the other hand, we may say that
any four hyperplanes not in the same bundle determine
the point.
Given
be the equation of four hyperplanes.If
|Ai Bo 03 DJ/ 0
this system has a point in common, where
IA 1 Bp c 3
a.
is written for the determinant of the coefficients of
Xjr ,xp,X 3 ,X4 ,. This point is called the vertex.
From
Art.5. we recall that the planes in (19) will be sim­
ply parallel when
|Al Bp C3
D& I =
0 and when not all
of the other fourth order determinants of the matrix
7. We shall say that systems with one,two, or three
essential parameters shall be called pencils, bundles,
and webs respectively.
8 . This abridged notation will be adopted in t’ is thesis.
-13-
IIA1 Bp
e JI
C3
(in the abridged notation)
are
not zero. If one of the fourth order determinants of
the matrix is different from zero and
4
|A-j--Bp C 3 £ 4)
'
/ 0 the system £kj
= 0 is called a web of hyper-
/
planes.
Article X
A System of Five Hyperplanes.
The
condition that five hyperplanes
all pass through a point is that five numbers
(x^xp
X 3X /tX5 ),not all zero, exist which satisfy the five
equations simultaneously. This condition requires that
D/i
|Aq Bp C3
l^J = O.
The hyperplanes are said to be independent,
should
this condition not be satisfied. When the given hyper­
planes are independent, five numbers k-^ kp IC3 k^ k$
can always be found such that every hyperplane can be
expressed by the equation
To show this let £a.j Xj = 0 be the equation of any
/
hyperplane. This last equation and
(21) will represent
the same hyperplane if their coefficients are proport­
ional. This condition is fulfilled when
-14a1 = k1 A1 + k2B1
k 2C1
+k4d1 + k 5E1
a2 =
k1A2 + k2B 2 +
k 3C 2 + k 4D 2 + k 5E 2
a3 =
k1A 3 +
kpB 3 +
k^C^ + k 4D 3 + kp^E^
a4 =
k^A 4 +
kc>B4 +
k 3C 4 + k 4D 4 + k 3E 4
a5 =
klA 5 + kr,B5 + k 3C 5 + k 4D 5 + k 5E 5
•••••••••••• (22 )
Since the determinant of the co-efficients is not zero,
and the five k* s can he determined so as to satisfy
these equations.
Article 11.
The matrix of a System of Hyperplanes.
The material of the previous articles of this section
may he summarized thus: The necessary and sufficient
condition that a system of hyperplanes
(a) have no point in common is that the matrix
formed of their corfficients is of rank five;
(h) belong to a web of hyperplanes if the matrix
is of rank four, i.e.,
planes on a point,
»
we have all the hyper­
or in our abbreviated notation
a system of S3 s on an S0$
(c)constitute a bundle of hyperplanes if the
9
matrix is of rank three, all the 33 s on Sj;
(d) form a pencil of hyperplanes if the matrix is
9
of rank two and we have all S 3 s on Sp
(e) are all coincident hyperplanes in the matrix
is of rank one.
-15-
Section 4.
Systems of Hyperspheres.
Article 12.
The Hypersphere.
The equation in the
four variables x1,x2,X 3,X 4 ,
is the equation of the hypersphere of radius r and
center(a1 ,a2,a 3 ,a4 ,).
It is the locus of all points
equidistant from the fixed point,its center. See e q . (12 )
Any equation of the form
may he written in the form
where a11 / 0.
By comparing with
(23) this is seen
to he the equation of a hypersphere provided the
right hand member is greater than zero, with center
(-a1 5 /a11 ,—a25 /a11, -a^g/a^ j , “a^pi/a^-j,) and radius
Article 13.
The Absolute.
If v.e write the equ­
a-i-j^xl + 2^>aj_^Xj X 5 m Qm,{2.Q
!
1
(al 1^ 0 ),the intersection of this hypersphere with the
ation of the hypersphere
infinite S 3 is represented by the equations
Since these equations are independent of the
coefficients a^-j, a ^ ,
(i=l,2,3,4,) which appear in
the equation of the hypersphere
(26), we conclude
-16-
that all hyperspheres intersect the infinite hyper­
plane in the same trace.
This trace is called the absolute and contains no
real points.
The equation of any second degree sur­
face which contains tha absolute may be written
If a-Q / 0 this is the equation of a hyper sphere.
»
If alx = 0
the locus of the equation is two S 3 s,
of which one, at least, is X 5 = 0.
In the latter case
we shall call the surface a hypersphere,
since it con­
tains, or passes through the absolute,and is of the second
degree#
It may also he called a composite hypersphere.
Conversely,
every surface of the second degree
that contains that absolute is a hypersphere.
s~
Any hyperplane '^Tujx^ = 0 other than X 5 = 0 in/
tersects the absolute in a circle which in turn con­
tain the circular points mentioned in ordinary plane
analytical geometry.
Article 14. Systems of Hyperspheres.
Let A =
ao'2 jxi + 2 ^ ai 5x ix 5 = 0 be the equation of a, hypersphere with similiar equations for other hyperspheres.
then
kxA + k ?B = 0 ......................... (28)
is the equation of all hyperspheres containing the
intersection of A = 0, B = 0,
the hypersphere
How if k^ao + kpb0= 0
(28) is composite and consists of the
infinite S 3 and the hyperplane
-17-
which intersects all the hyperspheres of the system
(28) in a fixed quadric common to A = 0, and B = 0.
The hyperplane
(29) is to he called the radical hyper­
plane.
But further, the radical hyperpit'ne is the locus
of the centers of the hyperspheres intersecting A = 0
and B = 0 orthogonally.
