research article a porism concerning cyclic...
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Hindawi Publishing CorporationGeometryVolume 2013 Article ID 483727 5 pageshttpdxdoiorg1011552013483727
Research ArticleA Porism Concerning Cyclic Quadrilaterals
Jerzy Kocik
Department of Mathematics Southern Illinois University Carbondale IL 62901 USA
Correspondence should be addressed to Jerzy Kocik jkociksiuedu
Received 26 April 2013 Accepted 15 July 2013
Academic Editor Michel Planat
Copyright copy 2013 Jerzy Kocik This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We present a geometric theorem on a porism about cyclic quadrilaterals namely the existence of an infinite number of cyclicquadrilaterals through four fixed collinear points once one exists Also a technique of proving such properties with the use ofpseudounitary traceless matrices is presented A similar property holds for general quadrics as well as for the circle
1 Introduction
Inscribe a butterfly-like quadrilateral in a circle and draw aline 119871 see Figure 1 The sides of the quadrilateral will cut theline at four (not necessarily distinct) points It turns out thatas we continuously deform the inscribed quadrilateral thepoints of intersection remain invariant
More precisely think of the quadrilateral as a path fromvertex119883 through collinear points119875119877 119878 and119876 back to119883 (seeFigure 4 left) If we redraw the path starting from anotherpoint on the circle but passing through the same points onthe line in the same order the path closes to form an inscribedpolygon that is we will arrive at the starting point
In spirit this startling property is similar to Steinerrsquosfamous porism [1 2] which states that once we find twocircles one inner to the other such that a closed chain ofneighbor-wise tangent circles inscribed in the region betweenthem is possible then an infinite number of such inscribedchains exist (Figure 2) One may set the initial circle at anyposition and the chain will close with tangency Yet anothergeometric phenomenon in the same category is Ponceletrsquosporism [3ndash5]
The property for the cyclic quadrilateral described at theoutset may be restated similarly if four points on a lineadmit a cyclic quadrilateral then an infinite number of suchquadrilaterals inscribed in the same circle exist Hence theterm porism is justified
In the next sections we restate the theorem and define amap of reversion through point whichmay be represented bypseudounitary matricesThe technique developed allows one
to prove the theorem aswell as a diagrammatic representationof the relativistic addition of velocities presented elsewhere[6] A slight modification to arbitrary two-dimensional Clif-ford algebras allows one to modify the result to hold forhyperbolas and provides a geometric realization of trigono-metric tangent-like addition
2 The Main Result
Let us present the result more formally Reversion of a point119860 on a circle through point 119875 gives point P(119860) on the circlesuch that points 119860 119875 and P(119860) are collinear (see Figure 3)We can define it more precisely as follows
Definition 1 Given a circle 119870 a reversion through point 119875 notin
119870 is a map P 119870 rarr 119870 119860 997891rarr P(119860) such that points 119860 119875and P(119860) are collinear and P(119860) = 119860 If 119875 isin 119870 then for any119860 isin 119870 one defines P(119860) = 119875
By boldP we denote the reversionmapdefined by point119875(not bold) Clearly reversion through 119875 notin 119870 is an involutionP2 = id and therefore invertibleThe product of reversions isin general not commutative PQ =QP The main result maynow be given a concise form as follows
Theorem 2 Let 119875 119876 119877 and 119878 be collinear points 119870 acircle and Z = SRQP the composition of the correspondingreversions Then
if exist119883 isin 119870 Z (119883) = 119883 then forall119883 isin 119870 Z (119883) = 119883(1)
2 Geometry
L
Figure 1 Porism on concyclic quadrilaterals through collinearpoints
Figure 2 Steinerrsquos porismThe chain may be redrawn starting fromarbitrary position of the starting circle
The quadrilateral [119883P(119883)QP(119883)RQP(119883)] may beviewed as a member of a family parametrized by 119883 As119883 moves along the circle the quadrilateral continuouslychanges its form while the points of intersection with the lineremain invariant The quadrilateral appears to be ldquorotatingrdquowith points 119875 119876 119877 and 119878 playing the role of ldquoaxesrdquo of thismotion For an animated demonstration see [7]
Here is an equivalent version of Theorem 2
Theorem 3 Let 119870 be a circle and 119871 a line with three points119875 119876 and 119877 Let 119883 be a point on the circle The point 119878 anintersection of line 119871 with line [RQP(119883) 119883] does not dependon 119883 isin 119870
In other words for any three collinear points 119875 119876 and119877 isin 119871 there exists a unique point 119878 on 119871 such that any cyclicquadrilateral inscribed in 119870 passing through 119877 119875 and 119876 (inthe same order) must also pass through 119878
A more general statement holds
Theorem 4 The composition of three point reversions PQRof a circle is a reversion if and only if points P Q and R arecollinear
Remark 5 The figures present quadrilaterals in the butterflyshape for convenience Clearly they can have untwistedshape and also the points of intersection can lie outside thecircle and the line does not need to intersect the circle
3 Reversion CalculusmdashMatrix Representation
In order to prove the theorem we develop a technique thatuses complex numbers and matrices Interpret each point 119875as a complex number 119901 isin C Without loss of generality wewill assume that 119870 is the unit circle 119870 = 119911 isin C |119911|
2=
1 Complex conjugation is denoted in two ways Here is ourmain tool
P
A
P(A)
Figure 3 Reversion of 119860 through 119875
X
P S Q R
K
L
QP(X)
X
P S Q R
P(X)
RQP(X)
Figure 4 Porism details
Theorem6 Reversion in the unit circle119870 through point119901 isin C
corresponds to a Mobius transformation
119911 997888rarr 1199111015840= [
1 minus119901
119901 minus1] sdot 119911 =
119911 minus 119901
119901119911 minus 1 (2)
Proof First we check that reversions leave the unit circleinvariant |119911|2 = 1 rArr |119911
1015840|2
= 1
10038161003816100381610038161003816119911101584010038161003816100381610038161003816
2
=119911 minus 119901
119901119911 minus 1(119911 minus 119901
119901119911 minus 1)
lowast
=|119911|2minus 119911119901 minus 119901119911 +
10038161003816100381610038161199011003816100381610038161003816
2
10038161003816100381610038161199011003816100381610038161003816
2
|119911|2minus 119901119911 minus 119911119901 + 1
=1 minus 119911119901 minus 119901119911 +
10038161003816100381610038161199011003816100381610038161003816
2
10038161003816100381610038161199011003816100381610038161003816
2
minus 119901119911 minus 119911119901 + 1
= 1
(3)
Next we show that for any 119911 isin 119870 points (119911 119901 1199111015840) arecollinear We may establish that by checking that (1199111015840 minus 119901)
differs from (119911 minus 119901) only by scaling by a real number Indeedtake their ratio
1199111015840minus 119901
119911 minus 119901=(119911 minus 119901) (119901119911 minus 1) minus 119901
119911 minus 119901=(119911 minus 119901) minus 119901 (119901119911 minus 1)
(119911 minus 119901) (119901119911 minus 1)
=1 minus
10038161003816100381610038161199011003816100381610038161003816
2
119901119911 + 119901119911 minus10038161003816100381610038161199011003816100381610038161003816
2
minus 1
=1 minus
10038161003816100381610038161199011003816100381610038161003816
