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Ramon González Calvet

Ramon González Calvet

The geometric algebra,
initially discovered by Hermann
Grassmann (1809-1877) was
reformulated by William Kingdon
Clifford (1845-1879) through the
synthesis of the Grassmann™s
extension theory and the
quaternions of Sir William Rowan
Hamilton (1805-1865). In this way
the bases of the geometric algebra
were established in the XIX
century. Notwithstanding, due to
the premature death of Clifford, the
vector analysis ’a remake of the
quaternions by Josiah Willard
Gibbs (1839-1903) and Oliver
Heaviside (1850-1925)’ became,
after a long controversy, the
geometric language of the XX century; the same vector analysis whose beauty attracted the
attention of the author in a course on electromagnetism and led him -being still
undergraduate- to read the Hamilton™s Elements of Quaternions. Maxwell himself already
applied the quaternions to the electromagnetic field. However the equations are not written
so nicely as with vector analysis. In 1986 Ramon contacted Josep Manel Parra i Serra,
teacher of theoretical physics at the Universitat de Barcelona, who acquainted him with the
Clifford algebra. In the framework of the summer courses on geometric algebra which they
have taught for graduates and teachers since 1994, the plan of writing some books on this
subject appeared in a very natural manner, the first sample being the Tractat de geometria
plana mitjan§ant l™àlgebra geomètrica (1996) now out of print. The good reception of the
readers has encouraged the author to write the Treatise of plane geometry through
geometric algebra (a very enlarged translation of the Tractat) and publish it at the Internet
site http://campus.uab.es/˜PC00018, writing it not only for mathematics students but also
for any person interested in geometry. The plane geometry is a basic and easy step to enter
into the Clifford-Grassmann geometric algebra, which will become the geometric language
of the XXI century.
Dr. Ramon González Calvet (1964) is high school teacher of mathematics since 1987,
fellow of the Societat Catalana de Matemàtiques (http://www-ma2.upc.es/˜sxd/scma.htm)
and also of the Societat Catalana de Gnomònica (http://www.gnomonica.org).


Dr. Ramon González Calvet

Mathematics Teacher
I.E.S. Pere Calders, Cerdanyola del Vallès

To my son Pere, born with the book.

© Ramon González Calvet ( rgonzal1@teleline.es )
This is an electronic edition by the author at the Internet site
http://campus.uab.es/˜PC00018. All the rights reserved. Any electronic or
paper copy cannot be reproduced without his permission. The readers are
authorised to print the files only for his personal use. Send your comments
or opinion about the book to ramon.gonzalezc@campus.uab.es .

ISBN: 84-699-3197-0

First Catalan edition: June 1996
First English edition: June 2000 to June 2001


The book I am so pleased to present represents a true innovation in the field of the
mathematical didactics and, specifically, in the field of geometry. Based on the long
neglected discoveries made by Grassmann, Hamilton and Clifford in the nineteenth
century, it presents the geometry -the elementary geometry of the plane, the space, the
spacetime- using the best algebraic tools designed specifically for this task, thus making
the subject democratically available outside the narrow circle of individuals with the
high visual imagination capabilities and the true mathematical insight which were
required in the abandoned classical Euclidean tradition. The material exposed in the
book offers a wide repertory of geometrical contents on which to base powerful,
reasonable and up-to-date reintroductions of geometry to present-day high school
students. This longed-for reintroductions may (or better should) take advantage of a
combined use of symbolic computer programs and the cross disciplinary relationships
with the physical sciences.
The proposed introduction of the geometric Clifford-Grassmann algebra in high
school (or even before) follows rightly from a pedagogical principle exposed by
William Kingdon Clifford (1845-1879) in his project of teaching geometry, in the
University College of London, as a practical and empirical science as opposed to
Cambridge Euclidean axiomatics: “ ... for geometry, you know, is the gate of science,
and the gate is so low and small that one can only enter it as a little child”. Fellow of the
Royal Society at the age of 29, Clifford also gave a set of Lectures on Geometry to a
Class of Ladies at South Kengsinton and was deeply concerned in developing with
MacMillan Company a series of inexpensive “very good elementary schoolbook of
arithmetic, geometry, animals, plants, physics ...”. Not foreign to this proposal are Felix
Klein lectures to teachers collected in his book Elementary mathematics from an
advanced standpoint1 and the advice of Alfred North Whitehead saying that “the hardest
task in mathematics is the study of the elements of algebra, and yet this stage must
precede the comparative simplicity of the differential calculus” and that “the
postponement of difficulty mis no safe clue for the maze of educational practice” 2.
Clearly enough, when the fate of pseudo-democratic educational reforms,
disguised as a back to basic leitmotifs, has been answered by such an acute analysis by
R. Noss and P. Dowling under the title Mathematics in the National Curriculum: The
Empty Set?3, the time may be ripen for a reappraisal of true pedagogical reforms based
on a real knowledge, of substantive contents, relevant for each individual worldview
construction. We believe that the introduction of the vital or experiential plane, space
and space-time geometries along with its proper algebraic structures will be a
substantial part of a successful (high) school scientific curricula. Knowing, telling,
learning why the sign rule, or the complex numbers, or matrices are mathematical
structures correlated to the human representation of the real world are worthy objectives
in mass education projects. And this is possible today if we learn to stand upon the
shoulders of giants such as Leibniz, Hamilton, Grassmann, Clifford, Einstein,
Minkowski, etc. To this aim this book, offered and opened to suggestions to the whole
world of concerned people, may be a modest but most valuable step towards these very
good schoolbooks that constituted one of the cheerful Clifford's aims.
Felix Klein, Elementary mathematics from an advanced standpoint. Dover (N. Y., 1924).
A.N. Whitehead, The aims of education. MacMillan Company (1929), Mentor Books (N.Y.,
P. Dowling, R. Noss, eds., Mathematics versus the National Curriculum: The Empty Set?. The
Falmer Press (London, 1990).

