The motion of billiards in ellipses

The motion of billiards in ellipses

Hellmuth Stachel

stacheldmgtuwienacat mdash httpswwwgeometrietuwienacatstachel

Differential Geometry Seminar

Dec 9 2020 Vienna University of Technology

Table of contents

1 Metric properties of confocal conics

2 Confocal conics and billiards

3 Periodic N-sided billiards

4 A switch to Analysis

Acknowledgement

Dan Reznik Data Science Consulting Rio de Janeiro Brazil

Ronaldo Garcia Universidade Federal de Goiaacutes Goiania Brazil

Dec 9 2020 Differential Geometry Seminar TU Wien 134


F1 F2

P

e

The optical property of ellipses is well known and also the equivalence

equal angles lArrrArr F1P + F2P = const lArrrArr P isin e

There is a generalization



e

c

P

Q1 Q2

If any ray is reflected in a conic e thenthe incoming and the outgoing ray aretangent to the same confocal conic c called caustic

F1 F2

P

e

c

Q1Q2

Charles Graves (1812-1899) bishop ofLimerick and mathematician

The locus of point P used to pull thestring taut around c is a confocal ellipse e

De = PQ1 + PQ2 minusQ1Q2 = const



c

e

c

e

Billiards in an ellipse e are always tangent to a confocal ellipse or hyperbola

If one billiard closes after N reflections then all billiards close independent of the initialpoint on c (Poncelet porism) and all these closed loops have the same length



For centuries billiards attracted the attention of mathe-maticians beginning with J-V Poncelet and A Cayley

S Tabachnikov Geometry and BilliardsAmerican Mathematical Society 2005

Recently Dan Reznik revitalized the interest bycomputer animations showing the variation of periodicbilliards He identified 40 invariants eg a constantsum of Cosines of interior angles

Sum of square altitudes to N-periodic tangents is invariant

The two types of self-intersected 7-periodics in the Elliptic Billiard



A family of confocal central conics

x2

a2 + k+y 2

b2 + k= 1

k isin R minusa2minusb2 sends througheach point P outside the axesone ellipse and one orthogonallyintersecting hyperbola

The parameters (ke kh) define theelliptic coordinates of P with

minusa2 lt kh lt minusb2 lt ke

P

F1 F2M

We specify the caustic c (semiaxes ac bc) as k = 0 and the ellipse e with semiaxesae be as k = ke =rArr ke = a

2e minus a2c = b2e minus b2c gt 0



From given (ke kh) follows

x2 =(a2 + ke)(a

2 + kh)

d2

y 2 = minus(b2 + ke)(b

2 + kh)

d2

Conversely P = (ae cos t be sin t)with tangent vector

te(t) = (minusae sin t be cos t)and tc(t)=(minusac sin t bc cos t) yield

P

F1 F2M

kh(t) =minus (a2c sin2 t + b2c cos2 t) = minustc(t)2 = minuste(t)2 + ke

Points on confocal ellipses e and c with the same parameter t have the same coordinatekh ie they belong to the same confocal hyperbola



P1tP1

P2P5

Q1

Q4

Q5

R1

R4

R5

OF1

F2

e

c cprimer2

l1r1

l5

θ12Given P = (ae cos t be sin t)the tangents from P to c (k = 0)include the angle θ(t) where

sin2θ(t)

2=

kete(t)2

tanθ(t)

2= plusmn

radicke

tc(t)

cos θ = 1minus 2kete(t)2

=kh(t) + kekh(t)minus ke



P1tP1

P2P5

Q1

Q4

Q5

R1

R4

R5

OF1

F2

e

c cprimer2

l1r1

l5

θ12The signed distances of the sidetQ = P1P2 from the center Oand the pole R1 wrt e have theconstant product minuske For Q1 = (ac cos t

prime1 bc sin t

prime1)

OtQ =minusacbctc(t prime1)

RtQ = R1Q1 =ketc(t prime1)acbc



P1tP1

P2P5

Q1

Q4

Q5

R1

R4

R5

OF1

F2

e

c cprimer2

l1r1

l5

θ12 The side P1P2 (parameters t1 t2)contacts the caustic c iff

sin2t1 minus t22=keaebe

∥∥∥ te(t1 + t22

)∥∥∥2

If t prime1 is the parameter of thetangency point Q1 isin c then

tan t prime1 =bcaeacbe

tant1 + t22

Half angle substitution yields



The half-angle substitution τi = tanti2

yields for P1 and P2 a symmetric biquadraticequation in projective coordinates on e namely

b2ekeτ21τ22 minus b2ca2e(τ21 + τ22 ) + 2(a2eke + a2cb2e)τ1τ2 + b2eke = 0

which defines a 2-2-correspondence between consecute points P1 P2 of a billiard Thesame holds after N iteration between P1 and PN+1

We recall a classical algebraic argument for the Poncelet porism

A 2-2-correspondence (6= id) has at most four fixed points However four fixed pointsare already known as contact points between e and the common (isotropic) tangentswith the caustic c Hence if one N-sided billiard closes then the correspondence isthe identity and each billiard inscribed in e with caustic c must close



P1

P2

P3

Q1Q2

O

θi2

θi2

ce

n1

n2

u1u2

ui = 1

S Tabachnikov the key result for the integrability of billiards is the Joachimsthal

integral Je = minus〈ui ni〉 with ui as unit vector of PiPi+1 and ni = (cos tae sin tbe)as a normal vector of e This holds in all dimensions (ni = Api = p

lowasti ) In the plane

Je = minus〈ui ni〉 = minus cos(π2+θi2

)ni = sin

θi2ni =

radickete

teaebe

=

radickeaebe



c

The extended sides of a billiard intersect atpoints of confocal ellipses and hyperbolasand form a Poncelet grid

affinely transformed 72-sided periodic billiard with

associated Poncelet grid (G Glaeser B Odehnal

HS The Universe of Conics 7 KB)



B1

A1

A2

B2

A3

B3

c

c1 t1

t2

t3

t4 Theorem

Given a quadrilateral t1 t4 oftangents to c from A1 B1 isin c1Then the range Rc spanned byc and c1 contains conics c2c3 passing through the remain-ing pairs of opposite vertices(A2 B2) and (A3 B3)

(range = lsquodual pencilrsquo)

= summary of results from M Chasles (1843) W Boumlhm (1961) Izmestiev amp Tabachnikov (2016)

Akopyan amp Bobenko (2017)



B1

A1

A2

B2

A3

B3

R

c

c1

c2

c3

t1

t2

t3

t4

Rt = conics tangent to t1 t4

Rc capRt = c =rArr theyspan a net N (2-parameter set)

In N the line elements of c1 atA1 and B1 span a range whichcontains the rank-1 conic R

R

c

A1B1

c1

c2

c3

A3B3

A2B2

Rc Rt


2 Confocal conics and billiardsreplacements

B1

A1

A2

B2

A3

B3

R

c

c1

c2

c3

dt1

t2

t3

t4In N the line pencils Ai Bi

and the pencil R (2-fold) spana range which intersects Rc atci The range contains conicssharing the line elements at Aiand Bi

The tangents to ci at Ai and Bipass through RConfocal c c1 =rArr concyclicquadrilateral

This holds also when B2 isin c(t1 = t2)



P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)

S(1)2 = [P1 P2] cap [P3 P4] on the confocal hyperbola through Q2

S(2)2 = [P0 P1] cap [P3 P4] isin e(2) on the confocal hyperbola through P2


P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)

S(1)i = [Piminus1 Pi ] cap [Pi+1 Pi+2] on the confocal ellipse e(1) for all i

and e(1) is invariant of the initial data ke|1 = ke

(2acbcaebea2cb

2c minus k2e

)2



P2

P3

Q2

Q3

Q1

S(1)2

S(1)3

S(2)3

ww

w

w1

w1

ce

e(1)

e(2)

r2

l2

r3r3

l3

l3Surprisingly the distance w

between S(1)2 and the contact

point with the incircle isinvariant the same for w1

Gravesrsquos construction =rArrDe = Qiminus1Pi + PiQi minus

Qiminus1Qi

= ri + li minusQiminus1Qi

is constantSimilarly for e(1) follows

De|1 =Q1S(1)2 + S

(1)2 Q3 minus

Q1Q3 = (r2+l2+w) + (r3+l3+w)minus

Q1Q3= 2De + 2w


P2 P3

P4P1

Q2Q4

Q1

e

c

e(1)

e(2)

The same invariants show upon the sphere All circulararcs in black have the samelength The same lsquoin greenrsquo

In the plane w =2aebe

radick3e

a2cb2c minus k2e



P2

P3P1

Qprime1

Qprime2

P prime2P prime1

Q2 Q1

R2

R1

O

e

cr3

l2r2

l1

θ22 For each billiard P0P1P2 in the ellipse e with causticc there exists a conjugate

billiard P prime0Pprime1Pprime2

The axial scaling with c rarr eα Qi 7rarr P primei Qprimeiminus1 7rarr Pi transforms tangents P1P2 ofc to tangents P prime1P

prime2 This

results from the symmetrybetween ti and t primei in theequation

bcae cos ti cos tprimei +acbe sin ti sin t

primei = acbc which expresses that Pi lies on the tangent

to c at Qi and P primei on the tangent at Qprimei



P prime1

P prime2P2P3

P1

Qprime1Qprime2

Q1

Q2

S(1)1

S(1)2

S(1)3

c

e(1)

e1

0

1234

14

7858

One might say point P primei ishalfway from Pi to Pi+1

In the sequence of para-meters t1 t

prime1 t2 t

prime2

for P1 Pprime1(Q1) P2 P

prime2(Q2)

the transition Pi 7rarr P primeimeans a shift ti 7rarr t primei

I Izmestiev S Tabachnikov (2017) There exists a canonical parametrization ofe such that the billiard transformation Pi rarr Pi+1 corresponds to a shiftui rarr ui+1 = ui + 2∆u

Above an example of canonical parameters P1 sim 0 P2 sim 12 P3 sim 1



P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

ce

e(1)

e(2)

The turning number τ of aperiodic billiard counts the loopsaround the center

Theorem

(i) A periodic billiard with even Nand odd τ is centrally symmetric

(ii) For odd N and odd τ thebilliard is centrally symmetric tothe conjugate billiard

(iii) For odd N and even τthe billiard conincides with theconjugate billiard



P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

l2

r7

ce

e(1)

e(2)

From Gravesrsquo theorem

De = Qiminus1Pi + PiQi minusQiminus1Qi

follows for the perimeter of the N-sided billiard

Le = N middotDe + τ middot Pcwith Pc as perimeter of e

Ivoryrsquos theorem implies for odd

N = 2n + 1 the length li equalssymmetric l primei+n of the conjugatebilliard and ri+n+1 of the originalone

Theoremsumli =

sumri = Le2



Piminus1

Pi

Pi+1

Qiminus1Qi

Fiminus1Fi

O

θi2

θi2

ce

PiFi + PiFiminus1 =2aebete(ti)2

radicke

Le =sumNi=1

(PiFi + PiFiminus1

)=rArr

Le =aeberadicke

Nsum

i=1

2kete(ti)2

=aeberadicke

Nsum

i=1

(1minus cos θi)