2
2Re (119901119911) minus 10038161003816100381610038161199011003816100381610038161003816
2
minus 1
isin R
(4)
where to get the first expression in the second line wemultiplied the numerator and the denominator by 119911 and usedthe fact that |119911|2 = 1 It is easy (and not necessary) to checkthat 2 = id (up to Mobius equivalence)
Geometry 3
A more abstract definition of the matrix representationemerges Namely denote the pseudounitary group of two-by-two complex matrices
119880 (1 1) = 119860 isin C otimes C | 119860lowast119869119860 = 119869 (5)
where 119869 denotes the diagonal matrix 119869 = diag (1 minus1) andthe star ldquolowastrdquo denotes usual Hermitian transpose Let sim be theequivalence relation among matrices 119860 sim 119861 if there exists120582 isin C 120582 = 0 such that 119860 = 120582119861 Then we have a projectiveversion of the pseudounitary group
119875119880 (1 1) = 119880 (1 1) sim (6)
which is equivalent to its use for Mobius transformationsNow any element of the group may be represented by amatrix up to a scale factor The essence of Theorem 2 is thatreversions correspond to such matrices with vanishing traceTr119860 = 0
Note that we do not follow the tradition of normalizingthe determinants as is typically done for the representationof the modular group 119875119878119871(2Z)
4 Algebraic Proof of the Porism
We will now prove the main results
Proof of the Main Theorem Consider the product of threeconsecutive reversions through points 119875 119876 and 119877 as rep-resented by matrices
119872 = [1 minus119903
119903 minus1] [
1 minus119902
119902 minus1] [
1 minus119901
119901 minus1]
= [1 minus 119901119902 + 119902119903 minus 119903119901 minus119901 + 119902 minus 119903 + 119901119902119903
119901 minus 119902 + 119903 minus 119901119902119903 minus1 + 119901119902 minus 119902119903 + 119903119901]
(7)
Note that 11987212
= minus11987221 but we still need to see whether
Tr119872 = 0 Recall the assumption that 119901 119902 and 119903 are collinearIf 119903 = 119898119901 + 119899119902 where 119898 + 119899 = 1 119898 119899 isin R we observe thatthe diagonal elements are real
11987211= 1 minus 119898 (119901119902 + 119901119902) + 119898
10038161003816100381610038161199011003816100381610038161003816
2
minus 11989910038161003816100381610038161199021003816100381610038161003816
2
isin R
11987222= minus119872
11
(8)
Dividing every entry by11987211(Mobius equivalence) we arrive
at
119872 =
[[[
[
1minus119901 + 119902 minus 119903 + 119901119902119903
1 minus 119901119902 minus 119902119903 + 119903119901
119901 minus 119902 + 119903 minus 119901119902119903
1 minus 119901119902 minus 119902119903 + 119903119901minus1
]]]
]
(9)
which is visibly a matrix of reversion defining uniquely thepoint of reversion
119904 =119901 minus 119902 + 119903 minus 119901119902119903
1 minus 119901119902 minus 119902119903 + 119903119901 (10)
What remains is to verify that 119904 is collinear with 119901 and 119902 (andtherefore 119903) This may be done algebraically by checking that
Figure 5 A porism of cyclic polygons
(119904 minus 119901)(119903 minus 119902) isin R but it also follows neatly by geometrythat there are two points on the circle collinear with 119875 119876and 119877 say 119860 and 119861 Since RQP(119860) = 119861 and therefore S(119860)must be 119861 thus 119878 lies on the same line We have proved thatgiven a circle119870 for any three points 119875 119876 and 119877 on a line 119871there exists a point 119878 on 119871 such that SPQR = id or S = PQRbecause S2 = id The main theorem follows
The theorem generalizes to cyclic 119899-gons (see Figure 5)
Theorem 7 Let P = P2119899P2119899minus1
P2P1be the composition of
point reversions of a circle 119870 for some collinear set of an evennumber of (not necessarily distinct) points P
1 P
2119899 Then
(exist119883 isin 119870 P (119883) = 119883) 997904rArr (forall119883 isin 119870 P (119883) = 119883) (11)
Proof By Theorem 4 any three consecutive maps in thestring Pmay be replaced by oneThis reduces the string of aneven number of reversions to a string of four which is the caseproven as Theorem 2 Further reduction to two reversionswould necessarily produce P
1P1= id
Remark 8 Theorem 7 suggests a concept of a ldquoternary semi-group of pointsrdquo on a (compactified) line where the productrequires three elements to produce one
119886 119887 119888 997888rarr 119886119887119888 (12)
The following may be considered as ldquoaxiomsrdquo
(1) (119886119887119888)119889119890 = 119886(119887119888119889)119890 = 119886119887(119888119889119890) (associativity)(2) 119886119887119887 = 119887119887119886 = 119886 (absorption of squares)(3) 119886119887119888 = 119888119887119886 (mirror inversion of triples)
The point reversions define a realization of this algebra
Yet another concept comes from these notes two pairs ofpoints (119875 119876) and (119878 119877) on a line are conjugate with respectto a circle if they belong to inscribed angles based on the samechord (see Figure 4 left) in other words if PQ = SR or ifthere exists an inscribed quadrilateral which intersects 119871 in119875 119876 119877 and 119878 In matrix terms
119872(119902)119872(119901) =[1 minus 119902119901 119902 minus 119901
119902 minus 119901 1 minus 119902119901]
=
[[[
[
1119902 minus 119901
1 minus 119902119901
119902 minus 119901
1 minus 119902119901
1 minus 119902119901
1 minus 119902119901
]]]
]
=
[[[
[
1119902 minus 119901
1 minus 119902119901
(119902 minus 119901
1 minus 119902119901)
lowast1 minus 119902119901
1 minus 119902119901
]]]
]
4 Geometry
119872(119903)119872 (119904) =[1 minus 119903119904 119903 minus 119904
119903 minus 119904 1 minus 119903119904]
=
[[[
[
1119903 minus 119904
1 minus 119903119904
119903 minus 119904
1 minus 119903119904
1 minus 119903119904
1 minus 119903119904
]]]
]
=
[[[
[
1119903 minus 119904
1 minus 119903119904
(119903 minus 119904
1 minus 119903119904)
lowast1 minus 119903119904
1 minus 119903119904
]]]
]
(13)
from which this convenient formula results119903 minus 119904
1 minus 119903119904=
119902 minus 119901
1 minus 119902119901 (14)
provided 119901 119902 119903 119904 are collinear
5 Application Relativistic Velocities
Take the real line and the unit circle as ingredients for themodel Consider a quadrilateral that goes through threecollinear points the origin (0) and two points representedby real numbers 119886 and 119887 FromTheorem 3 it follows that thefourth point on the real line must be
119872(119887)119872 (119886)119872 (0)
= [1 minus119887
119887 minus1] [
1 minus119886
119886 minus1] [
1 0
0 minus1]
= [1 + 119886119887 minus119886 minus 119887
119886 + 119887 minus1 minus 119886119887] sim
[[[
[
1 minus119886 + 119887
1 + 119886119887
119886 + 119887
1 + 119886119887minus1
]]]
]
(15)
We used the fact that conjugation does nothing to realnumbers Thus the fourth point on R has coordinate
119886 oplus 119887 =119886 + 119887
1 + 119886119887 (16)
but this happens to be the formula for relativistic addition ofvelocities (in the natural units in which the speed of light is1) Thus we obtain its geometric interpretation presented inFigure 6 A conventional derivation of this diagram may befound in [6] See also [7] for an interactive applet
The segment through the origin does not have to bevertical for the device to work (by Theorem 3) but is set sofor simplicity
6 Further Generalizations
The porism described here for circles is also valid for anyquadric see Figure 7 What is intriguing three cases (vizcircle ellipse and a pair of parallel lines) correspond to thethree possible 2-dimensional Clifford algebras each of theform of ldquogeneralized number planerdquo 119886 + 119887119890 where 119890 is anldquoimaginary unitrdquo whose square is 1 minus1 or 0 (see Table 1)