Finally, some words borrowed from Whitehead and Russell, that I am sure
convey some of the deepest feelings, thoughts and critical concerns that Dr. Ramon
González has had in mind while writing the book, and that fully justify a work that
appears to be quite removed from today high school teaching, at least in Catalunya, our

“Where attainable knowledge could have changed the issue, ignorance has
the guilt of vice”2.
“The uncritical application of the principle of necessary antecedence of
some subjects to others has, in the hands of dull people with a turn for
organisation, produced in education the dryness of the Sahara”2.
“When one considers in its length and in its breadth the importance of this
question of the education of a nation's young, the broken lives, the defeated
hopes, the national failures, which result from the frivolous inertia with
which it is treated, it is difficult to restrain within oneself a savage rage”2.
“A taste for mathematics, like a taste for music, can be generated in some
people, but not in others. ... But I think that these could be much fewer than
bad instruction makes them seem. Pupils who have not an unusually strong
natural bent towards mathematics are led to hate the subject by two
shortcomings on the part of their teachers. The first is that mathematics is
not exhibited as the basis of all our scientific knowledge, both theoretical
and practical: the pupil is convincingly shown that what we can understand
of the world, and what we can do with machines, we can understand and do
in virtue of mathematics. The second defect is that the difficulties are not
approached gradually, as they should be, and are not minimised by being
connected with easily apprehended central principles, so that the edifice of
mathematics is made to look like a collection of detached hovels rather than
a single temple embodying a unitary plan. It is especially in regard to this
second defect that Clifford's book (Common Sense of the Exact Sciences) is
valuable.(Russell)” 4.

An appreciation that Clifford himself had formulated, in his fundamental paper upon
which the present book relies, relative to the Ausdehnungslehre of Grassmann,
expressing “my conviction that its principles will exercise a vast influence upon the
future of mathematical science”.

Josep Manel Parra i Serra, June 2001

Departament de Física Fonamental
Universitat de Barcelona

W. K. Clifford, Common Sense of the Exact Sciences. Alfred A. Knopf (1946), Dover (N.Y.,

« On demande en second lieu, laquelle des deux qualit©s doit être pr©f©r©e
dans des ©l©mens, de la facilit©, ou de la rigour exacte. Je r©ponds que cette
question suppose una chose fausse; elle suppose que la rigour exacte puisse
exister sans la facilit© & c™est le contraire; plus une d©duction est
rigoureause, plus elle est facile à entendre: car la rigueur consiste à reduire
tout aux principes les plus simples. D™où il s™ensuit encore que la rigueur
proprement dit entra®ne n©cessairement la m©thode la plus naturelle & la
plus directe. Plus les principles seront dispos©s dans l™ordre convenable,
plus la d©duction sera rigourease; ce n™est pas qu™absolument elle ne p»t
l™être si on suivonit une m©thode plus compos©e, com a fait Euclide dans ses
©l©mens: mais alors l™embarras de la marche feroit ais©ment sentir que cette
rigueur pr©caire & forc©e ne seroit qu™improprement telle. »5

[“Secondly, one requests which of the two following qualities must be
preferred within the elements, whether the easiness or the exact rigour. I
answer that this question implies a falsehood; it implies that the exact rigour
can exist without the easiness and it is the other way around; the more
rigorous a deduction will be, the more easily it will be understood: because
the rigour consists of reducing everything to the simplest principles.
Whence follows that the properly called rigour implies necessarily the most
natural and direct method. The more the principles will be arranged in the
convenient order, the more rigorous the deduction will be; it does not mean
that it cannot be rigorous at all if one follows a more composite method as
Euclid made in his elements: but then the difficulty of the march will make
us to feel that this precarious and forced rigour will only be an improper

Jean le Rond D'Alembert (1717-1783)

«El©mens des sciences» in Encyclop©die, ou dictionaire raisonn© des sciences, des arts et des
m©tiers (París, 1755).


The first edition of the Treatise of Plane Geometry through Geometric Algebra
is a very enlarged translation of the first Catalan edition published in 1996. The good
reception of the book (now out of print) encouraged me to translate it to the English
language rewriting some chapters in order to make easier the reading, enlarging the
others and adding those devoted to the non-Euclidean geometry.
The geometric algebra is the tool which allows to study and solve geometric
problems through a simpler and more direct way than a purely geometric reasoning, that
is, by means of the algebra of geometric quantities instead of synthetic geometry. In
fact, the geometric algebra is the Clifford algebra generated by the Grassmann's outer
product in a vector space, although for me, the geometric algebra is also the art of
stating and solving geometric equations, which correspond to geometric problems, by
isolating the unknown geometric quantity using the algebraic rules of the vectors
operations (such as the associative, distributive and permutative properties). Following

“The geometric Calculus differs from the Cartesian Geometry in that
whereas the latter operates analytically with coordinates, the former
operates directly on the geometric entities”.

Initially proposed by Leibniz7 (characteristica geometrica) with the aim of
finding an intrinsic language of the geometry, the geometric algebra was discovered and
developed by Grassmann8, Hamilton and Clifford during the XIX century. However, it
did not become usual in the XX century ought to many circumstances but the vector
analysis -a recasting of the Hamilton quaternions by Gibbs and Heaviside- was
gradually accepted in physics. On the other hand, the geometry followed its own way
aside from the vector analysis as Gibbs9 pointed out:

“And the growth in this century of the so-called synthetic as opposed to
analytical geometry seems due to the fact that by the ordinary analysis
geometers could not easily express, except in a cumbersome and unnatural
manner, the sort of relations in which they were particularly interested”

Giuseppe Peano, «Saggio di Calcolo geometrico». Translated in Selected works of Giuseppe
Peano, 169 (see the bibliography).
C. I. Gerhardt, G. W. Leibniz. Mathematical Schriften V, 141 and Der Briefwechsel von
Gottfried Wilhelm Leibniz mit Mathematiker, 570.
In 1844 a prize (45 gold ducats for 1846) was offered by the Fürstlich Jablonowski'schen
Gessellschaft in Leipzig to whom was capable to develop the characteristica geometrica of
Leibniz. Grassmann won this prize with the memoir Geometric Analysis, published by this
society in 1847 with a foreword by August Ferdinand Möbius. Its contents are essentially those
of Die Ausdehnungslehre (1844).
Josiah Willard Gibbs, «On Multiple Algebra», reproduced in Scientific papers of J.W. Gibbs,
II, 98.

