Les\; deux\; trajets

J

M

A

O

I

V_1

V_2

V_2(α\;,\;t_2)

V_1(α\;,\;t_1)

On \;désigne\;par:\
*\;\alpha=\widehat{IOM}\;en\;rad\
*\;t_1:\;la\;longueur \;du\;trajet\;1:\;\;A→O→M\
*\;t_2:\;la\;longueur\;du\;trajet\;2:\;\;A→I→M\
*\;V_1: \;le\;point\;de\;coordonnées\;\left( \alpha\;,\;t_1  \right) \
*\;V_2:\;le\;point\;de\;coordonnées\;\left( \alpha\;,\;t_2  \right) 

Conjecture:\
*\;1^{ère}\;méthode: d'après\;les\;traces\; de \;V_1\;et\;V_2\
t_1≈t_2\;se\;réalise\;pour\;\alpha=1.0297rad \;\left( t_1=1.5150\;\;\;\;et\;\;\;\;\;t_2=1.5147  \right) \
*\;2^{ème}\;méthode:d'après\;le\;tableur\;dont\;le\;pas\;est\;égal\;à\;l'incrément\;du\;curseur\;\alpha   \
-\;A\;partir\;du\;menu\;''Affichage'',\;cacher\;Graphique\;2\;et\;afficher\;Tableur\
-\;A\;la\;ligne\;60\;de\;ce\;tableur\;on\;retrouve\;les\;mêmes\;résultats:\
            \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;  \left( \alpha=1.0297\;\;\;\;\;t_1=1.5150\;\;\;\;\;t_2=1.5147  \right)