The case of the circle corresponds to complex numbers asdescribed in the previous sections The case of the hyperbolacorresponds to ldquoduplex numbersrdquo called also hyperbolicnumbers or split-complex numbers
D = 119886 + 119887119868 119886 119887 isin R 1198682= 1 (17)
a b0 a oplus bR
Figure 6 Relativistic addition of velocities
Figure 7 Porism on quadrilaterals inscribed in quadrics
Table 1
Algebra ldquoSphererdquo Formula representedgeometrically
C (complex numbers) Circle Relativistic velocitiesD (hyperbolic numbers) Hyperbola Trigonometric tangentsG (dual numbers) Parallel lines Regular addition
They aremdashlike complex numbersmdasha two-dimensional unitalalgebra except that its ldquoimaginary unitrdquo 119868 is 1 when squaredAs complex numbers are related to rotations duplex numberscorrespond to hyperbolic rotations They were introduced in[8] as tessarines and represent a Clifford algebra R
01 They
found useful applications for example in [9] a ldquohyperbolicquantum mechanicsrdquo was introduced Here is the theoremcorresponding toTheorem 6
Proposition 9 The reversion through a point 119901 isin D withrespect to the hyperbola 1199092minus1199102 = 1 is represented by thematrix
119901 119911 997888rarr 1199111015840= [
1 minus119901
119901 minus1] sdot 119911 =
119911 minus 119901
119901119911 minus 1(|119911|2= 1) (18)
while for branch 1199092 minus 1199102 = minus1 it is
119901 119911 997888rarr 1199111015840= [
minus1 119901
119901 1] sdot 119911 =
minus119911 + 119901
119901119911 + 1(|119911|2= minus1) (19)
Proof For (18) follow the lines of the proof of the maintheorem To get (19) multiply the ingredients of (18) by 119868to switch the axes use (18) and then multiply by 119868 again torestore the original axes
For the last case of parallel lines we can use the sameprevious matrix calculus but replace the algebra by dualnumbers
G = 119886 + 119887119868 119886 119887 isin R 1198682= 0 (20)
Geometry 5
minus1 +1b a oplus b
0a
Figure 8 Trigonometric tangent formula visualized
P
P L
L
K
K
Figure 9 Porism under inversion extends to circles replacing linesand vice versa
Transformations (18) and (19) apply for two cases vertical andhorizontal lines respectively (|119911|2 = 1 versus |119911|2 = minus1 in thehyperbolic norm)
It is a rather pleasant surprise to find the algebra andgeometry interacting at such a basic level in an entirelynontrivial way Repeating a construction analogous to the onein Section 5 that results in a geometric tool for addition ofrelativistic velocities to the hyperbolic numbers gives a simi-lar geometric diagram representing addition of trigonometrictangents see Figure 8 Consider
119886 oplus 119887 =119886 + 119887
1 minus 119886119887= tan (arctan 119886 + arctan 119887) (21)
The final comment concerns the obvious extension ofthese results implied by inversion of the standard config-uration like in Figure 1 through a circle Mobius geometryremoves the distinction between circles and lines The linesof the quadrilateral and the line 119871 under inversion maybecome circles The point at infinity where these lines meetbecomes under inversion a point that is part of the porismrsquosconstruction
Thus the general statement is the following given circles119871 and 119870 and an odd number of points 119875 119875
1 1198752 119875
2119899on
119871 suppose that there exists an ordered 2119899-tuple (pencil) ofcircles119862
1 119862
2119899 through119875 such that119875
119894isin 119862119894for every 119894 and
the points of intersections119862119894cap119862119894+1
and1198622119899cap1198621different from
119875 all lie on the circle 119870 then there are infinitely many such2119899-tuples of circles Figure 9 illustrates the theorem for fourcircles For interactive version of these and more examplessee [7]
Acknowledgments
The author is grateful to Philip Feinsilver for his interest inthis work and for his helpful comments Special thanks goalso to creators ofCinderella awonderful software that allowsone to quickly test geometric conjectures and to create niceinteractive applets
References
[1] D Coxeter and S Greitzer Geometry Revisited MathematicalAssociation of America Washington DC USA 1967
[2] EWWeisstein ldquoSteinerChainrdquo FromMathWorldmdashAWolframWeb Resource httpmathworldwolframcomSteinerChainhtml
[3] W Barth and T Bauer ldquoPoncelet theoremsrdquo Expositiones Math-ematicae vol 14 no 2 pp 125ndash144 1996
[4] J-V Poncelet Traite des Proprietes Projectives des FiguresOuvrage Utile A Qui SrsquoOccupent des Applications de la GeometrieDescriptive et Drsquooperations Geometriques sur le Terrain vol 1-2Gauthier-Villars Paris France 2nd edition 1865
[5] E W Weisstein ldquoPonceletrsquos Porismrdquo From Math WorldmdashA Wolfram Web Resource httpmathworldwolframcomPonceletsPorismhtml
[6] J Kocik ldquoDiagram for relativistic addition of velocitiesrdquo Amer-ican Journal of Physics vol 80 no 8 p 720 2012
[7] J Kocik Interactive diagrams httpwwwmathsiueduKocikgeometryhtml
[8] J Cockle ldquoOn certain functions resembling quaternions andon a new imaginary in algebrardquo London-Edinburgh-DublinPhilosophical Magazine vol 33 pp 435ndash439 1848
[9] J Kocik ldquoDuplex numbers diffusion systems and general-ized quantum mechanicsrdquo International Journal of TheoreticalPhysics vol 38 no 8 pp 2221ndash2230 1999
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2 Geometry
L
Figure 1 Porism on concyclic quadrilaterals through collinearpoints
Figure 2 Steinerrsquos porismThe chain may be redrawn starting fromarbitrary position of the starting circle
The quadrilateral [119883P(119883)QP(119883)RQP(119883)] may beviewed as a member of a family parametrized by 119883 As119883 moves along the circle the quadrilateral continuouslychanges its form while the points of intersection with the lineremain invariant The quadrilateral appears to be ldquorotatingrdquowith points 119875 119876 119877 and 119878 playing the role of ldquoaxesrdquo of thismotion For an animated demonstration see [7]
Here is an equivalent version of Theorem 2
Theorem 3 Let 119870 be a circle and 119871 a line with three points119875 119876 and 119877 Let 119883 be a point on the circle The point 119878 anintersection of line 119871 with line [RQP(119883) 119883] does not dependon 119883 isin 119870
In other words for any three collinear points 119875 119876 and119877 isin 119871 there exists a unique point 119878 on 119871 such that any cyclicquadrilateral inscribed in 119870 passing through 119877 119875 and 119876 (inthe same order) must also pass through 119878
A more general statement holds
Theorem 4 The composition of three point reversions PQRof a circle is a reversion if and only if points P Q and R arecollinear
Remark 5 The figures present quadrilaterals in the butterflyshape for convenience Clearly they can have untwistedshape and also the points of intersection can lie outside thecircle and the line does not need to intersect the circle
3 Reversion CalculusmdashMatrix Representation
In order to prove the theorem we develop a technique thatuses complex numbers and matrices Interpret each point 119875as a complex number 119901 isin C Without loss of generality wewill assume that 119870 is the unit circle 119870 = 119911 isin C |119911|
2=
1 Complex conjugation is denoted in two ways Here is ourmain tool
P
A
P(A)
Figure 3 Reversion of 119860 through 119875
X
P S Q R
K
L
QP(X)
X
P S Q R
P(X)
RQP(X)
Figure 4 Porism details
Theorem6 Reversion in the unit circle119870 through point119901 isin C
corresponds to a Mobius transformation
119911 997888rarr 1199111015840= [
1 minus119901
