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22 e 23 + 10 e31 + 24 e12 1160
A§B = A§ B =
39 39

’ 60 e 23 + 60 e31 + 144 e12 27936
B §C = B §C =
169 169

’ 10 e 23 ’ 22 e31 + 24 e12 1160
C § A= C§A =
39
39

The cosines of the angles between planes are obtained through the analogous of scalar
product for bivectors:

(C § A) · ( A § B ) 136
cos ± = ’ ± = 1.6882
=’
C§A A§B 1160

( A § B ) · (B § C ) 2736
β = 2.0721
cos β = ’ =’
A§B B §C 1160 … 27936
RAMON GONZALEZ CALVET
256

(B § C ) · (C § A) 2736
γ = 2.0721
cos γ = ’ =’
B §C C§A 27936 … 1160

The area of the triangle is the spherical excess:

area = ± + β + γ ’ π = 2.6908

14.3 Let us calculate the Cartesian coordinates of Fastnet (point F), the point in the
middle Atlantic (point A) and Sandy Hook (point S). Identifying the geographical
longitude with the angle • and the colatitude with the angle θ we have:

x = sinθ cos • y = sinθ sin• z = cos θ

F = (θ = 38º 30' , • = ’9 º 35' ) = 0.6138 e1 ’ 0.1036 e 2 + 0.7826 e 3

A = (θ = 47 º 30' , • = ’47 º ) = 0.5028 e1 ’ 0.5392 e 2 + 0.6756 e 3

S = (θ = 49º 30' , • = ’74 º ) = 0.2096 e1 ’ 0.7309 e 2 + 0.6494 e3

The length of the arcs is obtained from the inner products of the position vectors:

cos FS = F · S = 0.7126 FS = 44º33' = 4950 km

cos FA = F · A = 0.8932 FA = 26º 43' = 2969 km

cos AS = A · S = 0.9382 AS = 20º14' = 2250 km

Now we see that the track is 269 km longer than the shortest path between Fastnet and
Sandy Hook.
Let us investigate whether the Titanic followed the obliged track, that is whether
the point T where the tragedy happened lies on the line AS, by calculating the
determinant of the three points:

T = (θ = 48º14' , • = ’50º14' ) = 0.4771 e1 ’ 0.5733 e 2 + 0.6661 e3

0.5028 0.4771 0.2096
A § T § S = ’ 0.5392 ’ 0.5733 ’ 0.7309 e123 = ’0.0050 e123
0.6756 0.6661 0.6494

The negative sign indicates that the
orientation of the three points is
clockwise (figure 16.6). In other words,
the Titanic was shipping at the South of
the obliged track. We can calculate the
distance from T to the track AS taking the Figure 16.6
right angle triangle. By the Napier™s rule we have:
TREATISE OF PLANE GEOMETRY THROUGH GEOMETRIC ALGEBRA 257


A§T § S 0.0050
sin d = sinσ sin ST = = 0.0144 d = 49' = 92 km
=
A§S 0.3461

14.4 After removing the margins and subtracting the coordinates of the centre of the
photograph in the bitmap file, I have obtained a pair of coordinates u' and v' in pixels,
which are proportional to u and v:

star u' v'
± Cassiopeia ’139
173
β Cassiopeia ’135
349
γ Cassiopeia 199 24
δ Cassiopeia 89 93
’240 ’320
Hale-Bopp comet

On the other hand, the right ascension A and declination D of these stars are
known data. From the spherical coordinates and taking θ = 90º ’ D and • = A one
obtains their Cartesian equatorial coordinates in the following way:

x = cos D cos A y = cos D sin A z = sin D

The Aries point has the equatorial coordinates (1,0,0). Using these formulas I have
found the coordinates:

star x y z

± Cassiopeia 0.543519 0.096098 0.833878

β Cassiopeia 0.512996 0.019711 0.858216

γ Cassiopeia 0.475011 0.119054 0.871889

δ Cassiopeia 0.462917 0.180840 0.867759


A photograph is a central projection. So the arch s between two stars A and B is related
with their coordinates (u, v) on the projection plane by:

1 + u A uB + v A vB
cos s AB =
2 2 2 2
1 + uA + vA 1 + uB + vB

The focal distance f is the distance between the projection plane (the photograph) and
the centre of projection. When the focal distance is unknown we only can measure
proportional coordinates u' and v' instead of u and v, and the foregoing formula
RAMON GONZALEZ CALVET
258

becomes:

f 2 + u' A u' B + v' A v' B
cos s AB =
2 2 2 2
f 2 + u' A + v' A f 2 + u' B + v' B

which is a second degree equation on f 2:

(f )( f )
+ u' B + v' B cos 2 s AB = ( f 2 + u' A u' B + v' A v' B )
2
2 2 2 2
+ u' A + v' A
2 2