O = (0, 0)
I = (1, 0)
J = (0, 1)
c = CircleArc(O, I, J)
a = Segment(O, I)
α = 0.7504914699999999 rad (slider from 0 to 1.5707963267948966, step 0.01745329)
O' = Rotate(O, α, O)
M = Rotate(I, α, O)
a' = Segment(O', M)
b = OrthogonalLine(M, a)
A = Intersect(a, b)
d = Segment(M, A)
e = CircleArc(O, I, M)
f = Segment(A, I)
g = Segment(A, O)
t_1: g + 1
t_2: e + f
V_1: (α, t_1)
V_2: (α, t_2)
h: x = α
A1 = Name(α)
B1 = Name(t_1)
C1 = Name(t_2)
A2 = 0.01745329 rad
B2 = 1.9998476952003703
C2 = 0.017605594799629688
A3 = 0.03490658 rad
B3 = 1.9993908271949852
C3 = 0.03551575280501482
A4 = 0.05235987 rad
B4 = 1.9986295351502248
C4 = 0.05373033484977521
A5 = 0.06981316 rad
B5 = 1.9975640509629535
C5 = 0.07224910903704636
A6 = 0.08726645 rad
B6 = 1.9961946991898831
C6 = 0.09107175081011687
A7 = 0.10471974 rad
B7 = 1.994521896948708
C7 = 0.11019784305129193
A8 = 0.12217303 rad
B8 = 1.9925461537910487
C8 = 0.1296268762089513
A9 = 0.13962632 rad
B9 = 1.9902680715472367
C9 = 0.14935824845276324
A10 = 0.15707961 rad
B10 = 1.9876883441429913
C10 = 0.16939126585700867
A11 = 0.1745329 rad
B11 = 1.9848077573880434
C11 = 0.1897251426119566
A12 = 0.19198619 rad
B12 = 1.9816271887367698
C12 = 0.21035900126323012
A13 = 0.20943948 rad
B13 = 1.9781476070209132
C13 = 0.23129187297908682
A14 = 0.22689277 rad
B14 = 1.9743700721544655
C14 = 0.2525226978455345
A15 = 0.24434606 rad
B15 = 1.9702957348108083
C15 = 0.2740503251891917
A16 = 0.26179935 rad
B16 = 1.9659258360722074
C16 = 0.2958735139277926
A17 = 0.27925264 rad
B17 = 1.9612617070517662
C17 = 0.31799093294823383
A18 = 0.29670593 rad
B18 = 1.9563047684879566
C18 = 0.3404011615120433
A19 = 0.31415922 rad
B19 = 1.9510565303118481
C19 = 0.363102689688152
A20 = 0.33161250999999997 rad
B20 = 1.9455185911871682
C20 = 0.3860939188128318
A21 = 0.3490658 rad
B21 = 1.9396926380233346
C21 = 0.4093731619766654
A22 = 0.36651909 rad
B22 = 1.9335804454616055
C22 = 0.4329386445383944
A23 = 0.38397238 rad
B23 = 1.9271838753345083
C23 = 0.4567885046654918
A24 = 0.40142567 rad
B24 = 1.9205048760987053
C24 = 0.48092079390129466
A25 = 0.41887896 rad
B25 = 1.913545482241478
C25 = 0.5053334777585221
A26 = 0.43633225 rad
B26 = 1.9063078136609994
C26 = 0.5300244363390004
A27 = 0.45378554 rad
B27 = 1.8987940750205963
C27 = 0.5549914649794038
A28 = 0.47123883 rad
B28 = 1.8910065550771842
C28 = 0.5802322749228157
A29 = 0.48869212 rad
B29 = 1.8829476259840927
C29 = 0.6057444940159071
A30 = 0.50614541 rad
B30 = 1.8746197425684832
C30 = 0.6315256674315167
A31 = 0.5235987 rad
B31 = 1.8660254415835857
C31 = 0.6575732584164143
A32 = 0.54105199 rad
B32 = 1.8571673409359786
C32 = 0.6838846490640214
A33 = 0.55850528 rad
B33 = 1.8480481388881511
C33 = 0.7104571411118488
A34 = 0.57595857 rad
B34 = 1.8386706132365842
C34 = 0.7372879567634157
A35 = 0.59341186 rad
B35 = 1.8290376204656085
C35 = 0.7643742395343915
A36 = 0.61086515 rad
B36 = 1.8191520948772921
C36 = 0.791713055122708
A37 = 0.62831844 rad
B37 = 1.8090170476976222
C37 = 0.8193013923023778
A38 = 0.64577173 rad
B38 = 1.7986355661592595
C38 = 0.8471361638407405
A39 = 0.6632250199999999 rad
B39 = 1.7880108125611347
C39 = 0.8752142074388652
A40 = 0.68067831 rad
B40 = 1.7771460233051835
C40 = 0.9035322866948166
A41 = 0.6981316 rad
B41 = 1.7660445079105074
C41 = 0.9320870920894927
A42 = 0.71558489 rad
B42 = 1.7547096480052615
C42 = 0.9608752419947385
A43 = 0.73303818 rad
B43 = 1.7431448962965799
C43 = 0.9898932837034201
A44 = 0.7504914699999999 rad
B44 = 1.7313537755188455
C44 = 1.0191376944811543
A45 = 0.76794476 rad
B45 = 1.7193398773606337
C45 = 1.0486048826393664
A46 = 0.78539805 rad
B46 = 1.7071068613706477
C46 = 1.0782911886293522
A47 = 0.80285134 rad
B47 = 1.6946584538429859
C47 = 1.108192886157014
A48 = 0.82030463 rad
B48 = 1.681998446682077
C48 = 1.138306183317923
A49 = 0.83775792 rad
B49 = 1.6691306962476287
C49 = 1.1686272237523712
A50 = 0.85521121 rad