If we define the angle bet­
ween two hyperspheres at a point P of their trace of
9
intersection, as the angle between their tangent S$ s
at the point P.
Thenby the same method as used in
three-space geometry,using the relation for the cosine
9
of the angle between two S 3 s.we have the relation for
two hyperspheres to be orthogonal
v/here A = 0, and B = 0, are the hyperspheres.
How if C = 0 be a hypersphere orthogonal to A = 0,
-18-
Likewise, if D = 0 is a hyper-sphere whose center
does not lie on the line of centers of A = 0 and B = 0
then every hypersphere of the system
k pA + kpB + k3D = 0 .................... ..(31)
passes through the intersection of the hyperspheres
A = 0, B = 0, D = 0#
Every hypersphere of the system (31) for which
kpa 0 + kpb 0 + k 3d 0 = 0 is composite and consists of
the infinite S 3 and the S 3 on the pencil given by the
radical hyperplanes
The S 9 (32) is called the radical plane of the system.
In the same manner as we showed that the radical
hyperplane was the locus of the centers of all hyper­
spheres orthogonal to the system (28), so the radical
plane is the locus of the centers of all hyperspheres
orthogonal to the system (31).
Again, if E = 0 is another hypersphere whose center
is not on the plane determined by the hyperspheres
A = 0, B = 0, D = 0, the relation
kpA + k^B + k 3D + k^E = 0 • • • • • • • • • • • • ......... ( 33)
is composite if kpa 0 + k 9b 0 + k 3 d 0 + k 4 e 0 = 0. The
centers of all spheres orthogonal to this system lie
on the radical axis defined by
-19-
After the same manner the radical center becomes the
center of the hypersphere cutting all the hyperspheres
of the linear system
k1A + k2B + K3D + k4E + k5F = 0........... (35)
orthogonally.
Article 15.
A System of Six Hyperspheres.
Given
six hyperspheres in S 4 , given by the equations
where p = 1 ,2 ,3,4,5, 6 ,.
How if we require that a hypersphere be a linear
combination of these so that
9
we must find k-j j k p ^ k ^ k ^ k ^ k g f
such that
where the q,* without the superscript is a coefficient
of Q, without the superscript.
The condition that these equations have a solution
in the k ’s other that zeros is that
Hence, any hypersphere can be written as the linear
combinaticm o f six independent hyperspheres in S4.
-20-
Section 5.
The Classification of Hyper­
quadrics in S 4 #
Article 16.
Definition of a Hyperquadric.
The
locus representing an equation of the second degree
in x-j_,xr>, X 3 ,X4 * (non-homogeneous co-ordinates)
is
called a hyperquadric in S 4 .
The most general equation of the second degree
in five homogeneous variables can he written
It is the purpose of this section to reduce this
equation to a simplier form and classify the surfaces
according to canonical forms.
Article 17.
ric and a Line.
The Intersection of a Hyper-quad­
Given the general equation of a
hyperquadric in S 4
and let us solve simultaneously with the line
Xj = y* + ljr .
(1=1,2,3,4,5,)......... (40$
v/here P 0 [y±) is a point on the line, and assuming that
the coefficients of the second degree terms are not
all zero and all the coefficients are real we have
-21-
S - F(y*)
(i-1>2,3,4 f5,)
The roots of (41) are the distances from the point
P 0 (yi) on the line
(40) to the points in which the
line intersects the hyperquadric*
If q / 0, equation
(41) is a quadratic in r.
Q, = 0, but R and S are not both zero,
If
(41) is still to
be considered a quadratic, with one or more infinite roots
I f Q = R = S = 0
(41) is satisfied for all values of
r and the corresponding line
surface©
(40) lies entirely on the
From which V3 cen conclude that every line
which does not 1 ie on a given hyperquadric has
( dis­
tinct or coincident) points in common with the surface
while if a given line has more than two points in
common with a given hyperquadric,
it lies entirely on
that surface.
Article 18.
Diametral Hyperplanes and Center.
Let P*£ and Pp be the points of intersection of the
line (40) with the hyperquadric.
ThenP^Pp will be
called a chord of the hyperquadric.
If r^ and rp
be the roots o f (41) so that P0P*£ = rp and P 0Pp = rp
and if we require that P0 be the middle point of P^Pp*
then r-| + rp = 0 and hence
-22-
If the parameters
vary the line
lines*
are constant hut y^ allowed to
(40) describes a system of parallel
Equation
(43) is the equation of the locus
of the middle points of this system of chords.
!\Tow
(43) is linear in yj, hence it is the equation of a
hyperplane.
We shall call it a hyper-diametral-plane,
or a diametral hyperplane.
If we consider (43) as a system we have the system
>
of all S 3 s on the intersection of four hyperplanes,
for all values of 1^
(43) passes through the inter­
section of the hyperplanes
Let us use the following notation for brevity,
If A 55 / 0 the hyperplanes intersect in a single
i
finite point, (y*) = (- 1 ) A 3;$/ Ar^
(i= 1 ,2 ,3,4,)
(non-homogeneous co-ordinates ), and (y^) is the
middle point of every chord through it.
If this
point does not lie on the surface it is called the
center of the hyperquadric.
of hyperplanes
In either case, the system
(44) is a three parameter system with
vertex at (y^).
Airticle 18.
but A ^
Discussion of A 53 .
If A 33 = 0,
are not all zero, the hyperplanes
sect in an infinitely distant point.
(44) inter­
See Art 5. And
-23-
the system is a parallel web .
The hyperquadric is now
said to he non-central.