[ 2 ( u' ]
( )
0 = f 4 (1 ’ cos 2 s AB ) + f u' B + v' A v' B ) ’ u' A + v' A + u' B + v' B cos 2 s AB
2 2 2 2
2
A



( )( u' )
+ ( u' A u' B + v' A v' B ) ’ u' A + v' A
2 2 2 2 2
+ v' B cos 2 s AB
B



All the coefficients of the equation are known, because cos sAB = A · B is calculated
from the Cartesian equatorial coordinates obtained from the right ascension and
declination of both stars. The focal distances so obtained are:

star f
± -β Cassiopeia 2012.7
± -γ Cassiopeia 2010.6
± -δ Cassiopeia 2017.2
β -γ Cassiopeia 2003.0
β -δ Cassiopeia 2019.1
γ -δ Cassiopeia 2049.1
mean value 2018.6

The width of the photograph is 750 pixels and the height 1166. If 750 pixels
corresponds to 24 mm and 1166 to 36 mm, we find a focal distance of the camera equal
to 64.6 mm and 62.3 mm respectively. However the author of the photograph indicated
me that his camera has a focal distance of 58 mm, what implies that the original image
was cut a 7 % in the photographic laboratory. This is a customary usage in photography,
so we do not know the enlargement proportion of a paper copy and we must calculate
the focal distance directly from the photograph, which is always somewhat higher than
that calculated from the focal distance of the camera. On the other hand, the calculus of
the focal distance is very sensitive to the errors and truncation of decimals. The best
performance is to write a program to calculate the mean focal distance from the original
data or to make us the paper copies without cutting the image. Known the focal
distance, we are able to calculate the normalised coordinates u and v:

star u v

± Cassiopeia ’0.068860
0.085703
TREATISE OF PLANE GEOMETRY THROUGH GEOMETRIC ALGEBRA 259


β Cassiopeia ’0.066878
0.172892

γ Cassiopeia 0.098583 0.011889

δ Cassiopeia 0.044090 0.046072

’0.118894 ’0.158526
Hale-Bopp comet

and from here the inner product between stars and the comet:

1 + u A uH + v A vH
cos( A, H ) =
1 + u A + v A 1 + uH + vH
2 2 2 2




1 + uB uH + vB vH
cos(B, H ) =
1 + uB + vB 1 + uH + vH
2 2 2 2




The unknown equatorial coordinates of the comet H are calculated from a system of
three equations:

± A · H = cos( A, H )

 B · H = cos(B, H )
 H 2 =1


Let us introduce the Clifford product instead of the inner product and multiply the first
equation by B and the second one by A on the right:

+H A AH B+H AB
±AH
= cos( A, H ) ’ = cos( A, H ) B
 2 2
B H +H B B H A+ H B A

= cos(B, H ) ’ = cos(B, H ) A

2 2

H 2 =1




u v w “ w v u = 2 u § v § w), the
Using the permutative property (in the space
difference of both equations is equal to:

A § H § B + H A § B = cos( A, H ) B ’ cos(B, H ) A

’ H § A § B + H A § B = cos( A, H ) B ’ cos(B, H ) A
or equivalently:

The outer product by H is the geometric product by its perpendicular component H⊥ , so
we have:

H || A § B = cos( A, H ) B ’ cos(B, H ) A
RAMON GONZALEZ CALVET
260

where we can isolate the coplanar component:

H || = (cos( A, H ) B ’ cos(B, H ) A) ( A § B )
’1




Observe that this result can be only obtained and expressed through the
geometric product, and not through the more usual inner and outer products. The
component perpendicular to the plane AB is proportional to the dual vector of the outer
product of this vectors. Known the coplanar component we may now fix the modulus of
the perpendicular component because H2=1:

e123 A § B
H⊥ = ± 1 ’ H ||2
A§B

There are two solutions, but according to the statement of the problem only one
of them is valid. Since each pair of stars gives two values of H (the position vector of
the comet), we may distinguish the true solution because it has proper values for
different pairs of stars. The mean value with the standard deviation so obtained is:

H = (0.661606 ± 0.000741) e1 + (0.236258 ± 0.001298) e 2 + (0.711659 ± 0.000302 ) e3

from where the following equatorial coordinates of the Hale-Bopp comet are obtained:

AH = 1h 18min 36s D H = 45º 22'12' '

In the ephemeris (http://www.xtec.es/recursos/astronom/hb/ephjan97.txt) for the comet
we find:
Date Right ascension (A) Declination (D)
1997 March 28 01h 09min 18.03s +45º 36™ 30.6”
1997 March 29 01h 19min 06.07s +45º 25™ 15.6”
1997 March 30 01h 28min 44.92s +45º 10™ 51.9”