B50 = 1.6560591221799443
C50 = 1.1991520878200557
A51 = 0.8726645 rad
B51 = 1.6427877062059622
C51 = 1.2298767937940378
A52 = 0.89011779 rad
B52 = 1.6293204909263839
C52 = 1.2607972990736163
A53 = 0.90757108 rad
B53 = 1.6156615785842585
C53 = 1.2919095014157413
A54 = 0.92502437 rad
B54 = 1.6018151298154013
C54 = 1.3232092401845987
A55 = 0.94247766 rad
B55 = 1.5877853623810232
C55 = 1.354692297618977
A56 = 0.95993095 rad
B56 = 1.5735765498829593
C56 = 1.3863544001170407
A57 = 0.97738424 rad
B57 = 1.559193020461891
C57 = 1.4181912195381092
A58 = 0.99483753 rad
B58 = 1.5446391554789511
C58 = 1.4501983745210487
A59 = 1.01229082 rad
B59 = 1.52991938818112
C59 = 1.48237143181888
A60 = 1.02974411 rad
B60 = 1.515038202350815
C60 = 1.514705907649185
A61 = 1.0471974 rad
B61 = 1.500000130940089
C61 = 1.547197269059911
A62 = 1.06465069 rad
B62 = 1.4848097546898475
C62 = 1.5798409353101524
A63 = 1.08210398 rad
B63 = 1.4694717007345126
C63 = 1.6126322792654872
A64 = 1.09955727 rad
B64 = 1.4539906411925538
C64 = 1.6455666288074462
A65 = 1.11701056 rad
B65 = 1.4383712917433138
C65 = 1.6786392682566862
A66 = 1.13446385 rad
B66 = 1.422618410190569
C66 = 1.711845439809431
A67 = 1.15191714 rad
B67 = 1.4067367950132563
C67 = 1.7451803449867436
A68 = 1.16937043 rad
B68 = 1.3907312839038104
C68 = 1.7786391460961897
A69 = 1.18682372 rad
B69 = 1.3746067522945566
C69 = 1.8122169677054434
A70 = 1.20427701 rad
B70 = 1.3583681118726068
C70 = 1.8459088981273932
A71 = 1.2217303 rad
B71 = 1.3420203090837117
C71 = 1.8797099909162882
A72 = 1.23918359 rad
B72 = 1.3255683236255285
C72 = 1.9136152663744714
A73 = 1.25663688 rad
B73 = 1.3090171669307538
C73 = 1.9476197130692463
A74 = 1.27409017 rad
B74 = 1.2923718806405962
C74 = 1.9817182893594039
A75 = 1.29154346 rad
B75 = 1.2756375350690419
C75 = 2.015905924930958
A76 = 1.30899675 rad
B76 = 1.2588192276583894
C76 = 2.0501775223416105
A77 = 1.3264500399999999 rad
B77 = 1.2419220814265195
C77 = 2.0845279585734806
A78 = 1.34390333 rad
B78 = 1.2249512434063738
C78 = 2.118952086593626
A79 = 1.36135662 rad
B79 = 1.2079118830781215
C79 = 2.1534447369218785
A80 = 1.37880991 rad
B80 = 1.1908091907944833
C80 = 2.1880007192055166
A81 = 1.3962632 rad
B81 = 1.1736483761997025
C81 = 2.2226148238002974
A82 = 1.4137164899999999 rad
B82 = 1.1564346666426353
C82 = 2.2572818233573644
A83 = 1.43116978 rad
B83 = 1.1391733055844517
C83 = 2.2919964744155483
A84 = 1.44862307 rad
B84 = 1.1218695510014265
C84 = 2.3267535189985735
A85 = 1.46607636 rad
B85 = 1.1045286737833089
C85 = 2.3615476862166913
A86 = 1.48352965 rad
B86 = 1.087155956127759
C86 = 2.396373693872241
A87 = 1.5009829399999999 rad
B87 = 1.06975668993134
C87 = 2.4312262500686597
A88 = 1.51843623 rad
B88 = 1.0523361751775553
C88 = 2.4661000548224448
A89 = 1.53588952 rad
B89 = 1.034899718322423
C89 = 2.500989801677577
A90 = 1.55334281 rad
B90 = 1.0174526306780782
C90 = 2.535890179321922
A91 = 1.5707961 rad
B91 = 1.0000002267948966
C91 = 2.5707958732051033
A92 = 1.5707963267948966 rad
B92 = 1
C92 = 2.5707963267948966
O_1 = (0.28851872201388873, 2.420346583910968)
I_1 = (1.2885187220138872, 2.420346583910968)
a_1 = Segment(O_1, I_1)
O'_1 = Rotate(O_1, α, O_1)
M_1 = Rotate(I_1, α, O_1)
b_1 = OrthogonalLine(M_1, a_1)
A_1 = Intersect(a_1, b_1)
d_1 = Segment(M_1, A_1)
e_1 = CircleArc(O_1, I_1, M_1)
f_1 = Segment(A_1, I_1)
g_1 = Segment(A_1, O_1)
t_3: g_1 + 1
V_3: (α, t_3)
h_1: x = α
A1_1 = Name(α)
B1_1 = Name(t_3)
β = Angle(A, O, M)
B = (0.2906800674324262, 0.08617093350595705)
texte8: "\alpha=" + (LaTeX[α]) + ""
C = Point(h)
D = Intersect(h, xAxis)
texte12: "α=" + (LaTeX[α]) + ""
i = OrthogonalLine(V_1, yAxis)
j = OrthogonalLine(V_2, yAxis)
E = Intersect(i, yAxis)
F = Intersect(j, yAxis)
texte13: "t_2=" + (LaTeX[t_2]) + ""
texte14: "t_1=" + (LaTeX[t_1]) + ""
k = Segment(V_1, E)
l = Segment(V_2, F)
m = Segment(V_1, D)
G = (0.12101583989701889, 0.14467583955264918)
γ = Angle(I, A, M)