The work of revision of the history and the sources (see J. M. Parra10) has
allowed us to synthesise the contributions of the different authors and completely
rebuild the evolution of the geometric algebra, removing the conceptual mistakes which
led to the vector analysis. This preface has not enough extension to explain all the
history11, but one must remember something usually forgotten: during the XIX century
several points of view over what should become the geometric algebra came into
competition. The Gibbs' vector analysis was one of these being not the better. In fact,
the geometric algebra is a field of knowledge where different formulations are possible
as Peano showed:

“Indeed these various methods of geometric calculus do not at all
contradict one another. They are various parts of the same science, or rather
various ways of presenting the same subject by several authors, each
studying it independently of the others.
It follows that geometric calculus, like any other method, is not a
system of conventions but a system of truth. In the same way, the methods
of indivisibles (Cavalieri), of infinitesimals (Leibniz) and of fluxions
(Newton) are the same science, more or less perfected, explained under
different forms.”12

The geometric algebra owns some fundamental geometric facts which cannot be
ignored at all and will be recognised to it, as Grassmann hoped:

“For I remain completely confident that the labour which I have
expanded on the science presented here and which has demanded a
significant part of my life as well as the most strenuous application of my
powers will not be lost. It is true that I am aware that the form which I have
given the science is imperfect and must be imperfect. But I know and feel
obliged to state (though I run the risk of seeming arrogant) that even if this
work should again remained unused for another seventeen years or even
longer, without entering into the actual development of science, still the
time will come when it will be brought forth from the dust of oblivion, and
when ideas now dormant will bring forth fruit. I know that if I also fail to
gather around me in a position (which I have up to now desired in vain) a
circle of scholars, whom I could fructify with these ideas, and whom I could
stimulate to develop and enrich further these ideas, nevertheless there will
come a time when these ideas, perhaps in a new form, will arise anew and
will enter into living communication with contemporary developments. For
truth is eternal and divine, and no phase in the development of truth,
however small may be the region encompassed, can pass on without leaving

Josep Manel Parra i Serra, «Geometric algebra versus numerical Cartesianism. The historical
trend behind Clifford™s algebra», in Brackx et al. ed., Clifford Algebras and their Applications
in Mathematical Physics, 307-316, .
A very complete reference is Michael J. Crowe, A History of Vector Analysis. The Evolution
of the Idea of a Vectorial System.
Giuseppe Peano, op. cit., 168.

a trace; truth remains, even though the garment in which poor mortals clothe
it may fall to dust.”13

As any other aspect of the human life, the history of the geometric algebra was
conditioned by many fortuitous events. While Grassmann deduced the extension theory
from philosophic concepts unintelligible for authors such as Möbius and Gibbs,
Hamilton identified vectors and bivectors -the starting point of the great tangle of vector
analysis- using a heavy notation14. Clifford had found the correct algebraic structure15
which integrated the systems of Hamilton and Grassmann. However due to the
premature death of Clifford in 1879, his opinion was not taken into account16 and a long
epistolary war was carried out by the quaternionists (specially Tait) against the
defenders of the vector analysis, created by Gibbs17, who did not recognise to be
influenced by Grassmann and Hamilton:

“At all events, I saw that the methods which I was using, while
nearly those of Hamilton, were almost exactly those of Grassmann. I
procured the two Ed. of the Ausdehnungslehre but I cannot say that I found
them easy reading. In fact I have never had the perseverance to get through
with either of them, and have perhaps got more ideas from his
miscellaneous memoirs than from those works.
I am not however conscious that Grassmann's writings exerted any
particular influence on my Vector Analysis, although I was glad enough in
the introductory paragraph to shelter myself behind one or two distinguished
names [Grassmann and Clifford] in making changes of notation which I felt
would be distasteful to quaternionists. In fact if you read that pamphlet
carefully you will see that it all follows with the inexorable logic of algebra
from the problem which I had set myself long before my acquaintance with
I have no doubt that you consider, as I do, the methods of Grassmann
to be superior to those of Hamilton. It thus seemed to me that it might [be]
interesting to you to know how commencing with some knowledge of
Hamilton's method and influenced simply by a desire to obtain the simplest
algebra for the expression of the relations of Geom. Phys. etc. I was led
essentially to Grassmann's algebra of vectors, independently of any
influence from him or any one else.”18

Hermann Gunther Grassmann. Preface to the second edition of Die Ausdehnungslehre (1861).
The first edition was published on 1844, hence the "seventeen years". Translated in Crowe, op.
cit. p. 89.
The Lectures on Quaternions was published in 1853, and the Elements of Quaternions
posthumously in 1866.
William Kingdon Clifford left us his synthesis in «Applications of Grassmann's Extensive
See «On the Classification of Geometric Algebras», unfinished paper whose abstract was
communicated to the London Mathematical Society on March 10th, 1876.
The first Vector Analysis was a private edition of 1881.
Draft of a letter sent by Josiah Willard Gibbs to Victor Schlegel (1888). Reproduced by
Crowe, op. cit. p. 153.

Before its beginning the controversy was already superfluous19. Notwithstanding
the epistolary war continued for twelve years.
The vector analysis is a provisional solution20 (which spent all the XX century!)
adopted by everybody ought to its easiness and practical notation but having many
troubles when being applied to three-dimensional geometry and unable to be
generalised to the Minkowski™s four-dimensional space. On the other hand, the
geometric algebra is, by its own nature, valid in any dimension and it offers the
necessary resources for the study and research in geometry as I show in this book.

The reader will see that the theoretical explanations have been completed with
problems in each chapter, although this splitting is somewhat fictitious because the
problems are demonstrations of geometric facts, being one of the most interesting
aspects of the geometric algebra and a proof of its power. The usual numeric problems,
which our pupils like, can be easily outlined by the teacher, because the geometric
algebra always yields an immediate expression with coordinates.
I'm indebted to professor Josep Manel Parra for encouraging me to write this
book, for the dialectic interchange of ideas and for the bibliographic support. In the
framework of the summer courses on geometric algebra for teachers that we taught
during the years 1994-1997 in the Escola d™estiu de secundària organised by the
Col·legi Oficial de Doctors i Llicenciats en Filosofia i Lletres i en Ciències de
Catalunya, the project of some books on this subject appeared in a natural manner. The
first book devoted to two dimensions already lies on your hands and will probably be
followed by other books on the algebra and geometry of the three and four dimensions.
Finally I also acknowledge the suggestions received from some readers.

Ramon González Calvet

Cerdanyola del Vallès, June 2001

See Alfred M. Bork «“Vectors versus quaternions””The letters in Nature».
The vector analysis bases on the duality of the geometric algebra of the three-dimensional
space: the fact that the orientation of lines and planes is determined by three numeric
components in both cases. However in the four-dimensional time-space the same orientations
are respectively determined by four and six numbers.