119901 minus1] sdot 119911 =
119911 minus 119901
119901119911 minus 1 (2)
Proof First we check that reversions leave the unit circleinvariant |119911|2 = 1 rArr |119911
1015840|2
= 1
10038161003816100381610038161003816119911101584010038161003816100381610038161003816
2
=119911 minus 119901
119901119911 minus 1(119911 minus 119901
119901119911 minus 1)
lowast
=|119911|2minus 119911119901 minus 119901119911 +
10038161003816100381610038161199011003816100381610038161003816
2
10038161003816100381610038161199011003816100381610038161003816
2
|119911|2minus 119901119911 minus 119911119901 + 1
=1 minus 119911119901 minus 119901119911 +
10038161003816100381610038161199011003816100381610038161003816
2
10038161003816100381610038161199011003816100381610038161003816
2
minus 119901119911 minus 119911119901 + 1
= 1
(3)
Next we show that for any 119911 isin 119870 points (119911 119901 1199111015840) arecollinear We may establish that by checking that (1199111015840 minus 119901)
differs from (119911 minus 119901) only by scaling by a real number Indeedtake their ratio
1199111015840minus 119901
119911 minus 119901=(119911 minus 119901) (119901119911 minus 1) minus 119901
119911 minus 119901=(119911 minus 119901) minus 119901 (119901119911 minus 1)
(119911 minus 119901) (119901119911 minus 1)
=1 minus
10038161003816100381610038161199011003816100381610038161003816
2
119901119911 + 119901119911 minus10038161003816100381610038161199011003816100381610038161003816
2
minus 1
=1 minus
10038161003816100381610038161199011003816100381610038161003816
2
2Re (119901119911) minus 10038161003816100381610038161199011003816100381610038161003816
2
minus 1
isin R
(4)
where to get the first expression in the second line wemultiplied the numerator and the denominator by 119911 and usedthe fact that |119911|2 = 1 It is easy (and not necessary) to checkthat 2 = id (up to Mobius equivalence)
Geometry 3
A more abstract definition of the matrix representationemerges Namely denote the pseudounitary group of two-by-two complex matrices
119880 (1 1) = 119860 isin C otimes C | 119860lowast119869119860 = 119869 (5)
where 119869 denotes the diagonal matrix 119869 = diag (1 minus1) andthe star ldquolowastrdquo denotes usual Hermitian transpose Let sim be theequivalence relation among matrices 119860 sim 119861 if there exists120582 isin C 120582 = 0 such that 119860 = 120582119861 Then we have a projectiveversion of the pseudounitary group
119875119880 (1 1) = 119880 (1 1) sim (6)
which is equivalent to its use for Mobius transformationsNow any element of the group may be represented by amatrix up to a scale factor The essence of Theorem 2 is thatreversions correspond to such matrices with vanishing traceTr119860 = 0
Note that we do not follow the tradition of normalizingthe determinants as is typically done for the representationof the modular group 119875119878119871(2Z)
4 Algebraic Proof of the Porism
We will now prove the main results
Proof of the Main Theorem Consider the product of threeconsecutive reversions through points 119875 119876 and 119877 as rep-resented by matrices
119872 = [1 minus119903
119903 minus1] [
1 minus119902
119902 minus1] [
1 minus119901
119901 minus1]
= [1 minus 119901119902 + 119902119903 minus 119903119901 minus119901 + 119902 minus 119903 + 119901119902119903
119901 minus 119902 + 119903 minus 119901119902119903 minus1 + 119901119902 minus 119902119903 + 119903119901]
(7)
Note that 11987212
= minus11987221 but we still need to see whether
Tr119872 = 0 Recall the assumption that 119901 119902 and 119903 are collinearIf 119903 = 119898119901 + 119899119902 where 119898 + 119899 = 1 119898 119899 isin R we observe thatthe diagonal elements are real
11987211= 1 minus 119898 (119901119902 + 119901119902) + 119898
10038161003816100381610038161199011003816100381610038161003816
2
minus 11989910038161003816100381610038161199021003816100381610038161003816
2
isin R
11987222= minus119872
11
(8)
Dividing every entry by11987211(Mobius equivalence) we arrive
at
119872 =
[[[
[
1minus119901 + 119902 minus 119903 + 119901119902119903
1 minus 119901119902 minus 119902119903 + 119903119901
119901 minus 119902 + 119903 minus 119901119902119903
1 minus 119901119902 minus 119902119903 + 119903119901minus1
]]]
]
(9)
which is visibly a matrix of reversion defining uniquely thepoint of reversion
119904 =119901 minus 119902 + 119903 minus 119901119902119903
1 minus 119901119902 minus 119902119903 + 119903119901 (10)
What remains is to verify that 119904 is collinear with 119901 and 119902 (andtherefore 119903) This may be done algebraically by checking that
Figure 5 A porism of cyclic polygons
(119904 minus 119901)(119903 minus 119902) isin R but it also follows neatly by geometrythat there are two points on the circle collinear with 119875 119876and 119877 say 119860 and 119861 Since RQP(119860) = 119861 and therefore S(119860)must be 119861 thus 119878 lies on the same line We have proved thatgiven a circle119870 for any three points 119875 119876 and 119877 on a line 119871there exists a point 119878 on 119871 such that SPQR = id or S = PQRbecause S2 = id The main theorem follows
The theorem generalizes to cyclic 119899-gons (see Figure 5)
Theorem 7 Let P = P2119899P2119899minus1
P2P1be the composition of
point reversions of a circle 119870 for some collinear set of an evennumber of (not necessarily distinct) points P
1 P
2119899 Then
(exist119883 isin 119870 P (119883) = 119883) 997904rArr (forall119883 isin 119870 P (119883) = 119883) (11)
Proof By Theorem 4 any three consecutive maps in thestring Pmay be replaced by oneThis reduces the string of aneven number of reversions to a string of four which is the caseproven as Theorem 2 Further reduction to two reversionswould necessarily produce P
1P1= id
Remark 8 Theorem 7 suggests a concept of a ldquoternary semi-group of pointsrdquo on a (compactified) line where the productrequires three elements to produce one
119886 119887 119888 997888rarr 119886119887119888 (12)
The following may be considered as ldquoaxiomsrdquo
(1) (119886119887119888)119889119890 = 119886(119887119888119889)119890 = 119886119887(119888119889119890) (associativity)(2) 119886119887119887 = 119887119887119886 = 119886 (absorption of squares)(3) 119886119887119888 = 119888119887119886 (mirror inversion of triples)
The point reversions define a realization of this algebra
Yet another concept comes from these notes two pairs ofpoints (119875 119876) and (119878 119877) on a line are conjugate with respectto a circle if they belong to inscribed angles based on the samechord (see Figure 4 left) in other words if PQ = SR or ifthere exists an inscribed quadrilateral which intersects 119871 in119875 119876 119877 and 119878 In matrix terms
119872(119902)119872(119901) =[1 minus 119902119901 119902 minus 119901
119902 minus 119901 1 minus 119902119901]
=
[[[
[
1119902 minus 119901
1 minus 119902119901
119902 minus 119901
1 minus 119902119901
1 minus 119902119901
1 minus 119902119901
]]]
]
=
[[[
[
1119902 minus 119901
1 minus 119902119901
(119902 minus 119901
1 minus 119902119901)
lowast1 minus 119902119901
1 minus 119902119901
]]]
]
4 Geometry
119872(119903)119872 (119904) =[1 minus 119903119904 119903 minus 119904
119903 minus 119904 1 minus 119903119904]
=
[[[
[
1119903 minus 119904
1 minus 119903119904
119903 minus 119904
1 minus 119903119904
1 minus 119903119904
1 minus 119903119904
]]]
]
=
[[[
[
1119903 minus 119904
1 minus 119903119904
(119903 minus 119904
1 minus 119903119904)
lowast1 minus 119903119904
1 minus 119903119904
]]]
]
(13)
from which this convenient formula results119903 minus 119904
1 minus 119903119904=
119902 minus 119901
1 minus 119902119901 (14)
provided 119901 119902 119903 119904 are collinear