So I conclude that this photograph was taken the March 29 of 1997.
On the other hand, I have also calculated the orientation O of the camera (of the
line passing through the centre of the photograph):

O = (0.513505 ± 0.000673) e1 + (0.201570 ± 0.000471) e 2 ± (0.834075 ± 0.000360 ) e3

AO = 1h 25min 44s DO = 56 0 31'11"


15. Hyperboloidal geometry in the pseudo-
Euclidean space (Lobachevsky™s geometry)

15.1 The circle being projection of a parallel line (geodesic
hyperbola) shown in the figure 15.9 passes through the
point (1, 0) which implies the equation:
TREATISE OF PLANE GEOMETRY THROUGH GEOMETRIC ALGEBRA 261



(u ’ 1)2 + (v ’ b)2 = b 2

According to this equation, the intersection of this line with the v-axis has the
)
(
coordinates 0, b ’ b ’ 1 (the other intersection point falls outside the Poincar©
2


disk). The inverse of the slope at this point with opposite sign is the tangent of the angle
of parallelism: Figure 15.9

« du  1
tg Π (s ) = ’¬ · = b 2 ’ 1 cos Π (s ) =

 dv  u = 0 b

Now let us calculate the distance s from the origin to this point. Its hyperbolic cosine is
the inner product of the unitary position vectors of the origin and the point of
intersection:

«« 

b ’ b2 ’ 1 b b
¬¬ e3 · =
·e +
cosh s = e 3 ·
¬¬ ’ b2 + 1 + b b2 ’ 1 · 2 b2 ’ 1 · b2 ’ 1
  

1 ’ cos Π (s ) Π (s )
b ’1
b +1
exp(’ s ) = = = tg
s = log
1 + cos Π (s )
b +1
b ’1 2

so we find the Lobachevsky™s formula for the angle of parallelism:

Π (s ) = 2 arctg(exp(’ s ))

15.2 The law of cosines in hyperboloidal trigonometry is:

cosh a = cosh b cosh c ’ sinh b sinh c cos a

For a right angle triangle ± = π/2 :

cosh a = cosh b cosh c

In the limit of small arcs it becomes the Pythagorean theorem:

« 4 « 
+ O (a ) = ¬1 + + O (b )· ¬1 + + O (c 4 )·
a2 b2 c2
1+ 4
¬ ·¬ ·
2 2 2
  

a 2 + O (a 4 ) = b 2 + c 2 + O (b 2 c 2 , b 4 , c 4 )

15.3 A «circle» is given by the equation system:
RAMON GONZALEZ CALVET
262

« z2 ’ x2 ’ y2 =1
¬
¬a x + b y + c = z


a) The intersection of a quadric (hyperboloid) with a plane is always a conic. Since the
curve is closed, it must be an ellipse (for horizontal planes it is a circle).
b) We hope that the centre of the ellipse be the intersection of the plane with its
perpendicular line passing through the origin (this occurs also in spherical geometry).
The plane has a bivector:

a e 23 + b e31 ’ e12

The vector perpendicular, the dual vector is:

a e1 + b e 2 + e 3

because the product of the vector and bivector is equal to a pure volume element:

(a e 23 + b e31 ’ e12 ) (a e1 + b e 2 + e3 ) = ( a 2 + b 2 ’ 1) e123

The axis of a circle is the line passing through the origin and perpendicular to the plane
containing this circle. The centre of the circle is the intersection point of the plane and
axis of this circle, given by the equation system:

±a x + b y + c = z
 ac bc c
’ x= y= z=
 x=y=z
1 ’ a 2 ’ b2 1 ’ a 2 ’ b2 1 ’ a 2 ’ b2
ab1


The distance from any point of the circle to its centre is constant:
2
2
«  « 
«  ac bc
c
¬z ’ · ’¬x ’ · ’¬y’ ·=
1 ’ a 2 ’ b2  1 ’ a 2 ’ b2  1 ’ a 2 ’ b2
   

c 2 (1 ’ a 2 ’ b 2 )
2 c2 c2
1’ + = 1’
(1 ’ a ’ b2 )
1’ a 2 ’ b2 1’ a 2 ’ b2
2
2




Since c > 1 and a + b < 1 this value is always negative, what means that the radius is
2 2


a distance comparable with distances on the Euclidean plane x-y. Taking the real value,
the radius of a circle is:

c2
r= ’1
1’ a ’ b
2 2




However this is not the radius measured on the hyperboloid and obtained from its
projections. The intersection of the axis with the hyperboloid gives its hyperboloidal
centre:
TREATISE OF PLANE GEOMETRY THROUGH GEOMETRIC ALGEBRA 263