Recalling the summary in Article 11, if A 55 is of
rank three,
the system (44) intersect in a line.
If
this line is finite and does not lie on the hyper
quadric it is called a line of centers, and the sur­
face is of the cylinder type.
If the line
lies on
the surface it is called a line of vertices.
If the system is of rank two, the diametral planes
9
(S3 s) lie on a plane (S2
), and we have a plane of
centers if the plane does not lie on the surface, and
a plane of vertices if the plane lies on the surface.
If the system is of rank one, the diametral hyper­
planes coincide and we have a hyperplane of centers or
vertices.
Article 20.
Discriminating Quartic.
The condition that
the diametral hyperplanes (43) he perpendicular to the
chord it bisects is that
If we denote the common ratio by k we may write in
place of
(45)
(all~:k)1l 4- a-^plp
+ a-^pl^
+ (app-k)I9 + ap^l^
+
+ s-o /iI a
a311I
+ a^ 9lp
+ (a.pp-lc) I 3 + 2-3414
a^l1!
+ a^plp
+ S-43I 3
0
= 0
'.......(46)
= 0
+ (aAA~k) 14= 0
-24-
In order that these equations have a solution in
1*L,lp, I 3 ,l^, other than 0 ,0 ,0 ,0 , we must have
which, developed and arranged in powers of k gives us
the quartic
In the determinant A 55 let us multiply the elements
of the first,
second, and third rows hy t0 , tq, tp,
respectively, and add the first three terms so formed
in each column and equate this sum to the last element
of the column.
Solving for
in terms of the other
relations we find
Therjif in Jp, the coefficient of the linear term in
the discriminating quartic (48) we substitue the relations
obtained above we find that
Jp =
A/4 ( t§ + t| + 2to ) rt
where A44 is the
cofactor of a 44 within A 55 . If Jpp vanishes two of the
roots of the quartic
(48) are zero and since t 0 ,t-j,tp,
-25-
are real numbers evidently A44 must vanish.
If A 44
vanishes we can write the row and column a31 a32 a 33
as a row and column of zeros.
Likewise If A 55 vanishes
(the condition that one root of the quartic vanish),the
row and column
a41 a 42 a43 a44
row and column of zeros.
can he written as a
Further,if A 55 and A44 vanish
simultaneously and hence two roots of the quartic he
zero it is necessary that
^. = 0
It is interesting to note that equation (48) has
only real roots,“ Also if we square J 0 and subtract Jq
we see that ajj =0 (i,j,= 1 ,2 ,3,4,) which is contrary
to hypothesis that the coefficients of the second degree
terms could not he zero..Looking further J0,Jq,Jq, are
invariant under rotation10* and translation1^*
Article 21.
The Hyperquadric and its Center,
Let the center of the hyperquadric
center) he
(y^)
(if it has a
(i=l,2 ,3,4,5,) and let us translate
9.Kov/aleski ,G. Einfuhrung in Die Determinenten Theori<
pg 126
10.Ibid page 245, Boclier,Maxime, Higher Algebra,pg. 154
ll.Schoute Dr.H.P.
Mehrdimensionale Geometrie,pg.131
-26-
By eliminating (yi) from (49) and (51) we obtain
If A 55 / 0 and A. = 0, thei\E = 0 and our equation is
homogeneous of the second degree in four variables and
represents a hypercone with vertex at the origin.
The
vertex of the hypercone and the hyperquadric are then the
same.
If A = 0 and E / 0, then
= 0.
Since
(y^)
(i=l,2,3,4,) was assumed to be a finite point,
= A53
= A<so = A53
=
0
(see Art I S ) ,
then A 54
the system (44) is
of rank three and the surface has a hyperplane of centers,
or it may be of rank two and the surface have a plane
of centers.
I f A = E=Aj55 = 0 the surface is composite and the
equation is factorable into the equations of two hyper­
planes.
Article 21.
The Discriminating Q,uartic again.
Let the equation of the quadric be so reduced that in
-27-
the homogeneous form it i
+ 2 X s ^jXjr ** 0. The
s
/
/
discriminating quartic now becomes
From this fact we can say further that if we translate
so that the center is taken as the new origin the equation
becomes
criminating quartic are the coefficients of the squared
terms and agg becomes
Article 22,
4/.A 55
if A 55 / 0 .
Reduction to Type Forms.
From the previous article we see that the hyper­
quadric can be written in the form
where the
are the roots of the discriminating quar­
tic and
The problem of reduction to type forms is divided
into several cases.
In our study we shall designate
each type by a number and rather that characterize each
by a name we shall characterize each by its projecting
*
cylinder in one of the co-ordinate S 3 s.
A tabulation
of these types will be made at the end of the section.
-
28 -
In the discussion of the various causes we shall use kj
to designate the absolute values of the roots of the
discriminating quartic for the sake of brevity and
ease of discussion.
Case 1.
If all the roots k^, k 9 , k;3, k 4 , are
different from zero and distinct, with A >
0 the
equation reduces to
»
If A were negative we would have obtained the same
types but slightly
shifted in their position with res­
pect to the co-ordinate system, i,e.,
type 1 , going over
into type 5, type 2 going over into type 4, a,nd type 3
going over into itself.
Case 2.
If in case 1, two of the roots,
say k-j and
k 9 , were alike we would have five additional types.
( A s t i l l positive and / 0 )
-29-
Case 3.
If in case 1, three of the roots say kq,
kp, kg, were alike we would obtain four additional cases
(^.still positive and
Case 4.
/ 0 )
If in case 1, all the roots are alike
(A>0 , A./ 0 )
we would obtain two types.
Case 5.