First Part: The vector plane and the complex numbers

1. The vectors and their operations. (June 24th 2000, updated March 17th 2002)
Vector addition, 1.- Product of a vector and a real number, 2.- Product of two vectors,
2.- Product of three vectors: associative property, 5. Product of four vectors, 7.- Inverse
and quotient of two vectors, 7.- Hierarchy of algebraic operations, 8.- Geometric algebra
of the vector plane, 8.- Exercises, 9.

2. A base of vectors for the plane. (June 24th 2000)
Linear combination of two vectors, 10.- Base and components, 10.- Orthonormal bases,
11.- Applications of the formulae for the products, 11.- Exercises, 12.

3. The complex numbers. (August 1st 2000, updated July 21st 2002)
Subalgebra of the complex numbers, 13.- Binomial, polar and trigonometric form of a
complex number, 13.- Algebraic operations with complex numbers, 14.- Permutation of
complex numbers and vectors, 17.- The complex plane, 18.- Complex analytic
functions, 19.- The fundamental theorem of algebra, 24.- Exercises, 26.

4. Transformations of vectors. (August 4th 2000, updated July 21st 2002)
Rotations, 27.- Reflections, 28.- Inversions, 29.- Dilatations, 30.- Exercises, 30

Second Part: The geometry of the Euclidean plane

5. Points and straight lines. (August 19th 2000, updated September 29th 2000)
Translations, 31.- Coordinate systems, 31.- Barycentric coordinates, 33.- Distance
between two points and area, 33.- Condition of alignment of three points, 35.- Cartesian
coordinates, 36.- Vectorial and parametric equations of a line, 36.- Algebraic equation
and distance from a point to a line, 37.- Slope and intercept equations of a line, 40.-
Polar equation of a line, 40.- Intersection of two lines and pencil of lines, 41.- Dual
coordinates, 43.- The Desargues theorem, 47.- Exercises, 50.

6. Angles and elemental trigonometry. (August 24th 2000, updated July 21st 2002)
Sum of the angles of a polygon, 53.- Definition of trigonometric functions and
fundamental identities, 54.- Angle inscribed in a circle and double angle identities, 55.-
Addition of vectors and sum of trigonometric functions, 56.- Product of vectors and
addition identities, 57.- Rotations and De Moivre's identity, 58.- Inverse trigonometric
functions, 59.- Exercises, 60.

7. Similarities and single ratio. (August 26th 2000, updated July 21st 2002)
Direct similarity, 61.- Opposite similarity, 62.- The theorem of Menelaus, 63.- The
theorem of Ceva, 64.- Homothety and single ratio, 65.- Exercises, 67.

8. Properties of the triangles. (September 3rd 2000, updated July 21st 2002)
Area of a triangle, 68.- Medians and centroid, 69.- Perpendicular bisectors and
circumcentre, 70.- Angle bisectors and incentre, 72.- Altitudes and orthocentre, 73.-
Euler's line, 76.- The Fermat's theorem, 77.- Exercises, 78.

9. Circles. (October 8th 2000, updated July 16th 2002)
Algebraic and Cartesian equations, 80.- Intersections of a line with a circle, 80.- Power
of a point with respect to a circle, 82.- Polar equation, 82.- Inversion with respect to a
circle, 83.- The nine-point circle, 85.- Cyclic and circumscribed quadrilaterals, 87.-
Angle between circles, 89.- Radical axis of two circles, 89.- Exercises, 91.

10. Cross ratios and related transformations. (October 18th 2000, updated July 21st
Complex cross ratio, 92.- Harmonic characteristic and ranges, 94.- The homography
(Möbius transformation), 96.- Projective cross ratio, 99.- The points at the infinity and
homogeneous coordinates, 102.- Perspectivity and projectivity, 103.- The projectivity as
a tool for theorems demonstration, 108.- The homology, 110.- Exercises, 115.

11. Conics (November 12th 2000, updated July 21st 2002)
Conic sections, 117.- Two foci and two directrices, 120.- Vectorial equation, 121.- The
Chasles' theorem, 122.- Tangent and perpendicular to a conic, 124.- Central equations
for the ellipse and hyperbola, 126.- Diameters and Apollonius' theorem, 128.- Conic
passing through five points, 131.- Conic equations in barycentric coordinates and
tangential conic, 132.- Polarities, 134.- Reduction of the conic matrix to a diagonal
form, 136.- Using a base of points on the conic, 137.- Exercises, 137.

Third Part: Pseudo-Euclidean geometry

12. Matrix representation and hyperbolic numbers. (November 22nd 2000, updated
May 31st 2002)
Rotations and the representation of complex numbers, 139.- The subalgebra of the
hyperbolic numbers, 140.- Hyperbolic trigonometry, 141.- Hyperbolic exponential and
logarithm, 143.- Polar form, powers and roots of hyperbolic numbers, 144.- Hyperbolic
analytic functions, 147.- Analyticity and square of convergence of the power series,
150.- About the isomorphism of Clifford algebras, 152.- Exercises, 153.

13. The hyperbolic or pseudo-Euclidean plane (January 1st 2001, updated July 21st
Hyperbolic vectors, 154.- Inner and outer products of hyperbolic vectors, 155.- Angles
between hyperbolic vectors, 156.- Congruence of segments and angles, 158.-
Isometries, 158.- Theorems about angles, 160.- Distance between points, 160.- Area on
the hyperbolic plane, 161.- Diameters of the hyperbola and Apollonius' theorem, 163.-
The law of sines and cosines, 164.- Hyperbolic similarity, 167.- Power of a point with
respect to a hyperbola with constant radius, 168.- Exercises, 169.

Fourth Part: Plane projections of tridimensional spaces

14. Spherical geometry in the Euclidean space. (March 3rd 2001, updated August 25th
The geometric algebra of the Euclidean space, 170.- Spherical trigonometry, 172.- The
dual spherical triangle, 175.- Right spherical triangles and Napier™s rule, 176.- Area of a
spherical triangle, 176.- Properties of the projections of the spherical surface, 177.- The

central or gnomonic projection, 177.- Stereographic projection, 180.- Orthographic
projection, 181.- Spherical coordinates and cylindrical equidistant (Plate Carr©)
projection, 182.- Mercator's projection, 183.- Peter's projection, 184.- Conic projections,
184.- Exercises, 185.