5 Application Relativistic Velocities
Take the real line and the unit circle as ingredients for themodel Consider a quadrilateral that goes through threecollinear points the origin (0) and two points representedby real numbers 119886 and 119887 FromTheorem 3 it follows that thefourth point on the real line must be
119872(119887)119872 (119886)119872 (0)
= [1 minus119887
119887 minus1] [
1 minus119886
119886 minus1] [
1 0
0 minus1]
= [1 + 119886119887 minus119886 minus 119887
119886 + 119887 minus1 minus 119886119887] sim
[[[
[
1 minus119886 + 119887
1 + 119886119887
119886 + 119887
1 + 119886119887minus1
]]]
]
(15)
We used the fact that conjugation does nothing to realnumbers Thus the fourth point on R has coordinate
119886 oplus 119887 =119886 + 119887
1 + 119886119887 (16)
but this happens to be the formula for relativistic addition ofvelocities (in the natural units in which the speed of light is1) Thus we obtain its geometric interpretation presented inFigure 6 A conventional derivation of this diagram may befound in [6] See also [7] for an interactive applet
The segment through the origin does not have to bevertical for the device to work (by Theorem 3) but is set sofor simplicity
6 Further Generalizations
The porism described here for circles is also valid for anyquadric see Figure 7 What is intriguing three cases (vizcircle ellipse and a pair of parallel lines) correspond to thethree possible 2-dimensional Clifford algebras each of theform of ldquogeneralized number planerdquo 119886 + 119887119890 where 119890 is anldquoimaginary unitrdquo whose square is 1 minus1 or 0 (see Table 1)
The case of the circle corresponds to complex numbers asdescribed in the previous sections The case of the hyperbolacorresponds to ldquoduplex numbersrdquo called also hyperbolicnumbers or split-complex numbers
D = 119886 + 119887119868 119886 119887 isin R 1198682= 1 (17)
a b0 a oplus bR
Figure 6 Relativistic addition of velocities
Figure 7 Porism on quadrilaterals inscribed in quadrics
Table 1
Algebra ldquoSphererdquo Formula representedgeometrically
C (complex numbers) Circle Relativistic velocitiesD (hyperbolic numbers) Hyperbola Trigonometric tangentsG (dual numbers) Parallel lines Regular addition
They aremdashlike complex numbersmdasha two-dimensional unitalalgebra except that its ldquoimaginary unitrdquo 119868 is 1 when squaredAs complex numbers are related to rotations duplex numberscorrespond to hyperbolic rotations They were introduced in[8] as tessarines and represent a Clifford algebra R
01 They
found useful applications for example in [9] a ldquohyperbolicquantum mechanicsrdquo was introduced Here is the theoremcorresponding toTheorem 6
Proposition 9 The reversion through a point 119901 isin D withrespect to the hyperbola 1199092minus1199102 = 1 is represented by thematrix
119901 119911 997888rarr 1199111015840= [
1 minus119901
119901 minus1] sdot 119911 =
119911 minus 119901
119901119911 minus 1(|119911|2= 1) (18)
while for branch 1199092 minus 1199102 = minus1 it is
119901 119911 997888rarr 1199111015840= [
minus1 119901
119901 1] sdot 119911 =
minus119911 + 119901
119901119911 + 1(|119911|2= minus1) (19)
Proof For (18) follow the lines of the proof of the maintheorem To get (19) multiply the ingredients of (18) by 119868to switch the axes use (18) and then multiply by 119868 again torestore the original axes
For the last case of parallel lines we can use the sameprevious matrix calculus but replace the algebra by dualnumbers
G = 119886 + 119887119868 119886 119887 isin R 1198682= 0 (20)
Geometry 5
minus1 +1b a oplus b
0a
Figure 8 Trigonometric tangent formula visualized
P
P L
L
K
K
Figure 9 Porism under inversion extends to circles replacing linesand vice versa
Transformations (18) and (19) apply for two cases vertical andhorizontal lines respectively (|119911|2 = 1 versus |119911|2 = minus1 in thehyperbolic norm)
It is a rather pleasant surprise to find the algebra andgeometry interacting at such a basic level in an entirelynontrivial way Repeating a construction analogous to the onein Section 5 that results in a geometric tool for addition ofrelativistic velocities to the hyperbolic numbers gives a simi-lar geometric diagram representing addition of trigonometrictangents see Figure 8 Consider
119886 oplus 119887 =119886 + 119887
1 minus 119886119887= tan (arctan 119886 + arctan 119887) (21)
The final comment concerns the obvious extension ofthese results implied by inversion of the standard config-uration like in Figure 1 through a circle Mobius geometryremoves the distinction between circles and lines The linesof the quadrilateral and the line 119871 under inversion maybecome circles The point at infinity where these lines meetbecomes under inversion a point that is part of the porismrsquosconstruction
Thus the general statement is the following given circles119871 and 119870 and an odd number of points 119875 119875
1 1198752 119875
2119899on
119871 suppose that there exists an ordered 2119899-tuple (pencil) ofcircles119862
1 119862
2119899 through119875 such that119875
119894isin 119862119894for every 119894 and
the points of intersections119862119894cap119862119894+1
and1198622119899cap1198621different from
119875 all lie on the circle 119870 then there are infinitely many such2119899-tuples of circles Figure 9 illustrates the theorem for fourcircles For interactive version of these and more examplessee [7]
Acknowledgments
The author is grateful to Philip Feinsilver for his interest inthis work and for his helpful comments Special thanks goalso to creators ofCinderella awonderful software that allowsone to quickly test geometric conjectures and to create niceinteractive applets
References
[1] D Coxeter and S Greitzer Geometry Revisited MathematicalAssociation of America Washington DC USA 1967
[2] EWWeisstein ldquoSteinerChainrdquo FromMathWorldmdashAWolframWeb Resource httpmathworldwolframcomSteinerChainhtml
[3] W Barth and T Bauer ldquoPoncelet theoremsrdquo Expositiones Math-ematicae vol 14 no 2 pp 125ndash144 1996
[4] J-V Poncelet Traite des Proprietes Projectives des FiguresOuvrage Utile A Qui SrsquoOccupent des Applications de la GeometrieDescriptive et Drsquooperations Geometriques sur le Terrain vol 1-2Gauthier-Villars Paris France 2nd edition 1865
[5] E W Weisstein ldquoPonceletrsquos Porismrdquo From Math WorldmdashA Wolfram Web Resource httpmathworldwolframcomPonceletsPorismhtml
[6] J Kocik ldquoDiagram for relativistic addition of velocitiesrdquo Amer-ican Journal of Physics vol 80 no 8 p 720 2012
[7] J Kocik Interactive diagrams httpwwwmathsiueduKocikgeometryhtml
[8] J Cockle ldquoOn certain functions resembling quaternions andon a new imaginary in algebrardquo London-Edinburgh-DublinPhilosophical Magazine vol 33 pp 435ndash439 1848
[9] J Kocik ldquoDuplex numbers diffusion systems and general-ized quantum mechanicsrdquo International Journal of TheoreticalPhysics vol 38 no 8 pp 2221ndash2230 1999
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Geometry 3
A more abstract definition of the matrix representationemerges Namely denote the pseudounitary group of two-by-two complex matrices
119880 (1 1) = 119860 isin C otimes C | 119860lowast119869119860 = 119869 (5)
where 119869 denotes the diagonal matrix 119869 = diag (1 minus1) andthe star ldquolowastrdquo denotes usual Hermitian transpose Let sim be theequivalence relation among matrices 119860 sim 119861 if there exists120582 isin C 120582 = 0 such that 119860 = 120582119861 Then we have a projectiveversion of the pseudounitary group
119875119880 (1 1) = 119880 (1 1) sim (6)
which is equivalent to its use for Mobius transformationsNow any element of the group may be represented by amatrix up to a scale factor The essence of Theorem 2 is thatreversions correspond to such matrices with vanishing traceTr119860 = 0