If in the four cases above A w e r e
each of the types
would become a cone
zero,
(hypercone) with
vertex at the origin.,
If A is negative we obtain in general,
the same
surfaces but differently situated with respect to the
axis. In the tabulation at the end of the section
this
possible change will be noted.
Case 6 .
If three of the roots are all distinct and
/ 0 while one is zero say k^, we obtain the types
where 845 =
2
allowing for the change of
9
sign in the kj s.
If any of the roots of the quartic
are equal the above types will show surfaces of revolution
in the projecting cylinders as we discussed then in cases !-«<
-30-
Case 7.
If in case 6 , IC4 = 0 a n d A = 0, we could
determine the type forms if we could evaluate the in­
determinate formz^k^kpk^k,! = 0/0.
The method of
evaluating the indeterminate form is developed in
section 8 , article 27,
The type forms resulting,
when C is the constant
obtained from the evaluation
of A/k-| kpk;r;k4 = 0/0 are
Again, if any of the kj s are equal, the pro­
jections or sections will he surfaces of revolution in
s3.
Case 8 .
When three roots of the discriminating
quartic are zero, two of the diametral hyperplanes
are undetermined since the third order determinant
apo a ^ i must vanish in this case.
A short study of the discriminant A. shows that
if we choose k 4 = k% = 0 , that by making six suc­
cessive two-dimensional rotations ( if all the co­
efficients of the second degree terms^in the products
of the variables taken two at 9. time, are not zero)
we can simplify the equation so that A n o w becomes,
-31-
after two translations to remove two of the linear
terms
Our equation accordingly has been reduced to the
k1x1 ++
2
k 2a35X 3X 5 + 2a4 5 x 4 x 5 = 0
form
where k^ and kp are the roots of the discriminating
quartic*
If both a *55 and ai^ are zero we have a two dim­
ensional cylinder, if either a,35 or
are zero
but
not both zero, we have a three dimensional quadric
cylinder in S4 .
Using then k j
and kp as the absolute values of
the roots of the quartice we can distinguish the types
-32-
Case 9.
If three roots of the discriminating quartic
are zero, the second degree terms for a perfect
square and the equation of the surface takes the form
If the hyperplanes
are not parallel, we may choose/^so that they are
perpendicular*
Taking these hyperplanes, f h e s / i s
so chosen as the Xq = 0 and xq = 0 hyperplanes,
the
equation reduces to
If the hyperplanes are parallel A%ay he so chosen
that
becomes
(<*15 ~ai^) = 0
(i=l,2,3,4,)
and the equation
32 (i)
Characterization of Type Forms.
We shall characterize the type forms by their
9
section in the various co-ordinate S 3 s forming the
co-ordinate System. In our tabulation vTe shall use the
following symbols.
= Real Ellipsoid
E
1
= Imaginary Ellipsoid
= Hyperboloid of one sheet
h2
= Hyperboloid of two sheets
Si = Imaginary Sphere
s
= Sphere
EP = Elliptic Paraboloid
HP = Hyperbolic Paraboloid
cr
= Real Q,uadric Cone
=
O
Imaginary Quadric Cone
EC = Elliptic Cylinder
HC = Hyperbolic Cylinder
PC = Parabolic Cylinder
E lc = Imaginary Elliptic Cylinder
2p = Two planes.
If an R is placed after any of these symbols it
indicates that the surface is a surface of
revolution in that S-z.
32. (ii)
Characterization of Type Forms
x1 = k
X2 = K
x3 = k
x4 = k
A >0
Ei
Ei
Ei
Ei
A <0
E
E
E
E
A >0 H
1
H1
E
A <0 H
2
H2
H1
H2
H1
H2
Hp
h2
H2
Type
1.
2.
>o
1
H
Ei
3.
<o
H1
>0
Ei
h2
<o
E
H1
H1
H1
>0
E
E
E
E
<o
Ei
Ei
Ei
Ei
A >0
Ei
Ei
E
EiR
EiR
ER
ER
HlR
ER
A
4.
1
H
Hp
5.
6.
<o
E
A >0 H
1
1
H
7.
<o
H2
H2
H2R
EiR
A >0
H1
H1
H2R
H2R
A<o
Hp
H2
HiR
H!R
A>0
E
E
ER
ER
A<0
Si
Si
E^
EiR
A >0
Ho
h2
E fR
HpR
A <0
%
%
ER
HXR
A<0
Ej-R
Ej_R
EiR
Si
A<0
ER
ER
ER
s
A>0
% R
H1R
HiR
s
A<0
HpR
HpR
HpR
si
A>o
ER
ER
ER
S
A<o
E^R
EiR
EiR
s
8.
9.
10.
11 .
12
13.
32,(iii)
Type.
X1 = k
x2 = k
X3 = k
>0
X4 = k
H 2R
Si
H1R
s
Si
Si
s
14.
<0
H1R
>0
Si
<0
s
>0
s
s
s
s
<0
Si
Si
Si
Si
17.
EP
EP
EP
Ci
18.
HP
HP
HP
Cr
19.
EP
HP
HP
cr
20.
EP
EP
EP
Ci
C>O
EC
EC
EC
Ei
C< O
E iC
E iC
EiC
E
C>0
HC
HC
EjC
Ha
C<0
HC
HC
EC
Hi
C>0
EC
HC
HC
Hi
C<0
EjC
HC
HC
Hp
oo
EC
EC
EC
E
C<0
E 1C
s iG
EiC
Ei
25.
PC
PC
PC
EC
26.
PC
PC
HP
HP
27.
PC
PC
.EP
EP
28.
PC
PC
2p
-
29
PC
PC
2p
-
Si
15.