15. Hyperboloidal geometry in the pseudo-Euclidean space (Lobachevsky's
geometry). (April 13th 2001, updated August 21st 2001)
The geometric algebra of the pseudo-Euclidean space, 188.- The hyperboloid of two
sheets, 190.- The central projection (Beltrami disk), 191.- Hyperboloidal
(Lobachevskian) trigonometry, 196.- Stereographic projection (Poincar© disk), 198.-
Azimuthal equivalent projection, 200.- Weierstrass coordinates and cylindrical
equidistant projection, 201.- Cylindrical conformal projection, 202.- Cylindrical
equivalent projection, 203.- Conic projections, 203.- On the congruence of geodesic
triangles, 205.- Comment about the names of the non-Euclidean geometry, 205.-
Exercises, 205.

16. Solutions of the proposed exercises. (April 28th 2001 and May 27th 2001, updated
July 20th 2002)
1. The vectors and their operations, 207.- 2. A base of vectors for the plane, 208.- 3. The
complex numbers, 209.- 4. Transformations of vectors, 213.- 5. Points and straight
lines, 214.- 6. Angles and elemental trigonometry, 223.- 7. Similarities and single ratio,
226.- 8. Properties of the triangles, 228.- 9. Circles, 236.- 10. Cross ratios and related
transformations, 240.- 11. Conics, 245.- 12. Matrix representation and hyperbolic
numbers, 250.- 13. The hyperbolic or pseudo-Euclidean plane, 251.- 14. Spherical
geometry in the Euclidean space, 254.- 15. Hyperboloidal geometry in the pseudo-
Euclidean space (Lobachevsky's geometry), 260.

Bibliography, 266.

Index, 270.

Chronology, 275.



Points and vectors are the main elements of the plane geometry. A point is
conceived (but not defined) as a geometric element without extension, infinitely small,
that has position and is located at a certain place on the plane. A vector is defined as an
oriented segment, that is, a piece of a straight line having length and direction. A vector
has no position and can be translated anywhere. Usually it is called a free vector. If we
place the end of a vector at a point, then its head determines another point, so that a
vector represents the translation from the first point to the second one.
Taking into account the distinction between points and vectors, the part of the
book devoted to the Euclidean geometry has been divided in two parts. In the first one
the vectors and their algebraic properties are studied, which is enough for many
scientific and engineering branches. In the second part the points are introduced and
then the affine geometry is studied.
All the elements of the geometric algebra (scalars, vectors, bivectors, complex
numbers) are denoted with lowercase Latin characters and the angles with Greek
characters. The capital Latin characters will denote points on the plane. As you will see,
the geometric product is not commutative, so that fractions can only be written for real
and complex numbers. Since the geometric product is associative, the inverse of a
certain element at the left and at the right is the same, that is, there is a unique inverse
for each element of the algebra, which is indicated by the superscript ’1. Also due to the
associative property, all the factors in a product are written without parenthesis. In order
to make easy the reading I have not numerated theorems, corollaries nor equations.
When a definition is introduced, the definite element is marked with italic characters,
which allows to direct attention and helps to find again the definition.


A vector is an oriented segment, having length and direction but no position, that
is, it can be placed anywhere without changing its orientation. The vectors can represent
many physical magnitudes such as a force, a celerity, and also geometric magnitudes
such as a translation.
Two algebraic operations for vectors are defined, the addition and the product,
that generalise the addition and product of the real numbers.

Vector addition

The addition of two vectors u + v
is defined as the vector going from the
end of the vector u to the head of v when
the head of u contacts the end of v (upper
triangle in the figure 1.1 ). Making the
construction for v + u, that is, placing the
end of u at the head of v (lower triangle
in the figure 1.1) we see that the addition
vector is the same. Therefore, the vector
addition has the commutative property:
Figure 1.1


and the parallelogram rule follows: the addition of two vectors is the diagonal of the
parallelogram formed by both vectors.
The associative property follows from
this definition because (u+v)+w or
u+(v+w) is the vector closing the
polygon formed by the three vectors as
shown in the figure 1.2.
The neutral element of the
vector addition is the null vector, which
has zero length. Hence the opposite
vector of u is defined as the vector ’u
with the same orientation but opposite
Figure 1.2
direction, which added to the initial
vector gives the null vector:

u + ( ’u) = 0

Product of a vector and a real number

One defines the product of a vector and a real number (or scalar) k, as a vector
with the same direction but with a length increased k times (figure 1.3). If the real
number is negative, then the direction is the opposite. The geometric definition implies
the commutative property:

ku=uk Figure 1.3
Two vectors u, v with the same
direction are proportional because there
is always a real number k such that v =
k u , that is, k is the quotient of both

k = u ’1 v = v u ’1

Two vectors with different
directions are said to be linearly

Product of two vectors

The product of two vectors will be called the geometric product in order to be
distinguished from other vector products currently used. Nevertheless I hope that these
other products will play a secondary role when the geometric product becomes the most
used, a near event which this book will forward. At that time, the adjective «geometric»
will not be necessary.
The following properties are demanded to the geometric product of two vectors:

1) To be distributive with regard to the vector addition:


2) The square of a vector must be equal to the square of its length. By
definition, the length (or modulus) of a vector is a positive number and it is
noted by | u |:

u2 = | u |2

3) The mixed associative property must exist between the product of vectors
and the product of a vector and a real number.

k(uv)=(ku)v= kuv

k(lu)=(kl)u= klu

where k, l are real numbers and u, v vectors. Therefore, parenthesis are not
These properties allows us to deduce the product. Let us suppose that c is the
addition of two vectors a, b and calculate its square applying the distributive property:


c2 = ( a + b )2 = ( a + b ) ( a + b ) = a2 + a b + b a + b2

We have to preserve the order of the factors because we do not know whether the
product is commutative or not.
If a and b are orthogonal vectors, the Pythagorean theorem applies and then:

c2 = a2 + b2
a⊥b ’ ’ ’ ab=’ba

That is, the product of two perpendicular vectors is anticommutative.
If a and b are proportional vectors then:

a || b ’ b = k a, k real ’ ab=aka=kaa=ba

because of the commutative and mixed associative properties of the product of a vector
and a real number. Therefore the product of two proportional vectors is commutative. If
c is the addition of two vectors a, b with the same direction, we have:


c2 = a2 + b2 + 2 | a | | b |

ab=|a||b| angle(a, b) = 0

But if the vectors have opposite directions:


c2 = a2 + b2 ’ 2 | a | | b |

ab=’|a||b| angle(a, b) = π

How is the product of two vectors with any directions? Due to the distributive
property the product is resolved into one product by the proportional component b|| and
another by the orthogonal component b⊥:

a b = a ( b|| + b⊥ ) = a b|| + a b⊥

The product of one vector by the proportional component of the other is called
the inner product (also scalar product) and noted by a point · (figure 1.4). Taking into
account that the projection of b onto a is proportional to the cosine of the angle between
both vectors, one finds:

a · b = a b|| = | a | | b | cos ±
Figure 1.4
The inner product is always a real
number. For example, the work made by a
force acting on a body is the inner product of
the force and the walked space. Since the
commutative property has been deduced for
the product of vectors with the same
direction, it follows also for the inner