Note that we do not follow the tradition of normalizingthe determinants as is typically done for the representationof the modular group 119875119878119871(2Z)
4 Algebraic Proof of the Porism
We will now prove the main results
Proof of the Main Theorem Consider the product of threeconsecutive reversions through points 119875 119876 and 119877 as rep-resented by matrices
119872 = [1 minus119903
119903 minus1] [
1 minus119902
119902 minus1] [
1 minus119901
119901 minus1]
= [1 minus 119901119902 + 119902119903 minus 119903119901 minus119901 + 119902 minus 119903 + 119901119902119903
119901 minus 119902 + 119903 minus 119901119902119903 minus1 + 119901119902 minus 119902119903 + 119903119901]
(7)
Note that 11987212
= minus11987221 but we still need to see whether
Tr119872 = 0 Recall the assumption that 119901 119902 and 119903 are collinearIf 119903 = 119898119901 + 119899119902 where 119898 + 119899 = 1 119898 119899 isin R we observe thatthe diagonal elements are real
11987211= 1 minus 119898 (119901119902 + 119901119902) + 119898
10038161003816100381610038161199011003816100381610038161003816
2
minus 11989910038161003816100381610038161199021003816100381610038161003816
2
isin R
11987222= minus119872
11
(8)
Dividing every entry by11987211(Mobius equivalence) we arrive
at
119872 =
[[[
[
1minus119901 + 119902 minus 119903 + 119901119902119903
1 minus 119901119902 minus 119902119903 + 119903119901
119901 minus 119902 + 119903 minus 119901119902119903
1 minus 119901119902 minus 119902119903 + 119903119901minus1
]]]
]
(9)
which is visibly a matrix of reversion defining uniquely thepoint of reversion
119904 =119901 minus 119902 + 119903 minus 119901119902119903
1 minus 119901119902 minus 119902119903 + 119903119901 (10)
What remains is to verify that 119904 is collinear with 119901 and 119902 (andtherefore 119903) This may be done algebraically by checking that
Figure 5 A porism of cyclic polygons
(119904 minus 119901)(119903 minus 119902) isin R but it also follows neatly by geometrythat there are two points on the circle collinear with 119875 119876and 119877 say 119860 and 119861 Since RQP(119860) = 119861 and therefore S(119860)must be 119861 thus 119878 lies on the same line We have proved thatgiven a circle119870 for any three points 119875 119876 and 119877 on a line 119871there exists a point 119878 on 119871 such that SPQR = id or S = PQRbecause S2 = id The main theorem follows
The theorem generalizes to cyclic 119899-gons (see Figure 5)
Theorem 7 Let P = P2119899P2119899minus1
P2P1be the composition of
point reversions of a circle 119870 for some collinear set of an evennumber of (not necessarily distinct) points P
1 P
2119899 Then
(exist119883 isin 119870 P (119883) = 119883) 997904rArr (forall119883 isin 119870 P (119883) = 119883) (11)
Proof By Theorem 4 any three consecutive maps in thestring Pmay be replaced by oneThis reduces the string of aneven number of reversions to a string of four which is the caseproven as Theorem 2 Further reduction to two reversionswould necessarily produce P
1P1= id
Remark 8 Theorem 7 suggests a concept of a ldquoternary semi-group of pointsrdquo on a (compactified) line where the productrequires three elements to produce one
119886 119887 119888 997888rarr 119886119887119888 (12)
The following may be considered as ldquoaxiomsrdquo
(1) (119886119887119888)119889119890 = 119886(119887119888119889)119890 = 119886119887(119888119889119890) (associativity)(2) 119886119887119887 = 119887119887119886 = 119886 (absorption of squares)(3) 119886119887119888 = 119888119887119886 (mirror inversion of triples)
The point reversions define a realization of this algebra
Yet another concept comes from these notes two pairs ofpoints (119875 119876) and (119878 119877) on a line are conjugate with respectto a circle if they belong to inscribed angles based on the samechord (see Figure 4 left) in other words if PQ = SR or ifthere exists an inscribed quadrilateral which intersects 119871 in119875 119876 119877 and 119878 In matrix terms
119872(119902)119872(119901) =[1 minus 119902119901 119902 minus 119901
119902 minus 119901 1 minus 119902119901]
=
[[[
[
1119902 minus 119901
1 minus 119902119901
119902 minus 119901
1 minus 119902119901
1 minus 119902119901
1 minus 119902119901
]]]
]
=
[[[
[
1119902 minus 119901
1 minus 119902119901
(119902 minus 119901
1 minus 119902119901)
lowast1 minus 119902119901
1 minus 119902119901
]]]
]
4 Geometry
119872(119903)119872 (119904) =[1 minus 119903119904 119903 minus 119904
119903 minus 119904 1 minus 119903119904]
=
[[[
[
1119903 minus 119904
1 minus 119903119904
119903 minus 119904
1 minus 119903119904
1 minus 119903119904
1 minus 119903119904
]]]
]
=
[[[
[
1119903 minus 119904
1 minus 119903119904
(119903 minus 119904
1 minus 119903119904)
lowast1 minus 119903119904
1 minus 119903119904
]]]
]
(13)
from which this convenient formula results119903 minus 119904
1 minus 119903119904=
119902 minus 119901
1 minus 119902119901 (14)
provided 119901 119902 119903 119904 are collinear
5 Application Relativistic Velocities
Take the real line and the unit circle as ingredients for themodel Consider a quadrilateral that goes through threecollinear points the origin (0) and two points representedby real numbers 119886 and 119887 FromTheorem 3 it follows that thefourth point on the real line must be
119872(119887)119872 (119886)119872 (0)
= [1 minus119887
119887 minus1] [
1 minus119886
119886 minus1] [
1 0
0 minus1]
= [1 + 119886119887 minus119886 minus 119887
119886 + 119887 minus1 minus 119886119887] sim
[[[
[
1 minus119886 + 119887
1 + 119886119887
119886 + 119887
1 + 119886119887minus1
]]]
]
(15)
We used the fact that conjugation does nothing to realnumbers Thus the fourth point on R has coordinate
119886 oplus 119887 =119886 + 119887
1 + 119886119887 (16)
but this happens to be the formula for relativistic addition ofvelocities (in the natural units in which the speed of light is1) Thus we obtain its geometric interpretation presented inFigure 6 A conventional derivation of this diagram may befound in [6] See also [7] for an interactive applet
The segment through the origin does not have to bevertical for the device to work (by Theorem 3) but is set sofor simplicity
6 Further Generalizations
The porism described here for circles is also valid for anyquadric see Figure 7 What is intriguing three cases (vizcircle ellipse and a pair of parallel lines) correspond to thethree possible 2-dimensional Clifford algebras each of theform of ldquogeneralized number planerdquo 119886 + 119887119890 where 119890 is anldquoimaginary unitrdquo whose square is 1 minus1 or 0 (see Table 1)
The case of the circle corresponds to complex numbers asdescribed in the previous sections The case of the hyperbolacorresponds to ldquoduplex numbersrdquo called also hyperbolicnumbers or split-complex numbers
D = 119886 + 119887119868 119886 119887 isin R 1198682= 1 (17)
a b0 a oplus bR
Figure 6 Relativistic addition of velocities
Figure 7 Porism on quadrilaterals inscribed in quadrics
Table 1
Algebra ldquoSphererdquo Formula representedgeometrically
C (complex numbers) Circle Relativistic velocitiesD (hyperbolic numbers) Hyperbola Trigonometric tangentsG (dual numbers) Parallel lines Regular addition
They aremdashlike complex numbersmdasha two-dimensional unitalalgebra except that its ldquoimaginary unitrdquo 119868 is 1 when squaredAs complex numbers are related to rotations duplex numberscorrespond to hyperbolic rotations They were introduced in[8] as tessarines and represent a Clifford algebra R
01 They