S
S
16.
21 .
22.
23.
24.
Other types are trivial.
-33-
Section 6 .
Generation of Hyperquadrics in S 4 .
Article 23,
Methods.
When studying the hyper­
quadric in S 4 it was quite natural that we should in­
quire about methods of generating the surface.
The meth­
od that first suggested itself was that of extending
some of the loci problems of two and three dimensional
space to the analgous problem in four-space.
Both
special and general surfaces resulted.
(1 )
.
The first method was that of finding the locus
of a point moving equidistant from a point P as
(0 ,0 ,0 ,a) .and from
an S 3 as X 4 =-fc.This locus is
x| + x 2 + x§ - 4aX 4 = 0
(2)
.
The locus of a point moving so that its
distance from a point say (-ae,0 ,0 ,0 ,) and its dist­
ance from a given S 3 , say Xj = -a/e are in a con­
stant ratio e, we obtain
x 2/ a 2 + (xo + x§ + x|)/a2 (l - e 2) = 1
(3).
(a) The locus of a point moving such that
the sum of the squares of its distances from the
points (ia,0 ,0 ,0 ,) be a constants resulted in a
hypershpere.
-34-
(Id )If we had ashed for the
difference in­
stead of the sum in (a) we would have obtained
X1 = C/-4a
(c) If dj and do designate the respective
distances in (a) and we require
kdj + ldo = C
we
obtain a hypersphere where k and 1 are real numbers.
(d) If in (a) we had taken the eight points
symmetric about the origin we would also have obtains
a hypersphere.
(4)
(a) The locus of the point moving such
Poir^xs
that the sum of it distances from twoA {± a, 0 ,0 ,0 ,)
is constantly 2k gives us the form
(b)If the difference is desired in (a)
instead of the sum we obtain the same form of the
equation.
(5)
A.The locus of the point moving so that its
distances from a point and a line have a constant
ratio gives
where the line is
= 0 , Xp = 0 , X 3 = 0 and the
point is (a,0 ,0 ,0 ,).
This equation can be discussed for the cases
where e is =,>,or< 1 .
B.If the line is the general line
- 35-
and the fixed point be (bi)
(i=l,2,3,4,)
the equation
becomes
which is the equation of a general hyper quadric*
(6).
The locus of the point that moves equi­
distant from .the two lines
is given by the equation
which is the equation of a general hyperquadric.
(7).
The locus of a point moving so that the distance
dq from a point say P= (a, 0 ,0 ,0 ,) and the distance
(dp)
from an S 3 say Xq = 0 have a constant ratio e, so that
dq
= e&p, is represented by the equation
(8 )
The loeus of a point moving such that the
*
sum of the squares of its distances from tv/o S 3 s be
constant is a hyperquadric surface.
byperplanes as
the equation of the locus becomes
If we take the
-36-
(9).
If we require that a line move such that
four fixed points on it remain,
one in each of the
9
four S 3 s forming the rectangular co-ordinate system,
and if the points are A1 , A2 , A 3 , A 4 , with distances
a1 ,a2 ,a 3 ,a4 ,
respectively from p(x*|_ , x
p ,X 3 , X 4 ,
) , the
locus of P becomes
(10).
This is the method of two related project­
ive pencils of hyperplanes.
and similiarly
Bx = 0,DX = 0, Ex = 0 he the equations of four hyper­
planes. Let us write the two pencils of hyperplanes
and require that they he related thus
m =(rn + s)/(tn + v)
where rv-st / 0
Ax + (rn + s)/(tn + v) Bx = 0
then
and this gives
n = - ( vAx + s Bx )/(tAx + rBx ) . Prom the second pen­
cil we obtain the quadratic in Xj
How if we denote this surface hy Q and different­
iate with respect to the x^ in turn we obtain the
four equations of the form
These diametral hyperplanes interesct in appoint
-37-
which is common to the four given hyperplanes, and since
the point satisfies the given hyperplanesit also
satisfies the hyperquadric.
a cone
Hence the hyperquadric is
(hypercone).It is also quite evident that there
9
are So s on this hypercone.
(ll).
(10).
This method is closely related to that in
Let thercbe two planes F-j and Pp and a line P 3
given by the equations
where
(59) in the S 3 determined V r <y) and Pj and (60)
is the S 3 determined by(y)and Pp.
between these equations
-38-
el
e2
e3
fl
Si
S3
e4
e5
f4
fR
Sd
S5
which, yields a second degree equation in
=
and it locus is
a hyperquadric surface.
We can show that this surface is a hypercone hy the
same argument used in (10)
0
-39-
Section 7.
Tangent Hyperplanes and Polar Hyperplanes.
Article 24.
Tangent Lines and Tangent Hyperplanes.
If, when the equation of the hyperquadric P(x-j ,xp,X 3 ,X4 ,)=0
(see Art 16) is solved with the line x^ = yj + 1 jr through
the point P 0 (y j) we require that the point P 0 lie
on the surface, it is necessary that "both roots of
the equation (41) vanish.
Since 5*(yq*yp»73 ,7 4 ,) = 0
because P 0 is to lie on the surface, we need only to
require that (eq 42)
R = l/22]8S,
/ ay1 )li = 0 ........................ (61)
/
If from the equation of the line we substitute the
values of 1 ^ in (61) we obtain
S
2 x i"yi) (9F/0yi) = 0 ............... (62)
/
which must be satisfied by the co-ordinates of every
point of every line tangent to the surface at P0.
Conversely,
if (xj) is any point distinct from P0
whose co-ordinates satisfy S in (42), the line deter­
mined by (xj) and P 0 is tangent to the surface at P0.