The product of one vector by the orthogonal component of the other is called the
outer product (also exterior product) and it is noted with the symbol § :

a § b = a b⊥

The outer product represents the area of the parallelogram formed by both vectors
(figure 1.5):

¦ a § b ¦= ¦ a b⊥ ¦= ¦a¦¦b¦¦sin ±¦
Figure 1.5
Since the outer product is a product of
orthogonal vectors, it is anticommutative:


Some example of physical
magnitudes which are outer products are the
angular momentum, the torque, etc.
When two vectors are permuted, the
sign of the oriented angle is changed. Then the cosine remains equal while the sine
changes the sign. Because of this, the inner product is commutative while the outer

product is anticommutative. Now, we can rewrite the geometric product as the sum of
both products:

Figure 1.6
From here, the inner and outer
products can be written using the geometric

a·b =

a §b=

In conclusion, the geometric product of two proportional vectors is commutative
whereas that of two orthogonal vectors is anticommutative, just for the pure cases of
outer and inner products. The outer, inner and geometric products of two vectors only
depend upon the moduli of the vectors and the angle between them. When both vectors
are rotated preserving the angle that they form, the products are also preserved (figure
How is the absolute value of the product of two vectors? Since the inner and
outer product are linearly independent and orthogonal magnitudes, the modulus of the
geometric product must be calculated through a generalisation of the Pythagorean

¦ a b ¦2 = ¦a · b¦2 +¦a § b¦2
ab=a·b+a§b ’

¦ a b ¦2 = ¦ a ¦2 ¦ b ¦2 ( cos2± + sin2± ) = ¦ a ¦2 ¦ b ¦2

That is, the modulus of the geometric product is the product of the modulus of
each vector:

¦ a b ¦=¦ a ¦¦ b ¦

Product of three vectors: associative property

It is demanded as the fourth property that the product of three vectors be

4) u(vw)=(uv)w=uvw

Hence we can remove parenthesis in multiple products and with the foregoing
properties we can deduce how the product operates upon vectors.
We wish to multiply a vector a by a product of two vectors b, c. We ignore the
result of the product of three vectors with different orientations except when two
adjacent factors are proportional. We have seen that the product of two vectors depends
only on the angle between them. Therefore the parallelogram formed by b and c can be

rotated until b has, in the new orientation, the same direction as a. If b' and c' are the
vectors b and c with the new orientation (figure 1.7) then:

b c = b' c'

a ( b c ) = a ( b' c' )
Figure 1.7
and by the associative property:

a ( b c ) = ( a b' ) c'

Since a and b' have the same
direction, a b' = ¦a¦¦b¦ is a real
number and the triple product is a
vector with the direction of c' whose
length is increased by this amount:

a ( b c ) = ¦a¦¦b¦ c'

It follows that the modulus of the product of three vectors is the product of their

¦a b c¦ = ¦a¦¦b¦¦c¦

On the other hand, a can be Figure 1.8
firstly multiplied by b, and after this we
can rotate the parallelogram formed by
both vectors until b has, in the new
orientation, the same direction as c
(figure 1.8). Then:

( a b ) c = a'' ( b'' c ) = a'' ¦b¦¦c¦

Although the geometric
construction differs from the foregoing
one, the figures clearly show that the
triple product yields the same vector, as expected from the associative property.

( a b ) c = a'' ¦b¦¦c¦ = ¦c¦¦b¦ a'' = c b'' a'' = c ( b a )

That is, the triple product has the property:


which I call the permutative property: every vector can be permuted with a vector
located two positions farther in a product, although it does not commute with the
neighbouring vectors. The permutative property implies that any pair of vectors in a
product separated by an odd number of vectors can be permuted. For example:


The permutative property is characteristic of the plane and it is also valid for the
space whenever the three vectors are coplanar. This property is related with the fact that
the product of complex numbers is commutative.

Product of four vectors

The product of four vectors can be deduced from the former reasoning. In order
to multiply two pair of vectors, rotate the parallelogram formed by a and b until b' has
the direction of c. Then the product is the parallelogram formed by a' and d but
increased by the modulus of b and c:

a b c d = a' b' c d = a' ¦ b¦¦ c¦ d = ¦b¦¦ c¦ a' d

Now let us see the special case when a = c and b = d. If both vectors a, b have the same
direction, the square of their product is a
positive real number:

( a b ) 2 = a2 b 2 > 0
a || b

If both vectors are perpendicular, we must
rotate the parallelogram through π/2 until b'
has the same direction as a (figure 1.9). Then
a' and b are proportional but having opposite
signs. Therefore, the square of a product of
two orthogonal vectors is always negative:
Figure 1.9
( a b )2 = a' b' a b = a'¦b¦¦a¦b = ’a2 b2 <0

Likewise, the square of an outer product of any two vectors is also negative.

Inverse and quotient of two vectors

The inverse of a vector a is that vector whose multiplication by a gives the unity.
Only the vectors which are proportional have a real product. Hence the inverse vector
has the same direction and inverse modulus:

a ’1 = a¦a¦ ’2 a ’1 a = a a ’1 = 1

The quotient of vectors is a product for an inverse vector, which depends on the
order of the factors because the product is not commutative:

a ’1 b ≠ b a ’1

Obviously the quotient of proportional vectors with the same direction and sense
is equal to the quotient of their moduli. When the vectors have different directions, their
quotient can be represented by a parallelogram, which allows to extend the concept of

vector proportionality. We say that a is proportional to c as b is to d when their moduli
are proportional and the angle between a and c is equal to the angle between b and d1:

a c ’1 = b d ’1 ¦a¦¦c¦’1 =¦b¦¦d¦’1 ±(a, c) = ±(b, d)
” and

Then the parallelogram formed by a and b is similar to that formed by c and d,
being ±(a, c) the angle of rotation from the first to the second one.
The inverse of a product of several vectors is the product of the inverses with the
exchanged order, as can be easily seen from the associative property:

( a b c ) ’1 = c ’1 b ’1 a ’1

Hierarchy of algebraic operations

Like the algebra of real numbers, and in order to simplify the algebraic notation,
I shall use the following hierarchy for the vector operations explained above:
1) The parenthesis, whose content will be firstly operated.
2) The power with any exponent (square, inverse, etc.).
3) The outer and inner product, which have the same hierarchy level but must
be operated before the geometric product.
4) The geometric product.
5) The addition.