found useful applications for example in [9] a ldquohyperbolicquantum mechanicsrdquo was introduced Here is the theoremcorresponding toTheorem 6
Proposition 9 The reversion through a point 119901 isin D withrespect to the hyperbola 1199092minus1199102 = 1 is represented by thematrix
119901 119911 997888rarr 1199111015840= [
1 minus119901
119901 minus1] sdot 119911 =
119911 minus 119901
119901119911 minus 1(|119911|2= 1) (18)
while for branch 1199092 minus 1199102 = minus1 it is
119901 119911 997888rarr 1199111015840= [
minus1 119901
119901 1] sdot 119911 =
minus119911 + 119901
119901119911 + 1(|119911|2= minus1) (19)
Proof For (18) follow the lines of the proof of the maintheorem To get (19) multiply the ingredients of (18) by 119868to switch the axes use (18) and then multiply by 119868 again torestore the original axes
For the last case of parallel lines we can use the sameprevious matrix calculus but replace the algebra by dualnumbers
G = 119886 + 119887119868 119886 119887 isin R 1198682= 0 (20)
Geometry 5
minus1 +1b a oplus b
0a
Figure 8 Trigonometric tangent formula visualized
P
P L
L
K
K
Figure 9 Porism under inversion extends to circles replacing linesand vice versa
Transformations (18) and (19) apply for two cases vertical andhorizontal lines respectively (|119911|2 = 1 versus |119911|2 = minus1 in thehyperbolic norm)
It is a rather pleasant surprise to find the algebra andgeometry interacting at such a basic level in an entirelynontrivial way Repeating a construction analogous to the onein Section 5 that results in a geometric tool for addition ofrelativistic velocities to the hyperbolic numbers gives a simi-lar geometric diagram representing addition of trigonometrictangents see Figure 8 Consider
119886 oplus 119887 =119886 + 119887
1 minus 119886119887= tan (arctan 119886 + arctan 119887) (21)
The final comment concerns the obvious extension ofthese results implied by inversion of the standard config-uration like in Figure 1 through a circle Mobius geometryremoves the distinction between circles and lines The linesof the quadrilateral and the line 119871 under inversion maybecome circles The point at infinity where these lines meetbecomes under inversion a point that is part of the porismrsquosconstruction
Thus the general statement is the following given circles119871 and 119870 and an odd number of points 119875 119875
1 1198752 119875
2119899on
119871 suppose that there exists an ordered 2119899-tuple (pencil) ofcircles119862
1 119862
2119899 through119875 such that119875
119894isin 119862119894for every 119894 and
the points of intersections119862119894cap119862119894+1
and1198622119899cap1198621different from
119875 all lie on the circle 119870 then there are infinitely many such2119899-tuples of circles Figure 9 illustrates the theorem for fourcircles For interactive version of these and more examplessee [7]
Acknowledgments
The author is grateful to Philip Feinsilver for his interest inthis work and for his helpful comments Special thanks goalso to creators ofCinderella awonderful software that allowsone to quickly test geometric conjectures and to create niceinteractive applets
References
[1] D Coxeter and S Greitzer Geometry Revisited MathematicalAssociation of America Washington DC USA 1967
[2] EWWeisstein ldquoSteinerChainrdquo FromMathWorldmdashAWolframWeb Resource httpmathworldwolframcomSteinerChainhtml
[3] W Barth and T Bauer ldquoPoncelet theoremsrdquo Expositiones Math-ematicae vol 14 no 2 pp 125ndash144 1996
[4] J-V Poncelet Traite des Proprietes Projectives des FiguresOuvrage Utile A Qui SrsquoOccupent des Applications de la GeometrieDescriptive et Drsquooperations Geometriques sur le Terrain vol 1-2Gauthier-Villars Paris France 2nd edition 1865
[5] E W Weisstein ldquoPonceletrsquos Porismrdquo From Math WorldmdashA Wolfram Web Resource httpmathworldwolframcomPonceletsPorismhtml
[6] J Kocik ldquoDiagram for relativistic addition of velocitiesrdquo Amer-ican Journal of Physics vol 80 no 8 p 720 2012
[7] J Kocik Interactive diagrams httpwwwmathsiueduKocikgeometryhtml
[8] J Cockle ldquoOn certain functions resembling quaternions andon a new imaginary in algebrardquo London-Edinburgh-DublinPhilosophical Magazine vol 33 pp 435ndash439 1848
[9] J Kocik ldquoDuplex numbers diffusion systems and general-ized quantum mechanicsrdquo International Journal of TheoreticalPhysics vol 38 no 8 pp 2221ndash2230 1999
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Geometry
119872(119903)119872 (119904) =[1 minus 119903119904 119903 minus 119904
119903 minus 119904 1 minus 119903119904]
=
[[[
[
1119903 minus 119904
1 minus 119903119904
119903 minus 119904
1 minus 119903119904
1 minus 119903119904
1 minus 119903119904
]]]
]
=
[[[
[
1119903 minus 119904
1 minus 119903119904
(119903 minus 119904
1 minus 119903119904)
lowast1 minus 119903119904
1 minus 119903119904
]]]
]
(13)
from which this convenient formula results119903 minus 119904
1 minus 119903119904=
119902 minus 119901
1 minus 119902119901 (14)
provided 119901 119902 119903 119904 are collinear
5 Application Relativistic Velocities
Take the real line and the unit circle as ingredients for themodel Consider a quadrilateral that goes through threecollinear points the origin (0) and two points representedby real numbers 119886 and 119887 FromTheorem 3 it follows that thefourth point on the real line must be
119872(119887)119872 (119886)119872 (0)
= [1 minus119887
119887 minus1] [
1 minus119886
119886 minus1] [
1 0
0 minus1]
= [1 + 119886119887 minus119886 minus 119887
119886 + 119887 minus1 minus 119886119887] sim
[[[
[
1 minus119886 + 119887
1 + 119886119887
119886 + 119887
1 + 119886119887minus1
]]]
]
(15)
We used the fact that conjugation does nothing to realnumbers Thus the fourth point on R has coordinate
119886 oplus 119887 =119886 + 119887
1 + 119886119887 (16)
but this happens to be the formula for relativistic addition ofvelocities (in the natural units in which the speed of light is1) Thus we obtain its geometric interpretation presented inFigure 6 A conventional derivation of this diagram may befound in [6] See also [7] for an interactive applet
The segment through the origin does not have to bevertical for the device to work (by Theorem 3) but is set sofor simplicity
6 Further Generalizations
The porism described here for circles is also valid for anyquadric see Figure 7 What is intriguing three cases (vizcircle ellipse and a pair of parallel lines) correspond to thethree possible 2-dimensional Clifford algebras each of theform of ldquogeneralized number planerdquo 119886 + 119887119890 where 119890 is anldquoimaginary unitrdquo whose square is 1 minus1 or 0 (see Table 1)
The case of the circle corresponds to complex numbers asdescribed in the previous sections The case of the hyperbolacorresponds to ldquoduplex numbersrdquo called also hyperbolicnumbers or split-complex numbers
D = 119886 + 119887119868 119886 119887 isin R 1198682= 1 (17)
a b0 a oplus bR
Figure 6 Relativistic addition of velocities
Figure 7 Porism on quadrilaterals inscribed in quadrics
Table 1
Algebra ldquoSphererdquo Formula representedgeometrically
C (complex numbers) Circle Relativistic velocitiesD (hyperbolic numbers) Hyperbola Trigonometric tangentsG (dual numbers) Parallel lines Regular addition
They aremdashlike complex numbersmdasha two-dimensional unitalalgebra except that its ldquoimaginary unitrdquo 119868 is 1 when squaredAs complex numbers are related to rotations duplex numberscorrespond to hyperbolic rotations They were introduced in[8] as tessarines and represent a Clifford algebra R
01 They
found useful applications for example in [9] a ldquohyperbolicquantum mechanicsrdquo was introduced Here is the theoremcorresponding toTheorem 6