Since
(62) is of the first degree in(xj) its equation
represents a hyperplane and is called the tangent
hyperplane at p0.
The equation (62) of the tangent hyperplane may be
simplified by multiplying out the equivalents of the
-40-
partial derivatives and adding a15y 1 + a25y 2 +
a 35yj + ^4 ^ 4
+ a 55y 5 to each side of the equation
and transposing the constant twrm to the right hand
side,
then we obtain
Article 25.
Polar Hyperplanes,
Two points P and
R are said to be conjugate with regard to a hyper­
quadric surface when they are divided hamonically by
the points where the line connecting them meets the
surface.
If through P and R a line
is drawn inter­
secting the surface in two points Q, and S, the line
given by the equations,
where (y^) andfz^)
respectively.
quadric
xj =
+ kyj
(1=1,2,3,4,5,)
are the co-ordinates of P and R
Solving this line with the hyper­
(39) and requiring the the cross-ratio (PQ,RS)
be -1, i.e., the condition that the roots in k,
kp = k p
we obtain
This equation is linear in xq and is called the polar
hyperplane* It is the locus of R.
-41-
Article 26.
Relations of Hyperplanes
and Planes
to a Hyperquadric surface.
If we have a tangent hyperplane to the hyperquadric
t
each given by the
respective equations
the equation of the polar
hyperplane of (yj) coincides
with the tangent hyperplane, which algebraically,
states
that the coefficients of the two linear equations
(66) and
(68) are proportional and
and if (y^) lies on the
hyperplane
(66) as we stated
in the hypothesis then
If the hyperquadric is of the cone type,
not be a true tangent hyperplane.
(66) will
If (y^) is the
vertex and since it lies on (66) equation is fulfilled.
Since equation (68)
is identically zero, equations
( 69)
will be fulfilled if we let t = 0. Thus if (66) is a
-42-
tangent plane to (67) there exists six constants yq,
yptY^fY/ifYsfk of which the first five are not all
zero and which satisfy the equations
(69) and 70)
i.e., the condition is fulfilled
The converse of (71) can readily
can state the proposition: Equation
be shown so that we
(71) is the
necessary and sufficient condition that the hyperplane
(66) be tangent to the hyperquadric
(67)
how if the three-fold (66) and that g :ven by the
equation
intersect in a plane
(So) and if this plane be tangent
to the hyperquadric (67) then there is a point
(i=l,2,3,4,
,) the point of contact,
(yq)
such that its
polar (68) contains the plane*. Then it mustbe roossible
to write the equation of this polar plane
how we may choose m and n so that the coefficients of
(73) may be proportional to the coefficients of (68),
-43-
and further,
the corfficients of (73) can he equal to
those of "68) so
that
(j—1,2,3,4,5,)
But
(y^)
(i=l, 2 , 3,4,5,) lie "both on (66 ( and <72)
so that
Since we have found seven constants,
(yj)
(i=l,
2,3,4,5,) and m, n, not all zero, then surely
This equation was deduced on the supposition
that the plane of intersection of (66) and (72) is a
true tangent to (67).
But in the same manner as we
argued for equation (71) for the case of a psuedo
tangent we can prove (76) holds if the plane is a
-44-
psuedo tangent or a ruling of (67).
also quite evident.
The converse is
Ourproposition then becomes:
The necessary and sufficient condition that the plane
of intersection of the hyperpla.ne (66) and
tangent plane
fulfilled.
ruling
of (67) is that
(72) he a
(76) he
-45-
Section 8 .
Some Theorems, Proofs,
and Numerical Examples.
Article 27.
An Indeterminate F orm Evaluated.
In case 7 Article 22 Scetion 5 a certain indeter­
minate form was mentioned: A/k-jkok^hA where A- 0 and
^ 4= 0 .
To evaluate
this form we take recourse to a
method of differential calculus.
It is quite evident that if we insert a cr in any­
one of the elements a-j^, a^o, agj, a ^ ,
of A. or k^, kg,
, k^, of the transformed 4 > that neither t h e A ^ o t 1 A 55
will vanish hut yet approach zero a s ^ a p p r o a c h e s zero*
Since we can choose any of the k^ s to he the root that
is zero let us choose kA as that root and accordingly
insert (fin the position of this element.
The we have
-46-
To establish the validity of this limit we must show
that the denominator cannot vanish.
Prom the conditions
that one root of the discriminating quartic "be zero y/e
have J? / 0
( See eq 48 pg 24).
But since A 55 = 0 we
can write
Prom these relations we find that
whence,
our limit must exist when </is placed in the
position of
Likewise it can he shown that ifc^is
placed in any of the other positions our result is still
true.
By the same token we can prove that this method will
-47-
not work for the case of two roots zero for then Jp = 0
and we have shown that la^j app
vanishes for this
.
case ( see art 20 ) and the limit does not exist.
12
Article 28. Miquel Theorem.
In space of four
dimensions., all three dimensional quadrics which pass
through eleven general points are expressible linearly,
when their equations are in point co-ordinates, by four
linearly independent quadrics parsing through these points.
All such quadrics therefore pass through five other points.
It is assumed of course that these four quadrics have no
common curve or surface and intersect in sixteen points.
Article 29.
Isotropic Lines.
Darboux calls the centers
of the two spheres of zero radius which pass through a
13.
circle in S 3 foci.
In like manner, in S 4 ,
the foci of
a hypersphere may be defined as the centers of two
hyperspheres of zero radius which pass through the sphere.
If two spheres lying in the same S 3 touch, then the line .
joining a focus of one to one of the foci of the other
is
an isotropic line,i,e., a line of zero length.