Theorem [Akopyan Schwartz Tabachnikov Bialy]sumNi=1 cos θi = N minus

radickeaebe

Le

WithsumNi=1

1

te(ti)2also

sumNi=1OtP

2and

sumNi=1 κe(ti)

23 are invariant



F2F1

P2 tP

Q1Q2

r2

l2

e

c

vt1vn1

vt2vn2

v2

ω1

ω2

θ22

With a billiard motion we denotea variation of the billiard and thePoncelet grid induced by the variationof a single vertex

According to Gravesrsquo constructionthe v2 of P2 can be decomposed as

v2 = vt1 + vn1 = vt2 + vn2

where due to the constant length

vt2 = vt1 and vn2 = vn1l2ω2 = r2ω1 hence

ω1ω2=l2r2

If the billiard is periodic then the product of all ratios liri yields

l1r1middot l2r2middot middot middot lNrN=ωNω1middot ω1ω2middot middot middot ωNminus1ωN

= 1 hence l1l2 lN = r1r2 rN



P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5

For periodic billiards a givenvelocity vector v2 of anyvertex defines all velocities

In terms of the exterior anglesθ1 θN we obtain

sinθ22=l2 ω2v2=r2ω1v2

and

cosθ22=vt|2v2

where

v2 = v2 vt|2 = vt|2From P1P2 perp Q1R1 follows

R1Q1 = l1 tanθ12= r2 tan

θ22



P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5

After some manipulationsfollows

vt|1 tan2 θ12= vt|2 tan

2 θ22

middot middot middot = vt|i ketc(ti)2

= C

Instead of a free choice of v2we set C = ke

vt|i=tc2 vi=tc tevn|i = vi sin

θ

2= tc

radicke

for all t = ti and for all confocalellipses e



c

e

To each point

p = (x y) = (ae cos t be sin t)

we assign a velocity vector

v = tc te =radica2c sin

2 t + b2c cos2 t

(minusaeybebex

ae

)

Theorem

This vector field defines an infinitesimal motion which preserves confocal ellipses andpermutes the confocal hyperbolas and the tangents of the caustic c

The infinitesimal motion generates a one-parameter Liegroup Γ which carries out thebilliard transformation along e and simultaneously that of the associated Poncelet grid



c

e

Q

tQ

vn We set v =d p

du= p Then u is a

canonical parameter of e and of Γie γ(u2) γ(u1) = γ(u1 + u2)v(t)=tc(t) te(t) = t te(t)

t =dt

du=

radica2c sin

2 t + b2c cos2 t

Proof Γ permutes hyperbolas since t (and kh) is independent of e (and ke)

The condition bcae cos t cos tprime + acbe sin t sin t prime = acbc is equivalent to the fact that

P = (ae cos t be sin t) lies on the tangent tQ of Q = (ac cos tprime + bc sin t prime) Differen-

tiation by u yields an identity Hence Γ permutes the tangents of c



In order to express the action of γ(u) isin Γ on (ae cos t be sin t) we integrate

dt

du=

radica2c sin

2 t + b2c cos2 t =

radica2c sin

2 t + (a2c minus d2) cos2 t = acradic1minusm2 cos2 t

withm = dac lt 1 as numeric eccentricity of the caustic c We substitute ϕ = tminusπ2

and get under the initial condition ϕ = 0 for u = 0

dϕradic1minusm2 sin2ϕ

= ac du hence ac u(ϕ) = F (ϕm) =

int ϕ

0


with F (ϕm) as the elliptic integral of the first kind with the modulusm This functionshows the canonical coordinate u in terms of ϕ with the quarter period

K = ac u(π

2

)=

int π2

0




For the sake of simplicity we define u(ϕ) = ac u(ϕ) as a new canonical coordinate

The inverse function of u = F (ϕm) the Jacobian amplitude ϕ = am(u ) leads tothe Jacobian elliptic functions

sn u = sin(am(u )) and cn u = cos(am(u ))

which can be extended in R to periodic functions with period 4K



Theorem

For the ellipse c with semiaxes (ac bc) and eccentricity d =radica2c minus b2c and all confocal

ellipses e with semiaxes (ae be) the inscribed billiards with caustic c can be canonicallyparametrized as (minusae sn u be cn u ) using the Jacobian elliptic functions to themodulus m = dac

If bc = be cn(∆u ) then the vertices of the billiard in e have the canonical parametersu = (u0 + 2k∆u ) for k isin Z and any given initial u0

Conversely we obtain an N-sided billiard withturning number τ where gcd(N τ) = 1 bythe choice ∆u =

2τK

Nwith K as the complete

elliptic integral of the first kind to the modulusm provided that be = bccn(∆u )

Q1(∆u )

P1(0)

P1(2∆u )

y

c

e


Schoumlnbrunn Castle Vienna

Thank you for your attention


References

References

[1] AW Akopyan AI Bobenko Incircular nets and confocal conics Trans AmerMath Soc Nov 16 2017 (arXiv160204637v2[mathDS]27Oct2017)

[2] A Akopyan R Schwartz S Tabachnikov Billiards in ellipses revisitedarXiv200102934v2[mathMG] 2020

[3] M Bialy S Tabachnikov Dan Reznikrsquos identities and morearXiv200108469v2[mathDG] 2020

[4] W Boumlhm Ein Analogon zum Satz von Ivory Ann Mat Pura Appl (4) 54221ndash225 (1961)


[5] W Boumlhm Verwandte Saumltze uumlber Kreisvierseitnetze Arch Math 21 326ndash330(1970)

[6] M Chasles Proprieacuteteacutes geacuteneacuterales des arcs drsquoune section conique dont la differenceest rectifiable Comptes Rendus hebdomadaires de seacuteances de lrsquoAcadeacutemie dessciences 17 838ndash844 (1843)

[7] M Chasles Reacutesumeacute drsquoune theacuteorie des coniques spheacuteriques homofocalesComptes Rendus des seacuteances de lrsquoAcadeacutemie des Sciences 50 623ndash633 (1860)

[8] A Chavez-Caliz More about areas and centers of Poncelet polygonsarXiv200405404 (2020)

[9] G Glaeser H Stachel B Odehnal The Universe of Conics Springer SpectrumBerlin Heidelberg 2016

[10] Ph Griffiths J Harris On Cayleyrsquos explicit solution to Ponceletrsquos porismLrsquoEnseignement Matheacutematique 241-2 31ndash40 (1978)


[11] L Halbeisen N Hungerbuumlhler A Simple Proof of Ponceletrsquos Theorem (on theoccasion of its bicentennial) Amer Math Monthly 1226 537ndash551 (2015)

[12] I Izmestiev S Tabachnikov Ivoryrsquos Theorem revisited J Integrable Syst 21xyx006 (2017) httpsdoiorg101093integrxyx006

[13] D Reznik R Garcia J Koiller Forty New Invariants of N-Periodics in the EllipticBilliard arXiv200412497[mathDS] 2020

[14] H Stachel Recalling Ivoryrsquos Theorem FME Transactions 47 No 2 355ndash359(2019)

[15] S Tabachnikov Geometry and Billiards American Mathematical SocietyProvidenceRhode Island 2005


Table of contents





Acknowledgement

Dan Reznik Data Science Consulting Rio de Janeiro Brazil

Ronaldo Garcia Universidade Federal de Goiaacutes Goiania Brazil



F1 F2

P

e






e

c

P

Q1 Q2


F1 F2

P

e

c

Q1Q2






c

e

c

e













x2

a2 + k+y 2

b2 + k= 1




P

F1 F2M






x2 =(a2 + ke)(a

2 + kh)

d2


2 + kh)

d2



P

F1 F2M





P1tP1

P2P5

Q1

Q4

Q5

R1

R4

R5

OF1

F2

e

c cprimer2

l1r1

l5


sin2θ(t)

2=

kete(t)2

tanθ(t)

2= plusmn

radicke

tc(t)





P1tP1

P2P5

Q1

Q4

Q5

R1

R4

R5

OF1

F2

e

c cprimer2

l1r1

l5


prime1 bc sin t

prime1)





P1tP1

P2P5

Q1

Q4

Q5

R1

R4

R5

OF1

F2

e

c cprimer2

l1r1

l5



∥∥∥ te(t1 + t22

)∥∥∥2



tant1 + t22












P1

P2

P3

Q1Q2

O

θi2

θi2

ce

n1

n2

u1u2

ui = 1





)ni = sin

θi2ni =

radickete

teaebe

=

radickeaebe



c







B1

A1

A2

B2

A3

B3

c

c1 t1

t2

t3

t4 Theorem







B1

A1

A2

B2

A3

B3

R

c

c1

c2

c3

t1

t2

t3

t4




R

c

A1B1

c1

c2

c3

A3B3

A2B2

Rc Rt



B1

A1

A2

B2

A3

B3

R

c

c1

c2

c3

dt1

t2

t3







P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)




P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)



(2acbcaebea2cb

2c minus k2e

)2



P2

P3

Q2

Q3

Q1

S(1)2

S(1)3

S(2)3

ww

w

w1

w1

ce

e(1)

e(2)

r2

l2

r3r3

l3





Qiminus1Qi



De|1 =Q1S(1)2 + S

(1)2 Q3 minus

Q1Q3 = (r2+l2+w) + (r3+l3+w)minus

Q1Q3= 2De + 2w


P2 P3

P4P1

Q2Q4

Q1

e

c

e(1)

e(2)



radick3e

a2cb2c minus k2e



P2

P3P1

Qprime1

Qprime2

P prime2P prime1

Q2 Q1

R2

R1

O

e

cr3

l2r2

l1




prime2 This







P prime1

P prime2P2P3

P1

Qprime1Qprime2

Q1

Q2

S(1)1

S(1)2

S(1)3

c

e(1)

e1

0

1234

14

7858



prime1 t2 t

prime2


prime2(Q2)






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

ce

e(1)

e(2)


Theorem






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

l2

r7

ce

e(1)

e(2)







Theoremsumli =

sumri = Le2



Piminus1

Pi

Pi+1

Qiminus1Qi

Fiminus1Fi

O

θi2

θi2

ce


radicke

Le =sumNi=1

(PiFi + PiFiminus1

)=rArr

Le =aeberadicke

Nsum

i=1

2kete(ti)2

=aeberadicke

Nsum

i=1

(1minus cos θi)


radickeaebe

Le

WithsumNi=1

1

te(ti)2also

sumNi=1OtP

2and

sumNi=1 κe(ti)

23 are invariant



F2F1

P2 tP

Q1Q2

r2

l2

e

c

vt1vn1

vt2vn2

v2

ω1

ω2

θ22



v2 = vt1 + vn1 = vt2 + vn2



ω1ω2=l2r2






P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5




and

cosθ22=vt|2v2

where



θ22



P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5



2 θ22


= C



θ

2= tc

radicke




c

e

To each point




2 t + b2c cos2 t

(minusaeybebex

ae

)