As an example, some algebraic expressions are given with the simplified
expression at the left hand and its meaning using parenthesis at the right hand:


a2 b § c + 3 = ( ( a2 ) ( b § c ) ) + 3


Geometric algebra of the vectorial plane

The set of all the vectors on the plane together with the operations of vector
addition and product of vectors by real numbers is a two-dimensional space usually
called the vector plane V2. The geometric product generates new elements (the complex
numbers) not included in the vector plane. So, the geometric (or Clifford) algebra of a
vectorial space is defined as the set of all the elements generated by products of vectors,
for which the geometric product is an inner operation. The geometric algebra of the
Euclidean vector plane is usually noted as Cl2,0(R) or simply as Cl2. Making a
parallelism with probability, the sample space is the set of elemental results of a certain
William Rowan Hamilton defined the quaternions as quotients of two vectors in the way that
similar parallelograms located at the same plane in the space represent the same quaternion
(Elements of Quaternions, posthumously edited in 1866, Chelsea Publishers 1969, vol I, see p.
113 and fig. 34). In the vectorial plane a quaternion is reduced to a complex number. The
quaternions were discovered by Hamilton (October 16th, 1843) before the geometric product by
Clifford (1878).

random experiment. From the sample space „¦, the union ∪ and intersection © generate
the Boole algebra A(„¦), which includes all the possible events. In the same manner, the
addition and geometric product generate the geometric algebra of the vectorial space.
Then both sample and vectorial space play similar roles as generators of the Boole and
geometric (Clifford) algebras respectively.


1.1 Prove that the sum of the squares of the diagonals of any parallelogram is equal to
the sum of the squares of the four sides. Think about the sides as vectors.

1.2 Prove the following identity:

( a · b )2 ’ ( a § b )2 = a2 b2

1.3 Prove that:

a § b c § d + a § c d § b+ a § d b § c = 0
1.4 Prove that:

1.5 Prove the permutative property resolving b and c into the components which are
proportional and orthogonal to the vector a.

1.6 Prove the Heron™s formula for the area of the triangle:

A = s (s ’ a ) ( s ’ b ) ( s ’ c )

where a, b and c are the sides and s the semiperimeter:



Linear combination of two vectors

Every vector w on the plane is always a linear combination of two independent
vectors u and v:

w=au+bv a, b real

Because of this, the plane has dimension equal to 2. In order to calculate the coefficients
of linear combination a and b, we multiply w by u and v at both sides and subtract the

u w = a u2 + b u v and w u = a u2 + b v u ’ uw’wu=b(uv’vu)

v w = a v u + b v2 and w v = a u v + b v2 ’ wv’vw=a(uv’vu)

to obtain:

w§v u§w
a= b=
u§v u§v

The resolution of a vector as a linear combination of two independent vectors is a
very frequent operation and also the foundation of the coordinates method.

Base and components

Any set of two independent vectors {e1 , e2} can be taken as a base of the vector
plane. Every vector u can be written as linear combination of the base vectors:

u = u1 e1 + u2 e2

The coefficients of this linear combination u1, u2 are the components of the vector in this
base. Then a vector will be represented as a pair of components:

u = ( u1, u2 )

The components depend on the base, so that a change of base leads to a change of the
components of the given vector.
We must only add components to add vectors:

v = ( v1, v2 )

u + v = ( u1 + v1 , u2 + v2)

The expression of the geometric product with components is obtained by means of
the distributive property:

u v = ( u1 e1 + u2 e2 ) ( v1 e1 + v2 e2 ) = u1 v1 e12 + u2 v2 e22 + u1 v2 e1 e2+ u2 v1 e2 e1

u v = u1 v1 | e1 |2 + u2 v2 | e2 |2 +( u1 v2 + u2 v1 ) e1 · e2 + ( u1 v2 ’ u2 v1 ) e1 § e2

Hence the expression of the square of the vector modulus written in components is:

| u |2 = u2 = u12 | e1 |2 + u22 | e2 |2 + 2 u1 u2 e1 · e2

Orthonormal bases

Any base is valid to describe vectors using
Figure 2.1
components, although the orthonormal bases, for
which both e1 and e2 are unitary and perpendicular
(such as the canonical base shown in the figure 2.1),
are the more convenient and suitable:

e1 ⊥ e2 ¦e1¦ = ¦e2¦ = 1

For every orthonormal base :

e12 = e22 = 1 e1 e2 = ’ e2 e1

The product e1 e2 represents a square of unity area. The square power of this product is
equal to ’1:

( e1 e2 )2 = e1 e2 e1 e2 = ’ e1 e1 e2 e2 = ’1

For an orthonormal base, the geometric product of two vectors becomes:

u v = u1 v1 + u2 v2 + ( u1 v2 ’ u2 v1 ) e1 e2

Note that the first and second summands are real while the third is an area. Therefore it
follows that they are respectively the inner and outer products:

u § v = ( u1 v2 ’ u2 v1 ) e1 e2
u · v = u1 v1 + u2 v2

Also, the modulus of a vector is calculated from the self inner product:

¦u¦2 = u12 + u22

Applications of the formulae for the products

The first application is the calculation of the angle between two vectors:

u1 v1 + u 2 v 2 u1 v 2 ’ u 2 v 1
cos ± = sin ± =
uv uv

The values of sine and cosine determine a unique angle ± in the range 0<±<2π.
The angle between two vectors is a sensed magnitude having positive sign if it is
counterclockwise and negative sign if it is clockwise. Thus this angle depends on the
order of the vectors in the outer (and geometric) product. For example, let us consider the
vectors (figure 2.2) u and v:

u =2 2 v =5
u = (’2, 2 ) v=(4,3)

1 7
cos ± (u, v ) = cos ± (v, u ) = ’ sin ± (u, v ) = ’sin ± (v, u ) = ’
52 52

± (u, v) = 4.5705... ±( v, u ) = 1.7127...