Article 30. On an Hyperquadric In S 5 . A hexahedron is
defined as that configuration formed by six points in a
space of five dimensions where each of the five points
9
determine a S 4 and the six points and six S 4 g are called
respectively the vertices and faces of,the hexahedron.
12.
13.
Baker H*F. Principles of Geometry Vol 4, page 9.
Bateman H. British Association for Advancement of
Science, 80th meet ,1910 pg 532-33
-48-
H*W.Richmond
14.
proves an interesting theorem:
If in
S 5 a quadric passes through all the vertices of a hexa­
hedron and touches all of its faces, it must touch the sixth
face also.
This theorem, algebraically,
states that if in
a symmetrical determinant of order six^, the elements
in the leading diagonal all vanish and the first minors
of five of these elements vanish,
the minor of the remaining
element must vanish.
Article 31.
On the Condition that Five Straight Lines
in S 4 Should lie on a Quadric.
9
A Quadric in S 5 has on it an infinite number of S 9 s
which form two distinct families, an
S 9 from each family
passing through every line that lies on the quadric. Two
9
So s from the same family always have a point in common
9
while two Sp s from different families have no common point
at all,generally>but have a line in common.
If we require five lines a,b,c,d,e,
to lie on a
quadric Q in S 5 we arrive at the theorem: In order that
five lines in S 4 should lie orj/a quadric, it is necessary
and sufficient that they should be sections of five
,
14.
Sp s in S 5 of which each two have one point in common.
Article 32.
Quadric Point Cone.
In space of three
16.
dimensions we have the two definitions of a conef
the
envelope of its tangent lines and the dual, the quadric
14. Richmond H,W. Proceeding of
ambridge Philos,Soc.
Vol.14 pg.475
15 Ibid vol 10 pg 210-13
16. Baker H,F, Principles of Geometry Vol 4.pg 121
49-
cone, consisiting of of its generating lines passing
through its vertex.
Ino-particular solid lying in a space
of four dimensions we may have an ordinary quadric sur­
face.
We may treat it as two sets of generating; lines,
where ajiy line of either set meets every line of the
second set.
The dual will be two sets of planes all
passing through some point V, so that any plane of
either set meets every plane of the other set in a
line, passing through V, with two planes of the same
set meeting in V only.
Through every general line
lying in a quadric surface in four dimensions there passes
two planes,
each containing two lines of the quadric
surface* two points on this
of the four generators meet.
line are those where two
Dually, an arbitrary plane
through V contains two lines passing through V, each of
which is the intersection of a plane of one set with a
plane of the other set* and there are two solids con­
taining this plane each of which contains two
four planes so arising.
of the
The aggregate of points lying
in these sets of planes is called a quadric point cone
with vertex V.
Just as two intersecting generating
lines determine a point of the quadric surface and a
tangent plane,
so also,
two generating planes which meet
in a line lie in a solid, the tangent solid of the point
cone, the line of contact of this tangent solid v;it>/the
-50-
point
cone being the line of intersection of the planes.
There are a double infinity of tangent solids,
of contact of each passing through the vertex.
the line
It is
clear that the generating planes of the point cone meet
in an arbitrary solid in the generating lines of a
quadric surface lying in that solid.
Conversely,
the
planes joining a point, not lying in the solid in which
the quadric surfa.ee lies, to the generating lines of a
quadric surface, generate a point cone.
17.
Article 53.
Quadric line Cone.
The dual of the
tangent lines of a conic in S 4 consists of the single
infinity of planes passing through a line 1 , two of these
planes lying in an arbitrary solid which contains the
line 1.
The aggregate of the points of these planes is
calles a quadric line cone.
17.
Baker H.F. Principles of Geometry,vol 4 pg 121
-51-
Article 34. Kunerical Cases.
(l)
Classify the hyperquadric given by the equation
equation becomes
(2) Classify the surface
The surface is composite since
(3) Classify the surface
surface becomes
(4) Classify the surface
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(5)
Classify the surface
2yz + 4x + 2y + 4z + 2t + 1 = 0
Her e
- 0 and the quartic becomes
k 4 -k2 = 0 with
roots k = 0,0,1,-1
Rotating by the formulas
x = x
y = ycos 45 - zcos 45
t = t'
z = ysin 45 + z cos 45
we obtain
and on completeing squares and translating
y " 2 - z"2 + 4xn + 2t" = 0
(6)
Classify the surface
x 2 + y 2 + 2yz + 4x + 6y + 5 = 0
By the method of article 27
The equation becomes
-53-
IS.
(7)
Given in S 4 a plane and a line parallel to this
plane represented by the equations in homogeneous co­
ordinates.
Let us determine the equations of the plane, which
is the locus of the line containing the distance, and let
us find the length of the distance from the line to the
plane.
It is immediately seen that the given plane contains
the infinite point of the line given by the Equation
The S 3 given by the equation
through the point
(2 ,4, 6 ,0 ,1 ) of the given
k = -7/12 since the line
line for
(Sq) may be written in the
form
of the S 3 relating the given elements, i.e., it is the
S 3 obtained by substituting k = -7/12 in
IS.Exercises 7,8,9, are taken from Schoute,H.P. Mehrdimensionale Geometrie page 171
-54-
Therefore,
the perpendicular from 0 to this S 3 .
How each line con­
taining the desired distance while it lies in this space
is perpendicular to this normal.
Moreover it is perpen­
dicular to the given plane and consequently also to the
S* given by
which is obtained by choosing the points
S 3 a 7 so passes through this perpendicular parallel to
the given plane. S 3 reduces to
line containing the distance from 0 to S 3 . This per­
pendicular is parallel to the given line.
given line and the infinite point of this normal "becomes
the locus of the line containing the distance.