±z 2 ’ x 2 ’ y 2 = 1
 a b 1
x= y= z=
xyz ’
 a=b=1 1 ’ a 2 ’ b2 1 ’ a 2 ’ b2 1 ’ a 2 ’ b2


The hyperboloidal radius is the arc length on the hyperboloid from any point of the
circle to its hyperboloidal centre, which should be constant. Making the inner product of
the position vectors of a point on the circle and its centre, we find:

« 
a b z
coshψ = ( x e1 + y e 2 + z e3 ) · ¬ e1 + e2 + e=
2 3·
1 ’ a ’ b 1’ a ’ b 1’ a ’ b
2 2 2 2 2

’a x’b y+ z c
=
1 ’ a 2 ’ b2 1 ’ a 2 ’ b2

So the hyperboloidal radius is:

c
ψ = arg cosh
1 ’ a 2 ’ b2

The figure 16.7 shows a lateral view of a
plane with a = 0 and equation b x + c = z.
The points P and P' lie on the circle. O is
the centre of the circle on its plane and H
is the centre on the hyperboloid. Then OH
is the axis of the circle, which is Figure 16.7
perpendicular (in a pseudo-Euclidean way)
to the circle plane, that is, their bisector has a unity slope. Note that according to a
known property of the hyperbola, O must be the midpoint of the chord PP'. The plane
radius is the distance PO or P'O, while the hyperboloidal radius is the arc length PH or
P'H.
c) The coordinates of the stereographic projection are:

1 + u2 + v2
2u 2v
x= y= z=
1’ u2 ’ v2 1 ’ u2 ’ v2 1’ u2 ’ v2

Let us make the substitution of these coordinates in the equation of the circle plane:

2au+2bv 1 + u2 + v2
+c=
1 ’ u2 ’ v2 1 ’ u2 ’ v2

After simplification we arrive to an equation of a circle:
2 2
a 2 + b2 + c 2 ’ 1
« a « b
¬u ’ · + ¬v ’ ·=
(c + 1)2
c + 1 c + 1
 

15.4 A horocycle is a circle having a + b = 1 . Therefore its equation in the
2 2

stereographic projection is:
RAMON GONZALEZ CALVET
264


2 2
c2 « a « b
= ¬u ’ · + ¬v ’ ·
(c + 1)2 c + 1 c + 1
 

Let us search the intersection points (if they exist) with the limit circle u + v = 1 .
2 2

Then we solve the system of both equations and find that it has a unique solution:

u=a v=b

Since they meet in a unique point and a + b = 1 , the horocycle projection is tangent
2 2

to the limit circle. The centre of the horocycle is the intersection of the hyperboloid with
its axis. However this axis has unity slope, so the intersection lies at the infinity.

15.5 By differentiation of the coordinates in the Beltrami projection we find:

( 1 ’ v ) du + u v dv (1 ’ u ) dv + u v du
2 2
dy =
dx =
(1 ’ u ’ v ) (1 ’ u ’ v ) 2 3/ 2
2 3/ 2 2
2


u du + v dv
dz =
(1 ’ u ’ v2 )
3/ 2
2




The differential of area is a bivector and we search its modulus:

du § dv
(dx § dy )2 ’ (dy § dz )2 ’ (dz § dx )2
dA = =
(1 ’ u ’ v2 )
3/ 2
2




Note the outer products of the differentials
of coordinates so dx § dy = ’ dy § dx .
Now we must integrate the differential of
area of a doubly asymptotic triangle with
angle • (figure 15.10a). We change to
polar coordinates r and θ in the Beltrami
disk:

v
tg θ =
r = u2 + v2
u Figure 15.10
cos(• / 2 )
Since the equation of the line is r = the integral becomes:
cos θ

cos (• / 2 ) cos (• / 2 )
• /2 • /2 • /2 • /2
®1
r dr § dθ
cos θ
cos θ
cos θ

« « (1 ’ r ) =« « « dθ =
dθ = dθ ’
A= 

2 3/ 2
1 ’ r2 »0
’• / 2 °
’ sin 2θ
• • •
’ /2 ’ /2 ’ /2
1 ’ cos 2
0

2
TREATISE OF PLANE GEOMETRY THROUGH GEOMETRIC ALGEBRA 265

• /2
® 
 
sinθ
’θ = π ’•
= arcsin

 
sin
° » ’• / 2
2

This expression is applied to each asymptotic triangle surrounding the central
triangle (figure 15.10b). Since the area of a triangle having the three vertices in the limit
circle is 3π’2π = π, the area of the central triangle follows:

A= π ’± ’ β ’γ

15.6 In the azimuthal projection, the radius r of a circle centred at the origin is:

2 2
r 2 = u 2 + v 2 = 2 (z ’ 1)
u=x v= y
z +1 z +1

Since the projection is equivalent, we can evaluate the area directly in the plane:

A = π r 2 = 2 π ( z ’ 1) = 2 π (coshψ ’ 1)

where ψ is the Weierstrass coordinate, that is, the radius of the circle on the
hyperboloid.
RAMON GONZALEZ CALVET
266

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George A. JENNINGS. Modern Geometry with Applications. Springer-Verlag (New York,
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Felix KLEIN. Vorlesungen über nicht-euklidische geometrie. Verlag von Julius Springer


1
In this edition, the date in the preface by H. Grassmann was misprinted as 1884 instead of
1844.
RAMON GONZALEZ CALVET
268

(Berlin, 1928).
Daniel LEHMANN, Rudolph BKOUCHE. Initiation à la g©om©trie. Presses universitaires de
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Pertii LOUNESTO. Clifford Algebras and Spinors. Cambridge University Press (Cambridge,
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Edwin E. MOISE. Elementary Geometry from an Advanced Standpoint, Addison-Wesley
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Junjiro NOGUCHI. Introduction to Complex Analysis. Translations of Mathematical
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Josep Manel PARRA i SERRA. «Clifford algebras. Towards a common language for
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J. M. PARRA I SERRA. «Geometrical algebra versus numerical Cartesianism. The historical
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Janks eds. Kluwer, Dordrecht (Aachen, 1996).
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TREATISE OF PLANE GEOMETRY THROUGH GEOMETRIC ALGEBRA 269


INTERNET BIBLIOGRAPHY

Personal pages

Rafal Ablamowicz http://math.tntech.edu/rafal/
William E. Baylis http://www.cs.uwindsor.ca/users/b/baylis/
Ramon González Calvet. http://www.terra.es/personal/rgonzal1
David Hestenes. http://modelingnts.la.asu.edu/
Jaime Keller http://www.tuwien.ac.at/cms/jkeller/
Pertii Lounesto. http://www.helsinki.fi/˜lounesto/
Josep Manel Parra i Serra. http://hermes.ffn.ub.es/˜jmparra


Software pages

CABRI http://www.cabri.com
CINDERELLA homepage http://www.cinderella.de/en/index.html
application to geometric algebra http://carol.wins.uva.nl/˜leo/cinderella/
CLICAL (Clifford Calculator) http://www.helsinki.fi/˜lounesto/CLICAL.htm
CLU http://www.perwass.de/cbup/start.html
GAIGEN http://carol.wins.uva.nl/˜fontijne/gaigen/about.html
GEOMA http://nklein.com/products/geoma/
GLUCAT http://glucat.sourceforge.net/
MAPLE V package http://math.tntech.edu/rafal/cliff5/
NONEUCLID http://math.rice.edu/˜joel/NonEuclid/
REDUCE packages http://hermes.ffn.ub.es/˜jmparra


Journal pages

Advances in Applied Clifford Algebras http://www.clifford.org/journals/jadvclfa.html


Subject pages

European Mathematical Information Service http://www.emis.de/
The Geometry Center http://www.geom.umn.edu/
The geometry of the sphere http://math.rice.edu/˜pcmi/sphere/
Geometric Calculus: Research and Development http://modelingnts.la.asu.edu/
The Geometric Algebra Research Group (Cambridge) http://www.mrao.cam.ac.uk/˜clifford/
Clifford Research Group (Gent) http://cage.rug.ac.be/˜fb/crg/
RAMON GONZALEZ CALVET
270

INDEX


bisectors,
A
of the sides, 70
addition identities, 57
of the angles, 72
affine plane, 31
affinity, 51 Bretschneider™s theorem 91, 238 (9.4)
algebra Brianchon™s theorem, 138, 248, (11.8)
fundamental theorem, 24
geometric, of the C
Euclidean space, 170 Cartesian equation of a circle, 80
pseudo-Euclidean space, 188 Cauchy-Riemann conditions, 19
algebraic equation of a line, 37 Cauchy™s
algebraic equations of a circle, 80 theorem, 20
alignment of three points, 35 integral formula, 21
altitudes of a triangle, 73 central
analytic equation for conics, 126
continuation, 151 projection
complex functions, 19 of the sphere, 177
hyperbolic functions, 147 of the hyperboloid, 191
analyticity conditions centroid, 69
complex, 19 Ceva™s theorem, 64
hyperbolic, 148 cevian lines, 69
angle Chasles' theorem, 122
between circles, 89 circle
between Euclidean vectors, 12 of the nine points, 85
between hyperbolic vectors, 156 inscribed in a triangle, 72, 73
of parallelism, 206, 260 (15.1) circumscribed to a triangle, 70, 73
inscribed in a circle, 55 circles, 80, 236
antigraphy, 116 radical axis, 89
Apollonius' theorem radical centre, 90
theorem for the ellipse, 128 circular functions, 54
theorem for the hyperbola, 163 circumcentre, 70
circumscribed quadrilateral, 88
lost theorem, 79, 229 (8.3)
Clifford, vi, viii, 8
area, 33
Clifford algebras
of a parallelogram, 4
isomorphism of, 152
of a triangle, 35, 68
complex analytic functions, 19
of a spherical triangle, 176
complex numbers, 13
of Lobachevskian triangle, 206,
binomial form, 13
263 (15.6)
polar and trigonometric form, 13
on the hyperbolic plane, 161
matrix representation of, 139
alignment of three points, 35
algebraic operations, 14
analytic functions, 19
complex plane, 18
angle inscribed in a circle, 55
conformal
associative property, 5
cylindrical projection, 202
azimuthal equivalent projection of the
transformations, 66
hyperboloid, 200
congruence
of hyperbolic segments, 158
B
congruence of Lobachevskian triangles,
base of vectors for the plane, 10
205
barycentric coordinates, 33
conic sections, 117
Beltrami™s disk, 191
central equations, 126
binomial form of a complex number, 15
TREATISE OF PLANE GEOMETRY THROUGH GEOMETRIC ALGEBRA 271