Theorem





c

e

Q

tQ

vn We set v =d p

du= p Then u is a


t =dt

du=

radica2c sin

2 t + b2c cos2 t








dt

du=

radica2c sin

2 t + b2c cos2 t =

radica2c sin






int ϕ

0



K = ac u(π

2

)=

int π2

0










Theorem





2τK



Q1(∆u )

P1(0)

P1(2∆u )

y

c

e





References

References




















F1 F2

P

e






e

c

P

Q1 Q2


F1 F2

P

e

c

Q1Q2






c

e

c

e













x2

a2 + k+y 2

b2 + k= 1




P

F1 F2M






x2 =(a2 + ke)(a

2 + kh)

d2


2 + kh)

d2



P

F1 F2M





P1tP1

P2P5

Q1

Q4

Q5

R1

R4

R5

OF1

F2

e

c cprimer2

l1r1

l5


sin2θ(t)

2=

kete(t)2

tanθ(t)

2= plusmn

radicke

tc(t)





P1tP1

P2P5

Q1

Q4

Q5

R1

R4

R5

OF1

F2

e

c cprimer2

l1r1

l5


prime1 bc sin t

prime1)





P1tP1

P2P5

Q1

Q4

Q5

R1

R4

R5

OF1

F2

e

c cprimer2

l1r1

l5



∥∥∥ te(t1 + t22

)∥∥∥2



tant1 + t22












P1

P2

P3

Q1Q2

O

θi2

θi2

ce

n1

n2

u1u2

ui = 1





)ni = sin

θi2ni =

radickete

teaebe

=

radickeaebe



c







B1

A1

A2

B2

A3

B3

c

c1 t1

t2

t3

t4 Theorem







B1

A1

A2

B2

A3

B3

R

c

c1

c2

c3

t1

t2

t3

t4




R

c

A1B1

c1

c2

c3

A3B3

A2B2

Rc Rt



B1

A1

A2

B2

A3

B3

R

c

c1

c2

c3

dt1

t2

t3







P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)




P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)



(2acbcaebea2cb

2c minus k2e

)2



P2

P3

Q2

Q3

Q1

S(1)2

S(1)3

S(2)3

ww

w

w1

w1

ce

e(1)

e(2)

r2

l2

r3r3

l3





Qiminus1Qi



De|1 =Q1S(1)2 + S

(1)2 Q3 minus

Q1Q3 = (r2+l2+w) + (r3+l3+w)minus

Q1Q3= 2De + 2w


P2 P3

P4P1

Q2Q4

Q1

e

c

e(1)

e(2)



radick3e

a2cb2c minus k2e



P2

P3P1

Qprime1

Qprime2

P prime2P prime1

Q2 Q1

R2

R1

O

e

cr3

l2r2

l1




prime2 This







P prime1

P prime2P2P3

P1

Qprime1Qprime2

Q1

Q2

S(1)1

S(1)2

S(1)3

c

e(1)

e1

0

1234

14

7858



prime1 t2 t

prime2


prime2(Q2)






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

ce

e(1)

e(2)


Theorem






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

l2

r7

ce

e(1)

e(2)







Theoremsumli =

sumri = Le2



Piminus1

Pi

Pi+1

Qiminus1Qi

Fiminus1Fi

O

θi2

θi2

ce


radicke

Le =sumNi=1

(PiFi + PiFiminus1

)=rArr

Le =aeberadicke

Nsum

i=1

2kete(ti)2

=aeberadicke

Nsum

i=1

(1minus cos θi)


radickeaebe

Le

WithsumNi=1

1

te(ti)2also

sumNi=1OtP

2and

sumNi=1 κe(ti)

23 are invariant



F2F1

P2 tP

Q1Q2

r2

l2

e

c

vt1vn1

vt2vn2

v2

ω1

ω2

θ22



v2 = vt1 + vn1 = vt2 + vn2



ω1ω2=l2r2






P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5




and

cosθ22=vt|2v2

where



θ22



P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5



2 θ22


= C



θ

2= tc

radicke




c

e

To each point




2 t + b2c cos2 t

(minusaeybebex

ae

)

Theorem





c

e

Q

tQ

vn We set v =d p

du= p Then u is a


t =dt

du=

radica2c sin

2 t + b2c cos2 t








dt

du=

radica2c sin

2 t + b2c cos2 t =

radica2c sin






int ϕ

0



K = ac u(π

2

)=

int π2

0










Theorem





2τK



Q1(∆u )

P1(0)

P1(2∆u )

y

c

e





References

References




















e

c

P

Q1 Q2


F1 F2

P

e

c

Q1Q2






c

e

c

e













x2

a2 + k+y 2

b2 + k= 1




P

F1 F2M






x2 =(a2 + ke)(a

2 + kh)

d2


2 + kh)

d2



P

F1 F2M





P1tP1

P2P5

Q1

Q4

Q5

R1

R4

R5

OF1

F2

e

c cprimer2

l1r1

l5


sin2θ(t)

2=

kete(t)2

tanθ(t)

2= plusmn

radicke

tc(t)





P1tP1

P2P5

Q1

Q4

Q5

R1

R4

R5

OF1

F2

e

c cprimer2

l1r1

l5


prime1 bc sin t

prime1)





P1tP1

P2P5

Q1

Q4

Q5

R1

R4

R5

OF1

F2

e

c cprimer2

l1r1

l5



∥∥∥ te(t1 + t22

)∥∥∥2



tant1 + t22












P1

P2

P3

Q1Q2

O

θi2

θi2

ce

n1

n2

u1u2

ui = 1





)ni = sin

θi2ni =

radickete

teaebe

=

radickeaebe



c







B1

A1

A2

B2

A3

B3

c

c1 t1

t2

t3

t4 Theorem







B1

A1

A2

B2

A3

B3

R

c

c1

c2

c3

t1

t2

t3

t4




R

c

A1B1

c1

c2

c3

A3B3

A2B2

Rc Rt



B1

A1

A2

B2

A3

B3

R

c

c1

c2

c3

dt1

t2

t3







P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)




P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)



(2acbcaebea2cb

2c minus k2e

)2



P2

P3

Q2

Q3

Q1

S(1)2

S(1)3

S(2)3

ww

w

w1

w1

ce

e(1)

e(2)

r2

l2

r3r3

l3





Qiminus1Qi



De|1 =Q1S(1)2 + S

(1)2 Q3 minus

Q1Q3 = (r2+l2+w) + (r3+l3+w)minus

Q1Q3= 2De + 2w


P2 P3

P4P1

Q2Q4

Q1

e

c

e(1)

e(2)



radick3e

a2cb2c minus k2e



P2

P3P1

Qprime1

Qprime2

P prime2P prime1

Q2 Q1

R2

R1

O

e

cr3

l2r2

l1




prime2 This







P prime1

P prime2P2P3

P1

Qprime1Qprime2

Q1

Q2

S(1)1

S(1)2

S(1)3

c

e(1)

e1

0

1234

14

7858



prime1 t2 t

prime2


prime2(Q2)






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

ce

e(1)

e(2)


Theorem






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

l2

r7

ce

e(1)

e(2)







Theoremsumli =

sumri = Le2



Piminus1

Pi

Pi+1

Qiminus1Qi

Fiminus1Fi

O

θi2

θi2

ce


radicke

Le =sumNi=1

(PiFi + PiFiminus1

)=rArr

Le =aeberadicke

Nsum

i=1

2kete(ti)2

=aeberadicke

Nsum

i=1

(1minus cos θi)


radickeaebe

Le

WithsumNi=1

1

te(ti)2also

sumNi=1OtP

2and

sumNi=1 κe(ti)

23 are invariant



F2F1

P2 tP

Q1Q2

r2

l2

e

c

vt1vn1

vt2vn2

v2

ω1

ω2

θ22



v2 = vt1 + vn1 = vt2 + vn2



ω1ω2=l2r2






P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5




and

cosθ22=vt|2v2

where



θ22



P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5



2 θ22


= C



θ

2= tc

radicke




c

e

To each point




2 t + b2c cos2 t

(minusaeybebex

ae

)

Theorem





c

e

Q

tQ

vn We set v =d p

du= p Then u is a


t =dt

du=

radica2c sin

2 t + b2c cos2 t








dt

du=

radica2c sin

2 t + b2c cos2 t =

radica2c sin






int ϕ

0



K = ac u(π

2

)=

int π2

0










Theorem





2τK



Q1(∆u )

P1(0)

P1(2∆u )

y

c

e





References

References




















c

e

c

e













x2

a2 + k+y 2

b2 + k= 1




P

F1 F2M






x2 =(a2 + ke)(a

2 + kh)

d2


2 + kh)

d2



P

F1 F2M





P1tP1

P2P5

Q1

Q4

Q5

R1

R4

R5

OF1

F2

e

c cprimer2

l1r1

l5


sin2θ(t)

2=

kete(t)2

tanθ(t)

2= plusmn

radicke

tc(t)





P1tP1

P2P5

Q1

Q4

Q5

R1

R4

R5

OF1

F2

e

c cprimer2

l1r1

l5


prime1 bc sin t

prime1)





P1tP1

P2P5

Q1

Q4

Q5

R1

R4

R5

OF1

F2

e

c cprimer2

l1r1

l5



∥∥∥ te(t1 + t22

)∥∥∥2



tant1 + t22












P1

P2

P3

Q1Q2

O

θi2

θi2

ce

n1

n2

u1u2

ui = 1





)ni = sin

θi2ni =

radickete

teaebe

=

radickeaebe



c







B1

A1

A2

B2

A3

B3

c

c1 t1

t2

t3

t4 Theorem







B1

A1

A2

B2

A3

B3

R

c

c1

c2

c3

t1

t2

t3

t4




R

c

A1B1

c1

c2

c3

A3B3

A2B2

Rc Rt



B1

A1

A2

B2

A3

B3

R

c

c1

c2

c3

dt1

t2

t3







P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)




P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)



(2acbcaebea2cb

2c minus k2e

)2



P2

P3

Q2

Q3

Q1

S(1)2

S(1)3

S(2)3

ww

w

w1

w1

ce

e(1)

e(2)

r2

l2

r3r3

l3





Qiminus1Qi



De|1 =Q1S(1)2 + S

(1)2 Q3 minus

Q1Q3 = (r2+l2+w) + (r3+l3+w)minus

Q1Q3= 2De + 2w


P2 P3

P4P1

Q2Q4

Q1

e

c

e(1)

e(2)



radick3e

a2cb2c minus k2e



P2

P3P1

Qprime1

Qprime2

P prime2P prime1

Q2 Q1

R2

R1

O

e

cr3

l2r2

l1




prime2 This







P prime1

P prime2P2P3

P1

Qprime1Qprime2

Q1

Q2

S(1)1

S(1)2

S(1)3

c

e(1)

e1

0

1234

14

7858



prime1 t2 t

prime2


prime2(Q2)






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

ce

e(1)

e(2)


Theorem






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

l2

r7

ce

e(1)

e(2)