Also we may take the angle ±( u, v ) = 4.5705...’2π = ’1.7127... The angle so obtained is
always that going from the first to the second vector, being unique within a period.
Other application of the outer product is the calculus of areas. Using the
expression with components, the area
(considered as a positive real number) Figura 2.2
of the parallelogram formed by u and v

A = ¦u § v¦ = ¦u1 v2 ’ u2 v1¦= 14

When calculating the area of
any triangle we must only divide the
outer product of any two sides by 2.


2.1 Let (u1, u2) and (v1, v2) be the
components of the vectors u and v in
the canonical base. Prove geometrically that the area of the parallelogram formed by both
vectors is the modulus of the outer product ¦u § v¦ = u1 v2 ’ u2 v1 .

2.2 Calculate the area of the triangle whose sides are the vectors (3, 5), (’2, ’3) and their
addition (1, 2).

2.3 Prove the permutative property using components: a b c = c b a.

2.4 Calculate the angle between the vectors u = 2 e1 + 3 e2 and v = ’3 e1 + 4 e2 in the
canonical base.

2.5 Consider a base where e1 has modulus 1, e2 has modulus 2 and the angle between both
vectors is π/3. Calculate the angle between u = 2 e1 + 3 e2 and v = ’3 e1 + 4 e2 .

2.6 In the canonical base v = (3, ’5). Calculate the components of this vector in a new
base { u1, u2 } if u1 = (2, ’1) and u2 = (5, ’3).


Subalgebra of the complex numbers

If {e1, e2} is the canonical base of the vector plane V2, its geometric algebra is
defined as the vector space generated by the elements {1, e1, e2, e1e2} together with the
geometric product, so that the geometric algebra Cl2 has dimension four. The unitary area
e1e2 is usually noted as e12. Due to the associative character of the geometric product, the
geometric algebra is an associative algebra with identity. The complete table for the
geometric product is the following:

1 e1 e2 e12
1 e1 e2 e12
¦ e1
e1 1 e12 e2
¦ e2 ’e12 1 ’e1
¦ e12 ’e2 ’1
e12 e1

Note that the subset of elements containing only real numbers and areas is closed
for the product:

1 e12
1 e12
¦ e12 ’1

This is the subalgebra of complex numbers. e12 , the imaginary unity, is usually noted as i.
They are called complex numbers because their product is commutative like for the real

Binomial, polar and trigonometric form of a complex number

Every complex number z written in the binomial form is:

z = a + b e12 a, b real

where a and b are the real and imaginary components respectively. The modulus of a
complex number is calculated in the same way as the modulus of any element of the
geometric algebra by means of the Pythagorean theorem:

¦z¦2 = ¦ a + b e12¦2 = a2 + b2

Since every complex number can be written as a product of two vectors u and v
forming a certain angle ±:

Figure 3.1
z = u v = ¦u¦ ¦v¦ ( cos ± + e12 sin ± )

we may represent a complex number as
a parallelogram with sides being the
vectors u and v. But there are infinite
pairs of vectors u' and v' whose product
is the complex z provided that:

± = ±'
¦u¦¦v¦ = ¦u'¦¦v'¦ and

All the parallelograms having
the same area and obliquity represent a
given complex (they are equivalent) independently of the length and orientation of one
side.(figure 3.1). The trigonometric and polar forms of a complex number z specifies its
modulus ¦z¦and argument ± :

z = ¦z¦ ( cos ± + e12 sin ± ) = ¦z¦ ±

A complex number can be written using the exponential function, but firstly we
must prove the Euler™s identity:

exp(± e12) = cos ± + e12 sin ± ± real

The exponential of an imaginary number is
defined in the same manner as for a real Figure 3.2

lim «1 + ± e12 
exp(± e12 ) = ¬ ·
n ’ ∞

As shown in the figure 3.2 (for n = 5), the limit
is a power of n rotations with angle ±,/n or
equivalently a rotation of angle ±.
Now a complex number written in
exponential form is:

z = ¦z¦ exp(± e12 )

Algebraic operations with complex numbers

Each algebraic operation is more easily calculated in a form than in another
according to the following scheme:

addition / subtraction binomial form

product / quotient binomial or polar form

powers / roots polar form

The binomial form is suitable for the addition because both real components must
be added and also the imaginary ones, e.g.:

t = 2 ’ 5 e12
z = 3 + 4 e12

z + t = 3 + 4 e12 + 2 ’ 5 e12 = 5 ’ e12

If the complex numbers are written in another form, they must be converted to the
binomial form, e.g.

z = 23π / 4 t = 4π / 6

« 
3π 3π « π
z + t = 2 3π / 4 + 4 π / 6 = 2 ¬ cos + e12 sin · + 4 ¬ cos + e12 sin ·
4 4 6 6

 

( )
« 2 «3 1
= 2 ¬’ ·+4¬ + e12 · = ’ 2 + 2 3 + e12
+ e12 2+2
¬2 2· ¬2 2·
   

In order to write the result in polar form, the modulus must be calculated:

¦ z + t ¦2 = ( ’ 2 + 2 3 )2 + ( 2 + 2 )2 = 20 + 4 ( 2 ’ 6 )

¦ z + t ¦ = 2 5 + 2 ’ 6 = 3.9823...

and also the argument from the cosine and sine obtained as quotient of the real and
imaginary components respectively divided by the modulus:

’ 2+2 3 2+2
± = 1.0301...
cos ± = sin± =
2 5+ 2 ’ 6 2 5+ 2 ’ 6

z + t = 3.98231.0301

When multiplying complex numbers in binomial form, we apply the distributive
property taken into account that the square of the imaginary unity is ’1, e.g.:

(’2 + 5 e12 ) ( 3 ’4 e12 ) = ’6 + 8 e12 + 15 e12 ’ 20 ( e12 )2 = ’6 + 20 + 23 e12 = 14 + 23 e12

The exponential of an addition of arguments (real or complex) is equal to the
product of exponential functions of each argument. Applying this characteristic property
to the product of complex numbers in exponential form, we have:

z t = ¦z¦ exp(± e12 ) ¦t¦ exp( β e12 ) = ¦z¦ ¦t¦ exp[ (± + β ) e12 ]

from where the product of complex numbers in polar form is obtained by multiplying
both moduli and adding both arguments.:

¦z¦± ¦t¦β = ¦z¦ ¦t¦ ± + β

One may subtract 2π to the
resulting argument in order to keep it

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