distortion, 177
in barycentric coordinates, 132
double angle identities, 55
directrices, 120
dual coordinates, 43
foci, 120
dual spherical triangle, 175
matrix of, 136
duality principle, 43
passing through five points, 131
perpendicular to
E
tangent to, 124
eccentricity, 118
tangential conic, 132
ellipse,
vectorial equation, 121
central equation, 126
conical projections
Apollonius™ diameters, 128
of the sphere, 184
equations
of the hyperboloid, 203
of a circle, 80
conjugate
of a line, 36-40
diameters, 129, 163
of conics, 119, 121, 126
of a complex number, 16
equivalent cylindrical projection
of a hyperbolic number, 141
of the sphere, 184
convergence
of the hyperboloid, 203
of complex powers series, 23
Euler's line, 76
of hyperbolic powers series, 150
Euclidean space, 170
radius of, 22
excess, spherical, 177
square of, 150
exponential
coordinate systems, 31
complex, 14
coordinates
hyperbolic, 143
barycentric, 33
exterior product, 4
Cartesian, 36
dual, 43
F
homogeneous, 102
Fermat's theorem, 77
correlation, 134
focal distance
cosines law
of a conic, 121
Euclidean, 60, 223 (6.1)
of a photograph, 186, 258 (14.4)
hyperbolic, 164
foci of a conic, 120
hyperboloidal, 198
function of Riemann, 150
spherical, 174, 175
cross ratio
G
complex, 92
geometric algebra
projective, 99
of the vectorial plane, 8
cylindrical projections
of the pseudo-Euclidean space, 188
of the sphere, 182
Gibbs, vi, viii
of the hyperboloid, 201
gnomonic projection, 177
cyclic quadrilaterals, 87
Grassmann, vii,viii
D
H
De Moivre's identity, 58
Hamilton, viii, 8, 171
Desargues™ theorem, 47
harmonic
diameters
characteristic, 94
of the ellipse, 128
ranges, 94
of the hyperbola, 163
dilatations, 30 Heron™s formula, 9, 208 (1.6)
direct similarity, 61 hierarchy of algebraic operations, 8
distance between two points Hilbert, 190
on the Euclidean plane, 33 homogeneous coordinates, 102
on the hyperbolic plane, 160 homography, 96
distance from a point to a line, 37 homologous sides, 61
RAMON GONZALEZ CALVET
272

lines on the Euclidean plane, 31
homology, 110
linear combination of two vectors, 10
homothety, 65
Liouville™s theorem, 24
horocycle, 206, 263 (15.4)
Lobachevskian
hyperbola
geometry, 188
central equation, 126
trigonometry, 196
hyperbolic
logarithm
analyticity, 150
complex, 20
analytic functions, 147
hyperbolic, 143
cosine, 142
exponential and logarithm, 143
M
numbers, 140
McLaurin series, 22
polar form, 144
matrix representation, 139
powers and roots, 144
medians, 69
plane
Menelaus™ theorem, 63
distance between points
Mercator's projection, 183
160
Möbius, viii, 33, 78
area on , 161
transformation, 96
sector, 142
modulus
similarity, 167
of a complex number, 13
sine, 142
of a hyperbolic number, 141
tangent, 142
Mollweide™s formulas, 60
trigonometry, 141
vectors, 154
N
hyperboloid of two sheets, 190
Napier™s rule, 176
hyperboloidal
Napoleon™s theorem, 78
arc, 193
Newton™s theorem, 138
defect, 206, 263 (15.5)
nine-point circle, 85
triangle, 196
non-Euclidean geometry, 205
trigonometry, 196