Theoremsumli =

sumri = Le2



Piminus1

Pi

Pi+1

Qiminus1Qi

Fiminus1Fi

O

θi2

θi2

ce


radicke

Le =sumNi=1

(PiFi + PiFiminus1

)=rArr

Le =aeberadicke

Nsum

i=1

2kete(ti)2

=aeberadicke

Nsum

i=1

(1minus cos θi)


radickeaebe

Le

WithsumNi=1

1

te(ti)2also

sumNi=1OtP

2and

sumNi=1 κe(ti)

23 are invariant



F2F1

P2 tP

Q1Q2

r2

l2

e

c

vt1vn1

vt2vn2

v2

ω1

ω2

θ22



v2 = vt1 + vn1 = vt2 + vn2



ω1ω2=l2r2






P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5




and

cosθ22=vt|2v2

where



θ22



P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5



2 θ22


= C



θ

2= tc

radicke




c

e

To each point




2 t + b2c cos2 t

(minusaeybebex

ae

)

Theorem





c

e

Q

tQ

vn We set v =d p

du= p Then u is a


t =dt

du=

radica2c sin

2 t + b2c cos2 t








dt

du=

radica2c sin

2 t + b2c cos2 t =

radica2c sin






int ϕ

0



K = ac u(π

2

)=

int π2

0










Theorem





2τK



Q1(∆u )

P1(0)

P1(2∆u )

y

c

e





References

References




























x2

a2 + k+y 2

b2 + k= 1




P

F1 F2M






x2 =(a2 + ke)(a

2 + kh)

d2


2 + kh)

d2



P

F1 F2M





P1tP1

P2P5

Q1

Q4

Q5

R1

R4

R5

OF1

F2

e

c cprimer2

l1r1

l5


sin2θ(t)

2=

kete(t)2

tanθ(t)

2= plusmn

radicke

tc(t)





P1tP1

P2P5

Q1

Q4

Q5

R1

R4

R5

OF1

F2

e

c cprimer2

l1r1

l5


prime1 bc sin t

prime1)





P1tP1

P2P5

Q1

Q4

Q5

R1

R4

R5

OF1

F2

e

c cprimer2

l1r1

l5



∥∥∥ te(t1 + t22

)∥∥∥2



tant1 + t22












P1

P2

P3

Q1Q2

O

θi2

θi2

ce

n1

n2

u1u2

ui = 1





)ni = sin

θi2ni =

radickete

teaebe

=

radickeaebe



c







B1

A1

A2

B2

A3

B3

c

c1 t1

t2

t3

t4 Theorem







B1

A1

A2

B2

A3

B3

R

c

c1

c2

c3

t1

t2

t3

t4




R

c

A1B1

c1

c2

c3

A3B3

A2B2

Rc Rt



B1

A1

A2

B2

A3

B3

R

c

c1

c2

c3

dt1

t2

t3







P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)




P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)



(2acbcaebea2cb

2c minus k2e

)2



P2

P3

Q2

Q3

Q1

S(1)2

S(1)3

S(2)3

ww

w

w1

w1

ce

e(1)

e(2)

r2

l2

r3r3

l3





Qiminus1Qi



De|1 =Q1S(1)2 + S

(1)2 Q3 minus

Q1Q3 = (r2+l2+w) + (r3+l3+w)minus

Q1Q3= 2De + 2w


P2 P3

P4P1

Q2Q4

Q1

e

c

e(1)

e(2)



radick3e

a2cb2c minus k2e



P2

P3P1

Qprime1

Qprime2

P prime2P prime1

Q2 Q1

R2

R1

O

e

cr3

l2r2

l1




prime2 This







P prime1

P prime2P2P3

P1

Qprime1Qprime2

Q1

Q2

S(1)1

S(1)2

S(1)3

c

e(1)

e1

0

1234

14

7858



prime1 t2 t

prime2


prime2(Q2)






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

ce

e(1)

e(2)


Theorem






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

l2

r7

ce

e(1)

e(2)







Theoremsumli =

sumri = Le2



Piminus1

Pi

Pi+1

Qiminus1Qi

Fiminus1Fi

O

θi2

θi2

ce


radicke

Le =sumNi=1

(PiFi + PiFiminus1

)=rArr

Le =aeberadicke

Nsum

i=1

2kete(ti)2

=aeberadicke

Nsum

i=1

(1minus cos θi)


radickeaebe

Le

WithsumNi=1

1

te(ti)2also

sumNi=1OtP

2and

sumNi=1 κe(ti)

23 are invariant



F2F1

P2 tP

Q1Q2

r2

l2

e

c

vt1vn1

vt2vn2

v2

ω1

ω2

θ22



v2 = vt1 + vn1 = vt2 + vn2



ω1ω2=l2r2






P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5




and

cosθ22=vt|2v2

where



θ22



P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5



2 θ22


= C



θ

2= tc

radicke




c

e

To each point




2 t + b2c cos2 t

(minusaeybebex

ae

)

Theorem





c

e

Q

tQ

vn We set v =d p

du= p Then u is a


t =dt

du=

radica2c sin

2 t + b2c cos2 t








dt

du=

radica2c sin

2 t + b2c cos2 t =

radica2c sin






int ϕ

0



K = ac u(π

2

)=

int π2

0










Theorem





2τK



Q1(∆u )

P1(0)

P1(2∆u )

y

c

e





References

References





















x2 =(a2 + ke)(a

2 + kh)

d2


2 + kh)

d2



P

F1 F2M





P1tP1

P2P5

Q1

Q4

Q5

R1

R4

R5

OF1

F2

e

c cprimer2

l1r1

l5


sin2θ(t)

2=

kete(t)2

tanθ(t)

2= plusmn

radicke

tc(t)





P1tP1

P2P5

Q1

Q4

Q5

R1

R4

R5

OF1

F2

e

c cprimer2

l1r1

l5


prime1 bc sin t

prime1)





P1tP1

P2P5

Q1

Q4

Q5

R1

R4

R5

OF1

F2

e

c cprimer2

l1r1

l5



∥∥∥ te(t1 + t22

)∥∥∥2



tant1 + t22












P1

P2

P3

Q1Q2

O

θi2

θi2

ce

n1

n2

u1u2

ui = 1





)ni = sin

θi2ni =

radickete

teaebe

=

radickeaebe



c







B1

A1

A2

B2

A3

B3

c

c1 t1

t2

t3

t4 Theorem







B1

A1

A2

B2

A3

B3

R

c

c1

c2

c3

t1

t2

t3

t4




R

c

A1B1

c1

c2

c3

A3B3

A2B2

Rc Rt



B1

A1

A2

B2

A3

B3

R

c

c1

c2

c3

dt1

t2

t3







P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)




P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)



(2acbcaebea2cb

2c minus k2e

)2



P2

P3

Q2

Q3

Q1

S(1)2

S(1)3

S(2)3

ww

w

w1

w1

ce

e(1)

e(2)

r2

l2

r3r3

l3





Qiminus1Qi



De|1 =Q1S(1)2 + S

(1)2 Q3 minus

Q1Q3 = (r2+l2+w) + (r3+l3+w)minus

Q1Q3= 2De + 2w


P2 P3

P4P1

Q2Q4

Q1

e

c

e(1)

e(2)



radick3e

a2cb2c minus k2e



P2

P3P1

Qprime1

Qprime2

P prime2P prime1

Q2 Q1

R2

R1

O

e

cr3

l2r2

l1




prime2 This







P prime1

P prime2P2P3

P1

Qprime1Qprime2

Q1

Q2

S(1)1

S(1)2

S(1)3

c

e(1)

e1

0

1234

14

7858



prime1 t2 t

prime2


prime2(Q2)






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

ce

e(1)

e(2)


Theorem






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

l2

r7

ce

e(1)

e(2)







Theoremsumli =

sumri = Le2



Piminus1

Pi

Pi+1

Qiminus1Qi

Fiminus1Fi

O

θi2

θi2

ce


radicke

Le =sumNi=1

(PiFi + PiFiminus1

)=rArr

Le =aeberadicke

Nsum

i=1

2kete(ti)2

=aeberadicke

Nsum

i=1

(1minus cos θi)


radickeaebe

Le

WithsumNi=1

1

te(ti)2also

sumNi=1OtP

2and

sumNi=1 κe(ti)

23 are invariant



F2F1

P2 tP

Q1Q2

r2

l2

e

c

vt1vn1

vt2vn2

v2

ω1

ω2

θ22



v2 = vt1 + vn1 = vt2 + vn2



ω1ω2=l2r2






P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5




and

cosθ22=vt|2v2

where



θ22



P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5



2 θ22


= C



θ

2= tc

radicke




c

e

To each point




2 t + b2c cos2 t

(minusaeybebex

ae

)

Theorem





c

e

Q

tQ

vn We set v =d p

du= p Then u is a


t =dt

du=

radica2c sin

2 t + b2c cos2 t








dt

du=

radica2c sin

2 t + b2c cos2 t =

radica2c sin






int ϕ

0



K = ac u(π

2

)=

int π2

0










Theorem





2τK



Q1(∆u )

P1(0)

P1(2∆u )

y

c

e





References

References




















P1tP1

P2P5

Q1

Q4

Q5

R1

R4

R5

OF1

F2

e

c cprimer2

l1r1

l5


sin2θ(t)

2=

kete(t)2

tanθ(t)

2= plusmn

radicke

tc(t)





P1tP1

P2P5

Q1

Q4

Q5

R1

R4

R5

OF1

F2

e

c cprimer2

l1r1

l5


prime1 bc sin t

prime1)





P1tP1

P2P5

Q1

Q4

Q5

R1

R4

R5

OF1

F2

e

c cprimer2

l1r1

l5



∥∥∥ te(t1 + t22

)∥∥∥2



tant1 + t22












P1

P2

P3

Q1Q2

O

θi2

θi2

ce

n1

n2

u1u2

ui = 1





)ni = sin

θi2ni =

radickete

teaebe

=

radickeaebe



c







B1

A1

A2

B2

A3

B3

c

c1 t1

t2

t3

t4 Theorem







B1

A1

A2

B2

A3

B3

R

c

c1

c2

c3

t1

t2

t3

t4




R

c

A1B1

c1

c2

c3

A3B3

A2B2

Rc Rt



B1

A1

A2

B2

A3

B3

R

c

c1

c2

c3

dt1

t2

t3







P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)




P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)



(2acbcaebea2cb

2c minus k2e

)2



P2

P3

Q2

Q3

Q1

S(1)2

S(1)3

S(2)3

ww

w

w1

w1

ce

e(1)

e(2)

r2

l2

r3r3

l3





Qiminus1Qi



De|1 =Q1S(1)2 + S

(1)2 Q3 minus

Q1Q3 = (r2+l2+w) + (r3+l3+w)minus

Q1Q3= 2De + 2w


P2 P3

P4P1

Q2Q4

Q1

e

c

e(1)

e(2)



radick3e

a2cb2c minus k2e



P2

P3P1

Qprime1

Qprime2

P prime2P prime1

Q2 Q1

R2

R1

O

e

cr3

l2r2

l1




prime2 This







P prime1

P prime2P2P3

P1

Qprime1Qprime2

Q1

Q2

S(1)1

S(1)2

S(1)3

c

e(1)

e1

0

1234

14

7858



prime1 t2 t

prime2


prime2(Q2)