-
55 -
And the length of the distance will he found as the
(2 ,4, 6 ,0 ) from the S 3
distance of the point
through the point (3,4,0,0,)
of the givei^lane per­
pendicular to the line containing the distance.
equation (i) is in the
upon substituting
(8 )
When the
normal form we have
(2 ,4, 6 ,0) for the variables.
Determine the projection of the given line
upon the given plane in (7)
From (7) the locus of the line containing the
distance is given by the planes
The plane given in the problem is
Dow Sq is parallel to Sq of exercise (7) and we
/
find that S-j is the section of the given plane with
the locus of the line containing the distance.
(9) Determine the distance between the parallel
lines
of exercise
Let us write
(7) and Sq of exercise
(8 ).
-56-
and then write
>
as the system of S 3 s per­
pendicular to the two;
parallel
lines and passing through
(2 ,4, 6 ,0 ) which gives us
how (1 ) and
determine the
point
(27/12,49/12,67/12,19/12) and hence
Article 34, exercise 10.
As a three-dimensional exercise illustrating the
generalized theorem of Schoute, mentioned in article 4
let us find the distance from the point
(2 ,2 ,2 ,) to the
plane x-2y-z+ 3= 0.
The equation of the normal to the plane through 0
is x/l = y/-2 = z/ -1
while the equation of the line
along which the distance is measured is
line along which the distance is measured.
intersects the plane in the point
This line
(13/6,5/3,11/6). The
distance between the two points is c/l2/ l 2
Bibliography
I*
Books referred to
in this thesis.
Baker, H. F. "Principles of Geometry? Vols,l-4,
Cambridge, University Press, 1922
Bttcher, Maxime,"Higher Algebra?
Hew York,Macmillan,1924.
Kowalewski, Dr. Gerhard,
"Einfuhrung in de
Determinenten Theorie”,Leipzig, von Veit,1909
Schoute, Dr. P. H. "Mehrdimensionale Geometrie"
1. teil, Die linearen Raume, Leipzig, Goschsche, 1902
Woods, Frederick, Higher Geometry, Hew York,
Ginn, 1922.
II. Periodical articles referred to in this thesis.
Batemen, H. "Foci of a circle in space and some
geometrical theorems connected therewithM
British Association for the Advancement of
Science, 80th m et, Sheffield,1910, Pp. 532-33
Richmond, H. W, " Condition that five straight
lines situated in S a lie on a quadric", Pro­
ceedings of Cambridge Philosophical Society,
Volume 10, pp. 210-13
Richmond, H.W, ” On the property of a double six
of lines and its meaning in hypergeometry”,
Proceedings of Cambridge Philosophical Society,
volume 14, pp475.
Ill, Other hooks consulted,
Jessop, C, M.
” A Treatise on the Line Complex” ,
Cambridge'*. University Press, 1903,
Jouffret, E. "Traite Elementalre de Geometrie
A Quatre Dimensions”, Paris, Macmillan, 1924,
Manning, Henry Parker,
’’Geometry of Pour Dimensions”
Hew York, Macmillan, 1914,
IY.
Other periodicals consulted,
9
Bromwich, T. J. I a,
” Cla.ssifica.tion of Quadrics”
Transactions American Mathematical Society, Hew
York, Volume 6, pp 275-85
Burnside, W.
” On a Configuration of 27 Hyper­
planes in Space of Pour Dimensions”, Proceedings
of Cambridge Philosophical Society, Cambridge,
Volume 15, 1909 pp 71-75
Cayley, Prof,
” On the Superlines of a Quadric
Surface in Five Dimensional Spa.ce”, Quarterly
Journal of Mathematics,London, volume 12,
(1873) pg. 176
Coolidge, J, L. Annals of Mathematics,
” On the
Condition that a Line Intersect a Quadric",
second series, volume 13, pg 3 et seq.
Princeton, 1917.
Franklin, Philip,
"The Classification of Quadrics
in Euclidean n-Space hy means of Co-variants.
American Mathematical Monthly, volume 34, pp453.
Greene, G. M.
"The Intersections of a Straight
Line and a Hyperquadric", Annals of Mathematics,
volume 19, Lancaster,Pa. and Princeton, 1919
page 207
Moreno, H. C.
" On Ruled Loci in Space of n-
Dimensions", American Academy of Arts and S
Science, Volume 37,pp 121 - 157, Boston,
Academy, 1902.
Richmond, H. ¥.
" Figure formed from Six Points
in Space of Four Dimensions", Mathematische
Annalen, volume 53,pg 161, Leipzig, Tuebner,1900
Schoute, Dr. P.H.
" Angles of Regular Polytopes
of S 4 ." American Journal of Mathematics,
volume 31, pp 303-10
Baltimore, John Hopkins,
1909.
Sharpe, F. R.
" A Special Linear Substitution"
Bulletin of the American Mathematical Society,
♦
volume 17, pg 15, Lancaster, Pa. and Hew York,
1911.
Snyder Virgil,
” Condition that the Common
Line to n-1 Planes in an n-Space May Pierce
a Given Quadric Surface in the Same Space”.
Bulletin of the American Mathematical Society,
volume 4, pg 68, Lancaster Pa. and Few York,
Macmillan,1898.
Sommer, J. ” Pocaleigenschaften Qu dratischer
Mannigfaltigkeiten im Vier Dimensional Raum”
Mathematische Annalen, volume 53, pg 113,
Leipzig, Tuehner, 1900.