O
I
opposite similarity, 62
imaginary unity, 13
orthocentre, 73
incentre, 72
orthogonal hyperbolic vectors, 157
infinity, points at, 102
orthographic projection of the sphere, 181
inner product
orthonormal bases, 11
of hyperbolic vectors, 155
outer product
intercept equation of a line, 40
of Euclidean vectors, 4
intersection
of hyperbolic vectors, 155
of a line with a circle, 80
of two lines, 41
P
inverse
parametric equations of a line, 36
of a vector, 7
Pascal™s theorem, 138, 248, (11.7)
trigonometric functions, 59
Pauli™s matrices, 153
inversions of vectors, 29
Peano, vi, vii, 128
inversion with respect to a circle, 83
pencil of lines, 41
isometries
permutation of complex and vector, 17
hyperbolic, 158
permutative property
isomorphism of Clifford algebras, 152
of Euclidean vectors, 6, 9, 207 (1.5)
isosceles triangle theorem, 158
of hyperbolic vectors, 155
perspectivity, 103
L
Peter's projection, 184
Lauren series, 22
Plate Carr© projection, 182
Leibniz, vi, 190
Poincar©™s
line at the infinity, 47
TREATISE OF PLANE GEOMETRY THROUGH GEOMETRIC ALGEBRA 273

quaternions, 8, 171
disk, 198
quotient
half-plane, 200
of two vectors, 7
points, 31
of complex numbers, 16
alignment, 31
at the infinity, 102
R
on the Euclidean plane, 214
radical axis of two circles, 89
polar equation of a line, 40
radical centre of three circles, 90
polar equation of a circle, 82
ratio
polar form of a complex number, 14
complex cross, 92
polarities, 134
of similarity, 61
poles, 23
projective cross, 99
polygon
single, 65
sum of the angles, 53
reflections of vectors, 28
power of a point
residue theorem, 24
with respect to a circle, 82
reversal, 168
with respect to a hyperbola, 168
Riemann™s function, 150
product
rotations of vectors, 27, 139
of a vector and a real number, 2
of two vectors, 2
S
of three vectors, 5
sector of a hyperbola, 142
of four vectors, 7
series
inner
Lauren, 22
of hyperbolic vectors, 155
McLaurin, 22
outer
Taylor, 21
of hyperbolic vectors, 155
similarity
projection
direct, 61
of the spherical surface
opposite, 62
central or gnomonic, 177
hyperbolic, 167
conical, 184
Mercator™s, 183 Simson™s theorem, 91, 237, (9.3)
Peter's projection, 184 sines law
Plate Carr©™s, 182 Euclidean, 60, 223 (6.1)
orthographic, 181 hyperbolic, 164
stereographic, 180 hyperboloidal, 197
of the hyperboloid spherical, 173
central (Beltrami™s), 191 single ratio, 65
conical, 203 singularities, 23
stereographic, 198 slope-intercept equation of a line, 40
azimuthal equivalent, 200 space
cylindrical equidistant, 201 Euclidean, 170
cylindrical conformal, 202 pseudo-Euclidean, 188
cylindrical equivalent, 203 spherical triangle
projective cross ratio, 99 excess of, 177
projectivity, 103 dual, 175
pseudo-Euclidean area of, 176
plane, 154 spherical trigonometry, 172
space, 188 spherical coordinates, 182
Ptomely™s theorem, 115, 240, (10.1) stereographic projection
of the sphere, 180
Q of the hyperboloid, 198
quadrilaterals strict triangles, 173
cyclic, 87 sum of the angles
circumscribed, 88 of a polygon, 53
RAMON GONZALEZ CALVET
274

spherical, 172
of a hyperbolic triangle, 160
sum of trigonometric functions, 56
V
vector
addition, 1
T
product, 2
tangents law 60, 169, 254 (13.5)
transformations, 213
tangent to a conic, 124
vectorial equation of a line, 36
Taylor series, 22
tetranions, 188
W
theorem
Weierstrass™
of the algebra, fundamental, 24
coordinates, 201
of Apollonius, 128, 138, 163,
model, 190
(11.3)
lost, 78, 229, (8.3)
of Brianchon, 138, 248, (11.8)
of Cauchy, 20
of Ceva, 64
of Chasles, 122
of Desargues, 47
of Fermat, 77
of Liouville, 24
of Menelaus, 63
of Napoleon, 78
of Newton, 138
of Pascal, 138, 248, (11.7)
of Ptolemy, 115, 240, (10.1)
of Simson, 91, 237, (9.3)
of the residue, 24
of the isosceles triangle, 158
theorems
projective demonstration, 108
about hyperbolic angles, 160
transformations
of vectors, 213
special conformal, 115, 242, (10.5)
translations, 31
triangle
dual spherical, 175
triangular inequality, 34
trigonometric
addition identities, 57
double angle identities, 55
form of a complex number, 14
functions, 54
fundamental identities, 54
inverse functions, 59
sum of functions, 56
product of vectors
trigonometry
circular, 53
hyperbolic, 141
hyperboloidal (Lobachevskian),
196
TREATISE OF PLANE GEOMETRY THROUGH GEOMETRIC ALGEBRA 275