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

ce

e(1)

e(2)


Theorem






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

l2

r7

ce

e(1)

e(2)







Theoremsumli =

sumri = Le2



Piminus1

Pi

Pi+1

Qiminus1Qi

Fiminus1Fi

O

θi2

θi2

ce


radicke

Le =sumNi=1

(PiFi + PiFiminus1

)=rArr

Le =aeberadicke

Nsum

i=1

2kete(ti)2

=aeberadicke

Nsum

i=1

(1minus cos θi)


radickeaebe

Le

WithsumNi=1

1

te(ti)2also

sumNi=1OtP

2and

sumNi=1 κe(ti)

23 are invariant



F2F1

P2 tP

Q1Q2

r2

l2

e

c

vt1vn1

vt2vn2

v2

ω1

ω2

θ22



v2 = vt1 + vn1 = vt2 + vn2



ω1ω2=l2r2






P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5




and

cosθ22=vt|2v2

where



θ22



P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5



2 θ22


= C



θ

2= tc

radicke




c

e

To each point




2 t + b2c cos2 t

(minusaeybebex

ae

)

Theorem





c

e

Q

tQ

vn We set v =d p

du= p Then u is a


t =dt

du=

radica2c sin

2 t + b2c cos2 t








dt

du=

radica2c sin

2 t + b2c cos2 t =

radica2c sin






int ϕ

0



K = ac u(π

2

)=

int π2

0










Theorem





2τK



Q1(∆u )

P1(0)

P1(2∆u )

y

c

e





References

References




















P1tP1

P2P5

Q1

Q4

Q5

R1

R4

R5

OF1

F2

e

c cprimer2

l1r1

l5


prime1 bc sin t

prime1)





P1tP1

P2P5

Q1

Q4

Q5

R1

R4

R5

OF1

F2

e

c cprimer2

l1r1

l5



∥∥∥ te(t1 + t22

)∥∥∥2



tant1 + t22












P1

P2

P3

Q1Q2

O

θi2

θi2

ce

n1

n2

u1u2

ui = 1





)ni = sin

θi2ni =

radickete

teaebe

=

radickeaebe



c







B1

A1

A2

B2

A3

B3

c

c1 t1

t2

t3

t4 Theorem







B1

A1

A2

B2

A3

B3

R

c

c1

c2

c3

t1

t2

t3

t4




R

c

A1B1

c1

c2

c3

A3B3

A2B2

Rc Rt



B1

A1

A2

B2

A3

B3

R

c

c1

c2

c3

dt1

t2

t3







P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)




P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)



(2acbcaebea2cb

2c minus k2e

)2



P2

P3

Q2

Q3

Q1

S(1)2

S(1)3

S(2)3

ww

w

w1

w1

ce

e(1)

e(2)

r2

l2

r3r3

l3





Qiminus1Qi



De|1 =Q1S(1)2 + S

(1)2 Q3 minus

Q1Q3 = (r2+l2+w) + (r3+l3+w)minus

Q1Q3= 2De + 2w


P2 P3

P4P1

Q2Q4

Q1

e

c

e(1)

e(2)



radick3e

a2cb2c minus k2e



P2

P3P1

Qprime1

Qprime2

P prime2P prime1

Q2 Q1

R2

R1

O

e

cr3

l2r2

l1




prime2 This







P prime1

P prime2P2P3

P1

Qprime1Qprime2

Q1

Q2

S(1)1

S(1)2

S(1)3

c

e(1)

e1

0

1234

14

7858



prime1 t2 t

prime2


prime2(Q2)






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

ce

e(1)

e(2)


Theorem






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

l2

r7

ce

e(1)

e(2)







Theoremsumli =

sumri = Le2



Piminus1

Pi

Pi+1

Qiminus1Qi

Fiminus1Fi

O

θi2

θi2

ce


radicke

Le =sumNi=1

(PiFi + PiFiminus1

)=rArr

Le =aeberadicke

Nsum

i=1

2kete(ti)2

=aeberadicke

Nsum

i=1

(1minus cos θi)


radickeaebe

Le

WithsumNi=1

1

te(ti)2also

sumNi=1OtP

2and

sumNi=1 κe(ti)

23 are invariant



F2F1

P2 tP

Q1Q2

r2

l2

e

c

vt1vn1

vt2vn2

v2

ω1

ω2

θ22



v2 = vt1 + vn1 = vt2 + vn2



ω1ω2=l2r2






P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5




and

cosθ22=vt|2v2

where



θ22



P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5



2 θ22


= C



θ

2= tc

radicke




c

e

To each point




2 t + b2c cos2 t

(minusaeybebex

ae

)

Theorem





c

e

Q

tQ

vn We set v =d p

du= p Then u is a


t =dt

du=

radica2c sin

2 t + b2c cos2 t








dt

du=

radica2c sin

2 t + b2c cos2 t =

radica2c sin






int ϕ

0



K = ac u(π

2

)=

int π2

0










Theorem





2τK



Q1(∆u )

P1(0)

P1(2∆u )

y

c

e





References

References




















P1tP1

P2P5

Q1

Q4

Q5

R1

R4

R5

OF1

F2

e

c cprimer2

l1r1

l5



∥∥∥ te(t1 + t22

)∥∥∥2



tant1 + t22












P1

P2

P3

Q1Q2

O

θi2

θi2

ce

n1

n2

u1u2

ui = 1





)ni = sin

θi2ni =

radickete

teaebe

=

radickeaebe



c







B1

A1

A2

B2

A3

B3

c

c1 t1

t2

t3

t4 Theorem







B1

A1

A2

B2

A3

B3

R

c

c1

c2

c3

t1

t2

t3

t4




R

c

A1B1

c1

c2

c3

A3B3

A2B2

Rc Rt



B1

A1

A2

B2

A3

B3

R

c

c1

c2

c3

dt1

t2

t3







P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)




P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)



(2acbcaebea2cb

2c minus k2e

)2



P2

P3

Q2

Q3

Q1

S(1)2

S(1)3

S(2)3

ww

w

w1

w1

ce

e(1)

e(2)

r2

l2

r3r3

l3





Qiminus1Qi



De|1 =Q1S(1)2 + S

(1)2 Q3 minus

Q1Q3 = (r2+l2+w) + (r3+l3+w)minus

Q1Q3= 2De + 2w


P2 P3

P4P1

Q2Q4

Q1

e

c

e(1)

e(2)



radick3e

a2cb2c minus k2e



P2

P3P1

Qprime1

Qprime2

P prime2P prime1

Q2 Q1

R2

R1

O

e

cr3

l2r2

l1




prime2 This







P prime1

P prime2P2P3

P1

Qprime1Qprime2

Q1

Q2

S(1)1

S(1)2

S(1)3

c

e(1)

e1

0

1234

14

7858



prime1 t2 t

prime2


prime2(Q2)






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

ce

e(1)

e(2)


Theorem






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

l2

r7

ce

e(1)

e(2)







Theoremsumli =

sumri = Le2



Piminus1

Pi

Pi+1

Qiminus1Qi

Fiminus1Fi

O

θi2

θi2

ce


radicke

Le =sumNi=1

(PiFi + PiFiminus1

)=rArr

Le =aeberadicke

Nsum

i=1

2kete(ti)2

=aeberadicke

Nsum

i=1

(1minus cos θi)


radickeaebe

Le

WithsumNi=1

1

te(ti)2also

sumNi=1OtP

2and

sumNi=1 κe(ti)

23 are invariant



F2F1

P2 tP

Q1Q2

r2

l2

e

c

vt1vn1

vt2vn2

v2

ω1

ω2

θ22



v2 = vt1 + vn1 = vt2 + vn2



ω1ω2=l2r2






P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5




and

cosθ22=vt|2v2

where



θ22



P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5



2 θ22


= C



θ

2= tc

radicke




c

e

To each point




2 t + b2c cos2 t

(minusaeybebex

ae

)

Theorem





c

e

Q

tQ

vn We set v =d p

du= p Then u is a


t =dt

du=

radica2c sin

2 t + b2c cos2 t








dt

du=

radica2c sin

2 t + b2c cos2 t =

radica2c sin






int ϕ

0



K = ac u(π

2

)=

int π2

0










Theorem





2τK



Q1(∆u )

P1(0)

P1(2∆u )

y

c

e





References

References




























P1

P2

P3

Q1Q2

O

θi2

θi2

ce

n1

n2

u1u2

ui = 1





)ni = sin

θi2ni =

radickete

teaebe

=

radickeaebe



c







B1

A1

A2

B2

A3

B3

c

c1 t1

t2

t3

t4 Theorem







B1

A1

A2

B2

A3

B3

R

c

c1

c2

c3

t1

t2

t3

t4




R

c

A1B1

c1

c2

c3

A3B3

A2B2

Rc Rt



B1

A1

A2

B2

A3

B3

R

c

c1

c2

c3

dt1

t2

t3







P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)




P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)



(2acbcaebea2cb

2c minus k2e

)2



P2

P3

Q2

Q3

Q1

S(1)2

S(1)3

S(2)3

ww

w

w1

w1

ce

e(1)

e(2)

r2

l2

r3r3

l3





Qiminus1Qi



De|1 =Q1S(1)2 + S

(1)2 Q3 minus

Q1Q3 = (r2+l2+w) + (r3+l3+w)minus

Q1Q3= 2De + 2w


P2 P3

P4P1

Q2Q4

Q1

e

c

e(1)

e(2)



radick3e

a2cb2c minus k2e



P2

P3P1

Qprime1

Qprime2

P prime2P prime1

Q2 Q1

R2

R1

O

e

cr3

l2r2

l1




prime2 This







P prime1

P prime2P2P3

P1

Qprime1Qprime2

Q1

Q2

S(1)1

S(1)2

S(1)3

c

e(1)

e1

0

1234

14

7858



prime1 t2 t

prime2


prime2(Q2)






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

ce

e(1)

e(2)


Theorem






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

l2

r7

ce

e(1)

e(2)







Theoremsumli =

sumri = Le2



Piminus1

Pi

Pi+1

Qiminus1Qi

Fiminus1Fi

O

θi2

θi2

ce


radicke

Le =sumNi=1

(PiFi + PiFiminus1

)=rArr

Le =aeberadicke

Nsum

i=1

2kete(ti)2

=aeberadicke

Nsum

i=1

(1minus cos θi)


radickeaebe

Le

WithsumNi=1

1

te(ti)2also

sumNi=1OtP

2and

sumNi=1 κe(ti)

23 are invariant



F2F1

P2 tP

Q1Q2

r2

l2

e

c

vt1vn1

vt2vn2

v2

ω1

ω2

θ22



v2 = vt1 + vn1 = vt2 + vn2



ω1ω2=l2r2






P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5




and

cosθ22=vt|2v2

where



θ22



P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5



2 θ22


= C



θ

2= tc

radicke




c

e

To each point




2 t + b2c cos2 t

(minusaeybebex

ae

)