Proposition 9 The reversion through a point 119901 isin D withrespect to the hyperbola 1199092minus1199102 = 1 is represented by thematrix
119901 119911 997888rarr 1199111015840= [
1 minus119901
119901 minus1] sdot 119911 =
119911 minus 119901
119901119911 minus 1(|119911|2= 1) (18)
while for branch 1199092 minus 1199102 = minus1 it is
119901 119911 997888rarr 1199111015840= [
minus1 119901
119901 1] sdot 119911 =
minus119911 + 119901
119901119911 + 1(|119911|2= minus1) (19)
Proof For (18) follow the lines of the proof of the maintheorem To get (19) multiply the ingredients of (18) by 119868to switch the axes use (18) and then multiply by 119868 again torestore the original axes
For the last case of parallel lines we can use the sameprevious matrix calculus but replace the algebra by dualnumbers
G = 119886 + 119887119868 119886 119887 isin R 1198682= 0 (20)
Geometry 5
minus1 +1b a oplus b
0a
Figure 8 Trigonometric tangent formula visualized
P
P L
L
K
K
Figure 9 Porism under inversion extends to circles replacing linesand vice versa
Transformations (18) and (19) apply for two cases vertical andhorizontal lines respectively (|119911|2 = 1 versus |119911|2 = minus1 in thehyperbolic norm)
It is a rather pleasant surprise to find the algebra andgeometry interacting at such a basic level in an entirelynontrivial way Repeating a construction analogous to the onein Section 5 that results in a geometric tool for addition ofrelativistic velocities to the hyperbolic numbers gives a simi-lar geometric diagram representing addition of trigonometrictangents see Figure 8 Consider
119886 oplus 119887 =119886 + 119887
1 minus 119886119887= tan (arctan 119886 + arctan 119887) (21)
The final comment concerns the obvious extension ofthese results implied by inversion of the standard config-uration like in Figure 1 through a circle Mobius geometryremoves the distinction between circles and lines The linesof the quadrilateral and the line 119871 under inversion maybecome circles The point at infinity where these lines meetbecomes under inversion a point that is part of the porismrsquosconstruction
Thus the general statement is the following given circles119871 and 119870 and an odd number of points 119875 119875
1 1198752 119875
2119899on
119871 suppose that there exists an ordered 2119899-tuple (pencil) ofcircles119862
1 119862
2119899 through119875 such that119875
119894isin 119862119894for every 119894 and
the points of intersections119862119894cap119862119894+1
and1198622119899cap1198621different from
119875 all lie on the circle 119870 then there are infinitely many such2119899-tuples of circles Figure 9 illustrates the theorem for fourcircles For interactive version of these and more examplessee [7]
Acknowledgments
The author is grateful to Philip Feinsilver for his interest inthis work and for his helpful comments Special thanks goalso to creators ofCinderella awonderful software that allowsone to quickly test geometric conjectures and to create niceinteractive applets
References
[1] D Coxeter and S Greitzer Geometry Revisited MathematicalAssociation of America Washington DC USA 1967
[2] EWWeisstein ldquoSteinerChainrdquo FromMathWorldmdashAWolframWeb Resource httpmathworldwolframcomSteinerChainhtml
[3] W Barth and T Bauer ldquoPoncelet theoremsrdquo Expositiones Math-ematicae vol 14 no 2 pp 125ndash144 1996
[4] J-V Poncelet Traite des Proprietes Projectives des FiguresOuvrage Utile A Qui SrsquoOccupent des Applications de la GeometrieDescriptive et Drsquooperations Geometriques sur le Terrain vol 1-2Gauthier-Villars Paris France 2nd edition 1865
[5] E W Weisstein ldquoPonceletrsquos Porismrdquo From Math WorldmdashA Wolfram Web Resource httpmathworldwolframcomPonceletsPorismhtml
[6] J Kocik ldquoDiagram for relativistic addition of velocitiesrdquo Amer-ican Journal of Physics vol 80 no 8 p 720 2012
[7] J Kocik Interactive diagrams httpwwwmathsiueduKocikgeometryhtml
[8] J Cockle ldquoOn certain functions resembling quaternions andon a new imaginary in algebrardquo London-Edinburgh-DublinPhilosophical Magazine vol 33 pp 435ndash439 1848
[9] J Kocik ldquoDuplex numbers diffusion systems and general-ized quantum mechanicsrdquo International Journal of TheoreticalPhysics vol 38 no 8 pp 2221ndash2230 1999
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Geometry 5
minus1 +1b a oplus b
0a
Figure 8 Trigonometric tangent formula visualized
P
P L
L
K
K
Figure 9 Porism under inversion extends to circles replacing linesand vice versa
Transformations (18) and (19) apply for two cases vertical andhorizontal lines respectively (|119911|2 = 1 versus |119911|2 = minus1 in thehyperbolic norm)
It is a rather pleasant surprise to find the algebra andgeometry interacting at such a basic level in an entirelynontrivial way Repeating a construction analogous to the onein Section 5 that results in a geometric tool for addition ofrelativistic velocities to the hyperbolic numbers gives a simi-lar geometric diagram representing addition of trigonometrictangents see Figure 8 Consider
119886 oplus 119887 =119886 + 119887
1 minus 119886119887= tan (arctan 119886 + arctan 119887) (21)
The final comment concerns the obvious extension ofthese results implied by inversion of the standard config-uration like in Figure 1 through a circle Mobius geometryremoves the distinction between circles and lines The linesof the quadrilateral and the line 119871 under inversion maybecome circles The point at infinity where these lines meetbecomes under inversion a point that is part of the porismrsquosconstruction
Thus the general statement is the following given circles119871 and 119870 and an odd number of points 119875 119875
1 1198752 119875
2119899on
119871 suppose that there exists an ordered 2119899-tuple (pencil) ofcircles119862
1 119862
2119899 through119875 such that119875
119894isin 119862119894for every 119894 and
the points of intersections119862119894cap119862119894+1
and1198622119899cap1198621different from
119875 all lie on the circle 119870 then there are infinitely many such2119899-tuples of circles Figure 9 illustrates the theorem for fourcircles For interactive version of these and more examplessee [7]
Acknowledgments
The author is grateful to Philip Feinsilver for his interest inthis work and for his helpful comments Special thanks goalso to creators ofCinderella awonderful software that allowsone to quickly test geometric conjectures and to create niceinteractive applets
References
[1] D Coxeter and S Greitzer Geometry Revisited MathematicalAssociation of America Washington DC USA 1967
[2] EWWeisstein ldquoSteinerChainrdquo FromMathWorldmdashAWolframWeb Resource httpmathworldwolframcomSteinerChainhtml
[3] W Barth and T Bauer ldquoPoncelet theoremsrdquo Expositiones Math-ematicae vol 14 no 2 pp 125ndash144 1996
[4] J-V Poncelet Traite des Proprietes Projectives des FiguresOuvrage Utile A Qui SrsquoOccupent des Applications de la GeometrieDescriptive et Drsquooperations Geometriques sur le Terrain vol 1-2Gauthier-Villars Paris France 2nd edition 1865
[5] E W Weisstein ldquoPonceletrsquos Porismrdquo From Math WorldmdashA Wolfram Web Resource httpmathworldwolframcomPonceletsPorismhtml
[6] J Kocik ldquoDiagram for relativistic addition of velocitiesrdquo Amer-ican Journal of Physics vol 80 no 8 p 720 2012
[7] J Kocik Interactive diagrams httpwwwmathsiueduKocikgeometryhtml
[8] J Cockle ldquoOn certain functions resembling quaternions andon a new imaginary in algebrardquo London-Edinburgh-DublinPhilosophical Magazine vol 33 pp 435ndash439 1848
[9] J Kocik ldquoDuplex numbers diffusion systems and general-ized quantum mechanicsrdquo International Journal of TheoreticalPhysics vol 38 no 8 pp 2221ndash2230 1999
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of