Fahrrad auf Kurvenfahrt
Author: Haftendorn/Riebesehl
Date: 5 November 2013

Das Fahhrad ist von Prof. Dr. Dieter Riebesehl.

Ableitung

Ha 2011

Tangente

t = 0.004 (slider from -1 to 1, step 0.001)
f(x) = (((t * (x + 2)^(2)) * (x - 1)) * (x - 7)^(2))
Fahrradgröße = 0.7 (slider from 0.4 to 1.2, step 0.1)
ny(x) = f(x) + (0.5 * Fahrradgröße / sqrt(1 + f'(x)^(2)))
nx(x) = x - ((0.5 * Fahrradgröße) * f'(x) / sqrt(1 + f'(x)^(2)))
P = Point(f)
e = Circle(P, (0.5 * Fahrradgröße))
a = Tangent(P, f)
c = OrthogonalLine(P, a)
C, D = Intersect(e, c)
M = If(y(D) > y(P), D, C)
P_1 = Root(sqrt((nx(x) - x(M))^(2) + (ny(x) - y(M))^(2)) - (1.3 * Fahrradgröße), x(P), x(P) + (2.5 * Fahrradgröße))
B: (x(P_1), f(x(P_1)))
g = Circle(B, (0.5 * Fahrradgröße))
b = Tangent(B, f)
d = OrthogonalLine(B, b)
E, F = Intersect(g, d)
N = If(y(F) > y(B), F, E)
G = Midpoint(M, N)
G' = Rotate(G, 60*°, M)
l = Segment(G', M)
m_3 = Line(G, l)
i = Segment(M, G)
n = Line(G', i)
H_2 = Intersect(m_3, n)
a_1 = Ray(G, H_2)
h_4 = Circle(M, P)
k = Circle(N, B)
α = Angle(G, M, G')
p = Segment(G', H_2)
r = Segment(H_2, N)
b_1 = Ray(G, G')
s = Circle(G, (0.8 * Fahrradgröße))
I = Intersect(s, b_1)
J = Intersect(s, a_1)
j = Segment(J, G)
q = Line(J, p)
c_1 = Circle(J, (0.2 * Fahrradgröße))
K, L = Intersect(c_1, q)
d_1 = Segment(K, L)
e_1 = Circle(M, (0.1 * Fahrradgröße))
g_1 = Circle(G, (0.15 * Fahrradgröße))
h_1 = Ray(G, M)
k_1 = Circle(G, (1.95 * Fahrradgröße))
O = Intersect(k_1, h_1)
i_1, j_1 = Tangent(O, g_1)
Q_1, R = Intersect(e_1, j_1)
S, T = Intersect(e_1, i_1)
U = Intersect(g_1, i_1)
V = Intersect(g_1, j_1)
l_1 = Segment(Q_1, V)
m_1 = Segment(S, U)
H' = Rotate(H_2, 90*°, N)
H'_1 = Rotate(H_2, 260*°, N)
n_1 = Ray(N, H')
p_1 = Ray(N, H'_1)
q_1 = Circle(N, (0.53 * Fahrradgröße))
W = Intersect(q_1, n_1)
Z = Intersect(q_1, p_1)
r_1 = CircleArc(N, Z, W)
G'' = Rotate(G', 130*°, M)
β = Angle(G', M, G'')
G''_1 = Rotate(G', 280*°, M)
s_1 = Ray(M, G'')
t_1 = Ray(M, G''_1)
c_2 = Circle(M, (0.53 * Fahrradgröße))
A_1 = Intersect(c_2, s_1)
B_1 = Intersect(c_2, t_1)
d_2 = CircleArc(M, B_1, A_1)
G''_2 = Rotate(G', (-4 * x(P)), G)
γ = Angle(G', G, G''_2)
a_2 = Line(G''_2, G)
e_2 = Circle(G, (0.2 * Fahrradgröße))
C_1, D_1 = Intersect(e_2, a_2)
b_2 = Segment(D_1, C_1)
f_2 = Line(C_1, i)
g_2 = Line(D_1, i)
h_2 = Circle(C_1, (0.05 * Fahrradgröße))
E_1, F_1 = Intersect(h_2, f_2)
i_2 = Segment(E_1, F_1)
k_2 = Circle(D_1, (0.05 * Fahrradgröße))
G_1, H_1 = Intersect(k_2, g_2)
j_2 = Segment(G_1, H_1)
p_2(x) = 3 - (x - 6)^(2) / 12 - (x - 6)^(3) / 54 + (2 * sin(x))
X_p: (x(P), 0)
n_2 = Segment(P, X_p)
H = Point(f)
t_2 = Line(P, H)
a_3 = OrthogonalLine(H, xAxis)
b_3 = OrthogonalLine(P, a_3)
J_1 = Intersect(a_3, b_3)
poly1, breit, hoch, j_3 = Polygon(P, J_1, H)
f_3(x) = f'(x)