Theorem





c

e

Q

tQ

vn We set v =d p

du= p Then u is a


t =dt

du=

radica2c sin

2 t + b2c cos2 t








dt

du=

radica2c sin

2 t + b2c cos2 t =

radica2c sin






int ϕ

0



K = ac u(π

2

)=

int π2

0










Theorem





2τK



Q1(∆u )

P1(0)

P1(2∆u )

y

c

e





References

References




















c







B1

A1

A2

B2

A3

B3

c

c1 t1

t2

t3

t4 Theorem







B1

A1

A2

B2

A3

B3

R

c

c1

c2

c3

t1

t2

t3

t4




R

c

A1B1

c1

c2

c3

A3B3

A2B2

Rc Rt



B1

A1

A2

B2

A3

B3

R

c

c1

c2

c3

dt1

t2

t3







P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)




P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)



(2acbcaebea2cb

2c minus k2e

)2



P2

P3

Q2

Q3

Q1

S(1)2

S(1)3

S(2)3

ww

w

w1

w1

ce

e(1)

e(2)

r2

l2

r3r3

l3





Qiminus1Qi



De|1 =Q1S(1)2 + S

(1)2 Q3 minus

Q1Q3 = (r2+l2+w) + (r3+l3+w)minus

Q1Q3= 2De + 2w


P2 P3

P4P1

Q2Q4

Q1

e

c

e(1)

e(2)



radick3e

a2cb2c minus k2e



P2

P3P1

Qprime1

Qprime2

P prime2P prime1

Q2 Q1

R2

R1

O

e

cr3

l2r2

l1




prime2 This







P prime1

P prime2P2P3

P1

Qprime1Qprime2

Q1

Q2

S(1)1

S(1)2

S(1)3

c

e(1)

e1

0

1234

14

7858



prime1 t2 t

prime2


prime2(Q2)






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

ce

e(1)

e(2)


Theorem






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

l2

r7

ce

e(1)

e(2)







Theoremsumli =

sumri = Le2



Piminus1

Pi

Pi+1

Qiminus1Qi

Fiminus1Fi

O

θi2

θi2

ce


radicke

Le =sumNi=1

(PiFi + PiFiminus1

)=rArr

Le =aeberadicke

Nsum

i=1

2kete(ti)2

=aeberadicke

Nsum

i=1

(1minus cos θi)


radickeaebe

Le

WithsumNi=1

1

te(ti)2also

sumNi=1OtP

2and

sumNi=1 κe(ti)

23 are invariant



F2F1

P2 tP

Q1Q2

r2

l2

e

c

vt1vn1

vt2vn2

v2

ω1

ω2

θ22



v2 = vt1 + vn1 = vt2 + vn2



ω1ω2=l2r2






P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5




and

cosθ22=vt|2v2

where



θ22



P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5



2 θ22


= C



θ

2= tc

radicke




c

e

To each point




2 t + b2c cos2 t

(minusaeybebex

ae

)

Theorem





c

e

Q

tQ

vn We set v =d p

du= p Then u is a


t =dt

du=

radica2c sin

2 t + b2c cos2 t








dt

du=

radica2c sin

2 t + b2c cos2 t =

radica2c sin






int ϕ

0



K = ac u(π

2

)=

int π2

0










Theorem





2τK



Q1(∆u )

P1(0)

P1(2∆u )

y

c

e





References

References




















B1

A1

A2

B2

A3

B3

c

c1 t1

t2

t3

t4 Theorem







B1

A1

A2

B2

A3

B3

R

c

c1

c2

c3

t1

t2

t3

t4




R

c

A1B1

c1

c2

c3

A3B3

A2B2

Rc Rt



B1

A1

A2

B2

A3

B3

R

c

c1

c2

c3

dt1

t2

t3







P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)




P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)



(2acbcaebea2cb

2c minus k2e

)2



P2

P3

Q2

Q3

Q1

S(1)2

S(1)3

S(2)3

ww

w

w1

w1

ce

e(1)

e(2)

r2

l2

r3r3

l3





Qiminus1Qi



De|1 =Q1S(1)2 + S

(1)2 Q3 minus

Q1Q3 = (r2+l2+w) + (r3+l3+w)minus

Q1Q3= 2De + 2w


P2 P3

P4P1

Q2Q4

Q1

e

c

e(1)

e(2)



radick3e

a2cb2c minus k2e



P2

P3P1

Qprime1

Qprime2

P prime2P prime1

Q2 Q1

R2

R1

O

e

cr3

l2r2

l1




prime2 This







P prime1

P prime2P2P3

P1

Qprime1Qprime2

Q1

Q2

S(1)1

S(1)2

S(1)3

c

e(1)

e1

0

1234

14

7858



prime1 t2 t

prime2


prime2(Q2)






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

ce

e(1)

e(2)


Theorem






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

l2

r7

ce

e(1)

e(2)







Theoremsumli =

sumri = Le2



Piminus1

Pi

Pi+1

Qiminus1Qi

Fiminus1Fi

O

θi2

θi2

ce


radicke

Le =sumNi=1

(PiFi + PiFiminus1

)=rArr

Le =aeberadicke

Nsum

i=1

2kete(ti)2

=aeberadicke

Nsum

i=1

(1minus cos θi)


radickeaebe

Le

WithsumNi=1

1

te(ti)2also

sumNi=1OtP

2and

sumNi=1 κe(ti)

23 are invariant



F2F1

P2 tP

Q1Q2

r2

l2

e

c

vt1vn1

vt2vn2

v2

ω1

ω2

θ22



v2 = vt1 + vn1 = vt2 + vn2



ω1ω2=l2r2






P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5




and

cosθ22=vt|2v2

where



θ22



P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5



2 θ22


= C



θ

2= tc

radicke




c

e

To each point




2 t + b2c cos2 t

(minusaeybebex

ae

)

Theorem





c

e

Q

tQ

vn We set v =d p

du= p Then u is a


t =dt

du=

radica2c sin

2 t + b2c cos2 t








dt

du=

radica2c sin

2 t + b2c cos2 t =

radica2c sin






int ϕ

0



K = ac u(π

2

)=

int π2

0










Theorem





2τK



Q1(∆u )

P1(0)

P1(2∆u )

y

c

e





References

References




















B1

A1

A2

B2

A3

B3

R

c

c1

c2

c3

t1

t2

t3

t4




R

c

A1B1

c1

c2

c3

A3B3

A2B2

Rc Rt



B1

A1

A2

B2

A3

B3

R

c

c1

c2

c3

dt1

t2

t3







P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)




P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)



(2acbcaebea2cb

2c minus k2e

)2



P2

P3

Q2

Q3

Q1

S(1)2

S(1)3

S(2)3

ww

w

w1

w1

ce

e(1)

e(2)

r2

l2

r3r3

l3





Qiminus1Qi



De|1 =Q1S(1)2 + S

(1)2 Q3 minus

Q1Q3 = (r2+l2+w) + (r3+l3+w)minus

Q1Q3= 2De + 2w


P2 P3

P4P1

Q2Q4

Q1

e

c

e(1)

e(2)



radick3e

a2cb2c minus k2e



P2

P3P1

Qprime1

Qprime2

P prime2P prime1

Q2 Q1

R2

R1

O

e

cr3

l2r2

l1




prime2 This







P prime1

P prime2P2P3

P1

Qprime1Qprime2

Q1

Q2

S(1)1

S(1)2

S(1)3

c

e(1)

e1

0

1234

14

7858



prime1 t2 t

prime2


prime2(Q2)






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

ce

e(1)

e(2)


Theorem






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

l2

r7

ce

e(1)

e(2)







Theoremsumli =

sumri = Le2



Piminus1

Pi

Pi+1

Qiminus1Qi

Fiminus1Fi

O

θi2

θi2

ce


radicke

Le =sumNi=1

(PiFi + PiFiminus1

)=rArr

Le =aeberadicke

Nsum

i=1

2kete(ti)2

=aeberadicke

Nsum

i=1

(1minus cos θi)


radickeaebe

Le

WithsumNi=1

1

te(ti)2also

sumNi=1OtP

2and

sumNi=1 κe(ti)

23 are invariant



F2F1

P2 tP

Q1Q2

r2

l2

e

c

vt1vn1

vt2vn2

v2

ω1

ω2

θ22



v2 = vt1 + vn1 = vt2 + vn2



ω1ω2=l2r2






P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5




and

cosθ22=vt|2v2

where



θ22



P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5



2 θ22


= C



θ

2= tc

radicke




c

e

To each point




2 t + b2c cos2 t

(minusaeybebex

ae

)

Theorem





c

e

Q

tQ

vn We set v =d p

du= p Then u is a


t =dt

du=

radica2c sin

2 t + b2c cos2 t








dt

du=

radica2c sin

2 t + b2c cos2 t =

radica2c sin






int ϕ

0



K = ac u(π

2

)=

int π2

0










Theorem





2τK



Q1(∆u )

P1(0)

P1(2∆u )

y

c

e





References

References




















B1

A1

A2

B2

A3

B3

R

c

c1

c2

c3

dt1

t2

t3







P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)




P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)



(2acbcaebea2cb

2c minus k2e

)2



P2

P3

Q2

Q3

Q1

S(1)2

S(1)3

S(2)3

ww

w

w1

w1

ce

e(1)

e(2)

r2

l2

r3r3

l3





Qiminus1Qi



De|1 =Q1S(1)2 + S

(1)2 Q3 minus

Q1Q3 = (r2+l2+w) + (r3+l3+w)minus

Q1Q3= 2De + 2w


P2 P3

P4P1

Q2Q4

Q1

e

c

e(1)

e(2)



radick3e

a2cb2c minus k2e



P2

P3P1

Qprime1

Qprime2

P prime2P prime1

Q2 Q1

R2

R1

O

e

cr3

l2r2

l1




prime2 This







P prime1

P prime2P2P3

P1

Qprime1Qprime2

Q1

Q2

S(1)1

S(1)2

S(1)3

c

e(1)

e1

0

1234

14

7858



prime1 t2 t

prime2


prime2(Q2)






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

ce

e(1)

e(2)


Theorem






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

l2

r7

ce

e(1)

e(2)







Theoremsumli =

sumri = Le2



Piminus1

Pi

Pi+1

Qiminus1Qi

Fiminus1Fi

O

θi2

θi2

ce


radicke

Le =sumNi=1

(PiFi + PiFiminus1

)=rArr

Le =aeberadicke

Nsum

i=1

2kete(ti)2

=aeberadicke

Nsum

i=1

(1minus cos θi)


radickeaebe

Le

WithsumNi=1

1

te(ti)2also

sumNi=1OtP

2and

sumNi=1 κe(ti)