CHRONOLOGY OF THE GEOMETRIC ALGEBRA

1679 Letters of Leibniz to Huygens on the characteristica geometrica.
1799 Publication of Om Directionens analytiske Betegning by Caspar Wessel with scarce
diffusion.
1805 Birth of William Rowan Hamilton at Dublin.
1806 Publication of Essai sur une manière de repr©senter les quantit©s imaginaires dans les
constructions g©om©triques by Jean Robert Argand.
1809 Birth of Hermann Günther Grassmann at Stettin.
1818 Death of Wessel
1822 Death of Argand.
1827 Publication of Der barycentrische Calcul by Möbius at Leipzig.
1831 Birth of James Clerk Maxwell at Edinburgh.
1831 Birth of Peter Guthrie Tait.
1839 Birth of Josiah Willard Gibbs at New Haven.
1843 Discovery of the quaternions by Hamilton.
1844 Publication of Die Lineale Ausdehnungslehre (first edition), where Grassmann presents the
anticommutative product of geometric unities (outer product).
1845 Birth of William Kingdon Clifford at Exeter.
1847 Publication of Geometric Analysis with a foreword by Möbius, memoir with which
Grassmann won the prize to whom developed the Leibniz™s characteristica geometrica.
1850 Birth of Oliver Heaviside at London.
1853 Publication of Lectures on Quaternions where Hamilton introduces the nabla operator
(gradient).
1862 Publication of the second edition of Die Ausdehnungslehre.
1864 Publication of A dynamical theory of the electromagnetic field by Maxwell, where he defines
the divergence and the curl.
1865 Death of Hamilton.
1866 Posthumous publication of Hamilton™s Elements of Quaternions.
1867 Publication of Elementary Treatise on Quaternions by Tait.
1873 Publication of Introduction to Quaternions by Kelland and Tait.
1873 Maxwell publishes the Treatise on Electricity and Magnetism where he writes the equations
of electromagnetism with quaternions.
1877 Publication of the Grassmann™s paper «Der Ort der Hamilton™schen Quaternionen in der
Ausdehnungslehre».
1877 Death of Grassmann.
1878 Publication of the paper «Applications of Grassmann's Extensive Algebra» by Clifford where
he makes the synthesis of the systems of Grassmann and Hamilton.
1879 Death of Maxwell.
1879 Death of Clifford.
1880 Publication of Lipschitz™ Principes d™un calcul algebraic.
1881 Private printing of Elements of Vector Analysis by Gibbs.
1886 Publication of Lipschitz™ Untersuchungen uber die Summen von Quadraten.
1886 Publication of the Gibbs™ paper «On multiple algebra».
1888 Publication of Peano™s Calcolo geometrico secondo l™Ausdehnungslehre di H. Grassmann
preceduto dalle operazione della logica deduttiva.
1891 Oliver Heaviside publishes «The elements of vectorial algebra and analysis» in The
Electrician Series.
1895 Publication of Peano™s «Saggio di Calcolo Geometrico».
1901 Death of Tait.
1901 Wilson publishes the Gibbs™ lessons in Vector Analysis.
1903 Death of Gibbs.
RAMON GONZALEZ CALVET
276

1925 Death of Heaviside.
1926 Wolfgang Pauli introduces his matrices to explain the electronic spin.
1928 Publication of the paper «The Quantum Theory of Electron», where Paul A. M. Dirac defines
a set of 4—4 anticommutative matrices built from the Pauli™s matrices.




This comparative diagram of the life and works of the authors of (or related with)
the geometric algebra visualises and summarises the chronology. The XIX century may be
properly called the century of the geometric algebra. Note the premature death of Clifford,
which caused the delay in the development of the geometric algebra along the XX
century.


1800 1820 1840 1860 1880 1900
| | | | | |
Wessel**************
Argand*************************
Hamilton *******************************
Grassmann ***********************************
Maxwell *************************
Tait ************************************
Gibbs *********************************
Clifford ******************
Heaviside **************************************


Essai sur ... |
|
Der barycentrische Calcul
|
Die Ausdehnungslehre
|
Lectures on Quaternions
|
Elements on Quaternions
|
Elementary Treatise on Quaternions
|
Treatise on Electricity and Magnetism
|
Applications of Grassmann's Extensive Algebra

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