23 are invariant



F2F1

P2 tP

Q1Q2

r2

l2

e

c

vt1vn1

vt2vn2

v2

ω1

ω2

θ22



v2 = vt1 + vn1 = vt2 + vn2



ω1ω2=l2r2






P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5




and

cosθ22=vt|2v2

where



θ22



P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5



2 θ22


= C



θ

2= tc

radicke




c

e

To each point




2 t + b2c cos2 t

(minusaeybebex

ae

)

Theorem





c

e

Q

tQ

vn We set v =d p

du= p Then u is a


t =dt

du=

radica2c sin

2 t + b2c cos2 t








dt

du=

radica2c sin

2 t + b2c cos2 t =

radica2c sin






int ϕ

0



K = ac u(π

2

)=

int π2

0










Theorem





2τK



Q1(∆u )

P1(0)

P1(2∆u )

y

c

e





References

References




















P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)




P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)



(2acbcaebea2cb

2c minus k2e

)2



P2

P3

Q2

Q3

Q1

S(1)2

S(1)3

S(2)3

ww

w

w1

w1

ce

e(1)

e(2)

r2

l2

r3r3

l3





Qiminus1Qi



De|1 =Q1S(1)2 + S

(1)2 Q3 minus

Q1Q3 = (r2+l2+w) + (r3+l3+w)minus

Q1Q3= 2De + 2w


P2 P3

P4P1

Q2Q4

Q1

e

c

e(1)

e(2)



radick3e

a2cb2c minus k2e



P2

P3P1

Qprime1

Qprime2

P prime2P prime1

Q2 Q1

R2

R1

O

e

cr3

l2r2

l1




prime2 This







P prime1

P prime2P2P3

P1

Qprime1Qprime2

Q1

Q2

S(1)1

S(1)2

S(1)3

c

e(1)

e1

0

1234

14

7858



prime1 t2 t

prime2


prime2(Q2)






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

ce

e(1)

e(2)


Theorem






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

l2

r7

ce

e(1)

e(2)







Theoremsumli =

sumri = Le2



Piminus1

Pi

Pi+1

Qiminus1Qi

Fiminus1Fi

O

θi2

θi2

ce


radicke

Le =sumNi=1

(PiFi + PiFiminus1

)=rArr

Le =aeberadicke

Nsum

i=1

2kete(ti)2

=aeberadicke

Nsum

i=1

(1minus cos θi)


radickeaebe

Le

WithsumNi=1

1

te(ti)2also

sumNi=1OtP

2and

sumNi=1 κe(ti)

23 are invariant



F2F1

P2 tP

Q1Q2

r2

l2

e

c

vt1vn1

vt2vn2

v2

ω1

ω2

θ22



v2 = vt1 + vn1 = vt2 + vn2



ω1ω2=l2r2






P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5




and

cosθ22=vt|2v2

where



θ22



P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5



2 θ22


= C



θ

2= tc

radicke




c

e

To each point




2 t + b2c cos2 t

(minusaeybebex

ae

)

Theorem





c

e

Q

tQ

vn We set v =d p

du= p Then u is a


t =dt

du=

radica2c sin

2 t + b2c cos2 t








dt

du=

radica2c sin

2 t + b2c cos2 t =

radica2c sin






int ϕ

0



K = ac u(π

2

)=

int π2

0










Theorem





2τK



Q1(∆u )

P1(0)

P1(2∆u )

y

c

e





References

References



















P1

P2

P3P4

P0

Q1

Q2

Q3

S(1)1

S(1)2

S(2)1

S(2)2

c

e e(1)

e(2)



(2acbcaebea2cb

2c minus k2e

)2



P2

P3

Q2

Q3

Q1

S(1)2

S(1)3

S(2)3

ww

w

w1

w1

ce

e(1)

e(2)

r2

l2

r3r3

l3





Qiminus1Qi



De|1 =Q1S(1)2 + S

(1)2 Q3 minus

Q1Q3 = (r2+l2+w) + (r3+l3+w)minus

Q1Q3= 2De + 2w


P2 P3

P4P1

Q2Q4

Q1

e

c

e(1)

e(2)



radick3e

a2cb2c minus k2e



P2

P3P1

Qprime1

Qprime2

P prime2P prime1

Q2 Q1

R2

R1

O

e

cr3

l2r2

l1




prime2 This







P prime1

P prime2P2P3

P1

Qprime1Qprime2

Q1

Q2

S(1)1

S(1)2

S(1)3

c

e(1)

e1

0

1234

14

7858



prime1 t2 t

prime2


prime2(Q2)






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

ce

e(1)

e(2)


Theorem






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

l2

r7

ce

e(1)

e(2)







Theoremsumli =

sumri = Le2



Piminus1

Pi

Pi+1

Qiminus1Qi

Fiminus1Fi

O

θi2

θi2

ce


radicke

Le =sumNi=1

(PiFi + PiFiminus1

)=rArr

Le =aeberadicke

Nsum

i=1

2kete(ti)2

=aeberadicke

Nsum

i=1

(1minus cos θi)


radickeaebe

Le

WithsumNi=1

1

te(ti)2also

sumNi=1OtP

2and

sumNi=1 κe(ti)

23 are invariant



F2F1

P2 tP

Q1Q2

r2

l2

e

c

vt1vn1

vt2vn2

v2

ω1

ω2

θ22



v2 = vt1 + vn1 = vt2 + vn2



ω1ω2=l2r2






P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5




and

cosθ22=vt|2v2

where



θ22



P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5



2 θ22


= C



θ

2= tc

radicke




c

e

To each point




2 t + b2c cos2 t

(minusaeybebex

ae

)

Theorem





c

e

Q

tQ

vn We set v =d p

du= p Then u is a


t =dt

du=

radica2c sin

2 t + b2c cos2 t








dt

du=

radica2c sin

2 t + b2c cos2 t =

radica2c sin






int ϕ

0



K = ac u(π

2

)=

int π2

0










Theorem





2τK



Q1(∆u )

P1(0)

P1(2∆u )

y

c

e





References

References




















P2

P3

Q2

Q3

Q1

S(1)2

S(1)3

S(2)3

ww

w

w1

w1

ce

e(1)

e(2)

r2

l2

r3r3

l3





Qiminus1Qi



De|1 =Q1S(1)2 + S

(1)2 Q3 minus

Q1Q3 = (r2+l2+w) + (r3+l3+w)minus

Q1Q3= 2De + 2w


P2 P3

P4P1

Q2Q4

Q1

e

c

e(1)

e(2)



radick3e

a2cb2c minus k2e



P2

P3P1

Qprime1

Qprime2

P prime2P prime1

Q2 Q1

R2

R1

O

e

cr3

l2r2

l1




prime2 This







P prime1

P prime2P2P3

P1

Qprime1Qprime2

Q1

Q2

S(1)1

S(1)2

S(1)3

c

e(1)

e1

0

1234

14

7858



prime1 t2 t

prime2


prime2(Q2)






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

ce

e(1)

e(2)


Theorem






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

l2

r7

ce

e(1)

e(2)







Theoremsumli =

sumri = Le2



Piminus1

Pi

Pi+1

Qiminus1Qi

Fiminus1Fi

O

θi2

θi2

ce


radicke

Le =sumNi=1

(PiFi + PiFiminus1

)=rArr

Le =aeberadicke

Nsum

i=1

2kete(ti)2

=aeberadicke

Nsum

i=1

(1minus cos θi)


radickeaebe

Le

WithsumNi=1

1

te(ti)2also

sumNi=1OtP

2and

sumNi=1 κe(ti)

23 are invariant



F2F1

P2 tP

Q1Q2

r2

l2

e

c

vt1vn1

vt2vn2

v2

ω1

ω2

θ22



v2 = vt1 + vn1 = vt2 + vn2



ω1ω2=l2r2






P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5




and

cosθ22=vt|2v2

where



θ22



P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5



2 θ22


= C



θ

2= tc

radicke




c

e

To each point




2 t + b2c cos2 t

(minusaeybebex

ae

)

Theorem





c

e

Q

tQ

vn We set v =d p

du= p Then u is a


t =dt

du=

radica2c sin

2 t + b2c cos2 t








dt

du=

radica2c sin

2 t + b2c cos2 t =

radica2c sin






int ϕ

0



K = ac u(π

2

)=

int π2

0










Theorem





2τK



Q1(∆u )

P1(0)

P1(2∆u )

y

c

e





References

References



















P2 P3

P4P1

Q2Q4

Q1

e

c

e(1)

e(2)



radick3e

a2cb2c minus k2e



P2

P3P1

Qprime1

Qprime2

P prime2P prime1

Q2 Q1

R2

R1

O

e

cr3

l2r2

l1




prime2 This







P prime1

P prime2P2P3

P1

Qprime1Qprime2

Q1

Q2

S(1)1

S(1)2

S(1)3

c

e(1)

e1

0

1234

14

7858



prime1 t2 t

prime2


prime2(Q2)






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

ce

e(1)

e(2)


Theorem






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

l2

r7

ce

e(1)

e(2)







Theoremsumli =

sumri = Le2



Piminus1

Pi

Pi+1

Qiminus1Qi

Fiminus1Fi

O

θi2

θi2

ce


radicke

Le =sumNi=1

(PiFi + PiFiminus1

)=rArr

Le =aeberadicke

Nsum

i=1

2kete(ti)2

=aeberadicke

Nsum

i=1

(1minus cos θi)


radickeaebe

Le

WithsumNi=1

1

te(ti)2also

sumNi=1OtP

2and

sumNi=1 κe(ti)

23 are invariant



F2F1

P2 tP

Q1Q2

r2

l2

e

c

vt1vn1

vt2vn2

v2

ω1

ω2

θ22



v2 = vt1 + vn1 = vt2 + vn2



ω1ω2=l2r2






P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5




and

cosθ22=vt|2v2

where



θ22



P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5



2 θ22


= C



θ

2= tc

radicke




c

e

To each point




2 t + b2c cos2 t

(minusaeybebex

ae

)

Theorem





c

e

Q

tQ

vn We set v =d p

du= p Then u is a


t =dt

du=

radica2c sin

2 t + b2c cos2 t








dt

du=

radica2c sin

2 t + b2c cos2 t =

radica2c sin






int ϕ

0



K = ac u(π

2

)=

int π2

0










Theorem





2τK



Q1(∆u )

P1(0)

P1(2∆u )

y

c

e





References

References




















P2

P3P1

Qprime1

Qprime2

P prime2P prime1

Q2 Q1

R2

R1

O

e

cr3

l2r2

l1




prime2 This







P prime1

P prime2P2P3

P1

Qprime1Qprime2

Q1

Q2

S(1)1

S(1)2

S(1)3

c

e(1)

e1

0

1234

14

7858



prime1 t2 t

prime2


prime2(Q2)






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

ce

e(1)

e(2)


Theorem






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

l2

r7

ce

e(1)

e(2)







Theoremsumli =

sumri = Le2



Piminus1

Pi

Pi+1

Qiminus1Qi

Fiminus1Fi

O

θi2

θi2

ce


radicke

Le =sumNi=1

(PiFi + PiFiminus1

)=rArr

Le =aeberadicke

Nsum

i=1

2kete(ti)2

=aeberadicke

Nsum

i=1

(1minus cos θi)


radickeaebe

Le

WithsumNi=1

1

te(ti)2also

sumNi=1OtP

2and

sumNi=1 κe(ti)

23 are invariant



F2F1

P2 tP

Q1Q2

r2

l2

e

c

vt1vn1

vt2vn2

v2

ω1

ω2

θ22



v2 = vt1 + vn1 = vt2 + vn2



ω1ω2=l2r2






P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5




and

cosθ22=vt|2v2

where



θ22



P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5



2 θ22


= C



θ

2= tc

radicke




c

e

To each point




2 t + b2c cos2 t

(minusaeybebex

ae

)

Theorem





c

e

Q

tQ

vn We set v =d p

du= p Then u is a


t =dt

du=

radica2c sin

2 t + b2c cos2 t








dt

du=

radica2c sin

2 t + b2c cos2 t =

radica2c sin






int ϕ

0



K = ac u(π

2

)=

int π2

0










Theorem





2τK



Q1(∆u )

P1(0)

P1(2∆u )

y

c

e





References

References




















P prime1

P prime2P2P3

P1

Qprime1Qprime2

Q1

Q2

S(1)1

S(1)2

S(1)3

c

e(1)

e1

0

1234

14

7858



prime1 t2 t

prime2


prime2(Q2)






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

ce

e(1)

e(2)


Theorem






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

l2

r7

ce

e(1)

e(2)







Theoremsumli =

sumri = Le2



Piminus1

Pi

Pi+1

Qiminus1Qi

Fiminus1Fi

O

θi2

θi2

ce


radicke

Le =sumNi=1

(PiFi + PiFiminus1

)=rArr

Le =aeberadicke

Nsum

i=1

2kete(ti)2

=aeberadicke

Nsum

i=1

(1minus cos θi)


radickeaebe

Le

WithsumNi=1

1

te(ti)2also

sumNi=1OtP

2and

sumNi=1 κe(ti)

23 are invariant



F2F1

P2 tP

Q1Q2

r2

l2

e

c

vt1vn1

vt2vn2

v2

ω1

ω2

θ22



v2 = vt1 + vn1 = vt2 + vn2



ω1ω2=l2r2






P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5




and

cosθ22=vt|2v2

where



θ22



P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5



2 θ22


= C



θ

2= tc

radicke




c

e

To each point




2 t + b2c cos2 t

(minusaeybebex

ae

)

Theorem





c

e

Q

tQ

vn We set v =d p

du= p Then u is a


t =dt

du=

radica2c sin

2 t + b2c cos2 t








dt

du=

radica2c sin

2 t + b2c cos2 t =

radica2c sin






int ϕ

0



K = ac u(π

2

)=

int π2

0










Theorem





2τK



Q1(∆u )

P1(0)

P1(2∆u )

y

c

e





References

References




















P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

ce

e(1)

e(2)


Theorem






P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

l2

r7

ce

e(1)

e(2)







Theoremsumli =

sumri = Le2



Piminus1

Pi

Pi+1

Qiminus1Qi

Fiminus1Fi

O

θi2

θi2

ce


radicke

Le =sumNi=1

(PiFi + PiFiminus1

)=rArr

Le =aeberadicke

Nsum

i=1

2kete(ti)2

=aeberadicke

Nsum

i=1

(1minus cos θi)


radickeaebe

Le

WithsumNi=1

1

te(ti)2also

sumNi=1OtP

2and

sumNi=1 κe(ti)

23 are invariant



F2F1

P2 tP

Q1Q2

r2

l2

e

c

vt1vn1

vt2vn2

v2

ω1

ω2

θ22



v2 = vt1 + vn1 = vt2 + vn2



ω1ω2=l2r2






P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5




and

cosθ22=vt|2v2

where



θ22



P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5



2 θ22


= C



θ

2= tc

radicke




c

e

To each point




2 t + b2c cos2 t

(minusaeybebex

ae

)

Theorem





c

e

Q

tQ

vn We set v =d p

du= p Then u is a


t =dt

du=

radica2c sin

2 t + b2c cos2 t








dt

du=

radica2c sin

2 t + b2c cos2 t =

radica2c sin






int ϕ

0



K = ac u(π

2

)=

int π2

0










Theorem





2τK



Q1(∆u )

P1(0)

P1(2∆u )

y

c

e





References

References




















P1

P2

P3

P7

P8

P9

Q1

Q2

Q6Q7

Q8

Q9

S(1)1

S(1)2

S(1)7

S(1)8

S(1)9

S(2)1

S(2)9

S(2)8

l2

r7

ce

e(1)

e(2)







Theoremsumli =

sumri = Le2



Piminus1

Pi

Pi+1

Qiminus1Qi

Fiminus1Fi

O

θi2

θi2

ce


radicke

Le =sumNi=1

(PiFi + PiFiminus1

)=rArr

Le =aeberadicke

Nsum

i=1

2kete(ti)2

=aeberadicke

Nsum

i=1

(1minus cos θi)


radickeaebe

Le

WithsumNi=1

1

te(ti)2also

sumNi=1OtP

2and

sumNi=1 κe(ti)

23 are invariant



F2F1

P2 tP

Q1Q2

r2

l2

e

c

vt1vn1

vt2vn2

v2

ω1

ω2

θ22



v2 = vt1 + vn1 = vt2 + vn2



ω1ω2=l2r2






P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5




and

cosθ22=vt|2v2

where



θ22



P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5



2 θ22


= C



θ

2= tc

radicke




c

e

To each point




2 t + b2c cos2 t

(minusaeybebex

ae

)

Theorem





c

e

Q

tQ

vn We set v =d p

du= p Then u is a


t =dt

du=

radica2c sin

2 t + b2c cos2 t








dt

du=

radica2c sin

2 t + b2c cos2 t =

radica2c sin






int ϕ

0



K = ac u(π

2

)=

int π2

0










Theorem





2τK



Q1(∆u )

P1(0)

P1(2∆u )

y

c

e





References

References




















Piminus1

Pi

Pi+1

Qiminus1Qi

Fiminus1Fi

O

θi2

θi2

ce


radicke

Le =sumNi=1

(PiFi + PiFiminus1

)=rArr

Le =aeberadicke

Nsum

i=1

2kete(ti)2

=aeberadicke

Nsum

i=1

(1minus cos θi)


radickeaebe

Le

WithsumNi=1

1

te(ti)2also

sumNi=1OtP

2and

sumNi=1 κe(ti)

23 are invariant



F2F1

P2 tP

Q1Q2

r2

l2

e

c

vt1vn1

vt2vn2

v2

ω1

ω2

θ22



v2 = vt1 + vn1 = vt2 + vn2



ω1ω2=l2r2






P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5




and

cosθ22=vt|2v2

where



θ22



P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5



2 θ22


= C



θ

2= tc

radicke




c

e

To each point




2 t + b2c cos2 t

(minusaeybebex

ae

)

Theorem





c

e

Q

tQ

vn We set v =d p

du= p Then u is a


t =dt

du=

radica2c sin

2 t + b2c cos2 t








dt

du=

radica2c sin

2 t + b2c cos2 t =

radica2c sin






int ϕ

0



K = ac u(π

2

)=

int π2

0










Theorem





2τK



Q1(∆u )

P1(0)

P1(2∆u )

y

c

e





References

References




















F2F1

P2 tP

Q1Q2

r2

l2

e

c

vt1vn1

vt2vn2

v2

ω1

ω2

θ22



v2 = vt1 + vn1 = vt2 + vn2



ω1ω2=l2r2






P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5




and

cosθ22=vt|2v2

where



θ22



P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5



2 θ22


= C



θ

2= tc

radicke




c

e

To each point




2 t + b2c cos2 t

(minusaeybebex

ae

)

Theorem





c

e

Q

tQ

vn We set v =d p

du= p Then u is a


t =dt

du=

radica2c sin

2 t + b2c cos2 t








dt

du=

radica2c sin

2 t + b2c cos2 t =

radica2c sin






int ϕ

0



K = ac u(π

2

)=

int π2

0










Theorem





2τK



Q1(∆u )

P1(0)

P1(2∆u )

y

c

e





References

References




















P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5




and

cosθ22=vt|2v2

where



θ22



P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5



2 θ22


= C



θ

2= tc

radicke




c

e

To each point




2 t + b2c cos2 t

(minusaeybebex

ae

)

Theorem





c

e

Q

tQ

vn We set v =d p

du= p Then u is a


t =dt

du=

radica2c sin

2 t + b2c cos2 t








dt

du=

radica2c sin

2 t + b2c cos2 t =

radica2c sin






int ϕ

0



K = ac u(π

2

)=

int π2

0










Theorem





2τK



Q1(∆u )

P1(0)

P1(2∆u )

y

c

e





References

References




















P1

P2

P3

P4

P5

Q1Q2

Q3

Q4

Q5

R1

R2

R3

R4

R5

e

c

l1

r2l2

r3

θ12

θ12

θ22

θ22

ω1

ω2

ω3ω4

ω5

v2

vt2vn|2

v1

v3

v4

v5



2 θ22


= C



θ

2= tc

radicke




c

e

To each point




2 t + b2c cos2 t

(minusaeybebex

ae

)

Theorem





c

e

Q

tQ

vn We set v =d p

du= p Then u is a


t =dt

du=

radica2c sin

2 t + b2c cos2 t








dt

du=

radica2c sin

2 t + b2c cos2 t =

radica2c sin






int ϕ

0



K = ac u(π

2

)=

int π2

0










Theorem





2τK



Q1(∆u )

P1(0)

P1(2∆u )

y

c

e





References

References




















c

e

To each point




2 t + b2c cos2 t

(minusaeybebex

ae

)

Theorem





c

e

Q

tQ

vn We set v =d p

du= p Then u is a


t =dt

du=

radica2c sin

2 t + b2c cos2 t








dt

du=

radica2c sin

2 t + b2c cos2 t =

radica2c sin






int ϕ

0



K = ac u(π

2

)=

int π2

0










Theorem





2τK



Q1(∆u )

P1(0)

P1(2∆u )

y

c

e





References

References




















c

e

Q

tQ

vn We set v =d p

du= p Then u is a


t =dt

du=

radica2c sin

2 t + b2c cos2 t








dt

du=

radica2c sin

2 t + b2c cos2 t =

radica2c sin






int ϕ

0



K = ac u(π

2

)=

int π2

0










Theorem





2τK



Q1(∆u )

P1(0)

P1(2∆u )

y

c

e





References

References





















dt

du=

radica2c sin

2 t + b2c cos2 t =

radica2c sin






int ϕ

0



K = ac u(π

2

)=

int π2

0










Theorem





2τK



Q1(∆u )

P1(0)

P1(2∆u )

y

c

e





References

References


























Theorem





2τK



Q1(∆u )

P1(0)

P1(2∆u )

y

c

e





References

References






















References

References



















The motion of billiards in ellipses

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Transcript of The motion of billiards in ellipses