m_sH: (x(H), (y(H) - y(P)) / (x(H) - x(P)))
h = Tangent(P, f)
o = Slope(h)
A: (x(P), Slope[h])
m = Segment((x(P) + 1, y(P)), (x(P) + 1, y(P) + Slope[h]))
c_3 = Segment(A, X_p)
u = false
Q = Point(xAxis)
M_1 = Point(xAxis)
N_1 = Point(xAxis)
O_1 = Point(xAxis)
e_3 = OrthogonalLine(Q, xAxis)
g_3 = OrthogonalLine(M_1, xAxis)
h_3 = OrthogonalLine(N_1, xAxis)
i_3 = OrthogonalLine(O_1, xAxis)
Text2: "f(x)=" + f + ""
Strich1 = AttachCopyToView(PolyLine[(-3.6644636351897573, -0.17087677827792944), (-4.094532124038132, -0.7729726626656531), (-4.137538972922969, -0.858986360435328), (-4.202049246250225, -0.9234966337625841), (-4.245056095135062, -0.9880069070898401), (-4.2880629440199, -1.0525171804170963), (NaN, NaN), (-3.2774019952262203, -0.23538705160518553), (-3.8579944551715255, -1.1170274537443523), (-3.8149876062866883, -1.0525171804170963), (-3.7719807574018507, -0.9880069070898401), (-3.685967059632176, -0.9019932093201654), (NaN, NaN), (-2.868836930820265, -0.17087677827792944), (-3.384919117438314, -1.1385308781867711), (-3.320408844111058, -1.074020604859515), (NaN, NaN), (-2.4387684419718907, -0.17087677827792944), (-2.933347204147521, -1.224544575956446), (NaN, NaN), (-2.2452376219901224, -0.5579384182414661), (-2.1592239242204476, -0.45042129602937253), (-2.8043266574930086, -2.063178129210775), (-2.8258300819354276, -2.1706952514228686), (NaN, NaN), (-1.6001348887175608, 1.0548184149399367), (-1.8366725575841667, 0.302198559455282), (NaN, NaN), (-1.2345766731964427, 1.16233553715203), (-1.7721622842569105, 0.13017116391593242), (NaN, NaN), (-0.8690184576753246, 1.0763218393823553), (-1.1700663998691867, 0.3452054083401194), (-1.1270595509843493, 0.40971568166737554), (NaN, NaN), (-0.5679705154814626, 1.0118115660550993), (-0.46045339326936907, 0.7752738971884935), (-0.43894996882695037, 0.6462533505339813), (-0.696991062135975, 1.0763218393823553), (-0.9980390043298369, 0.0441574661462576), (NaN, NaN), (-0.3314328466148568, 0.7752738971884935), (0.8297520732757537, -0.02035280718099851), (0.09863564223351745, -1.4825856692654702), (0.12013906667593617, -1.3965719714957954), (NaN, NaN), (1.345834259893803, -0.17087677827792944), (0.6792281021788227, -1.4825856692654702), (0.6362212532939853, -1.5470959425927264), (NaN, NaN), (1.7543993242997584, -0.4074144471445352), (0.8942623466030099, -1.783633611459332), (0.958772619930266, -1.6976199136896573), (0.9802760443726847, -1.6331096403624012), (1.0232828932575222, -1.568599367035145), (NaN, NaN), (2.313488359802645, -0.19238020272034811), (2.89408081974795, -0.19238020272034811), (3.216632186384231, -0.32140074937486035), (3.4746732796932553, -0.21388362716276682), (2.5070191797844132, -0.47192472047179124), (1.8834198709542707, -1.5901027914775638), (1.9694335687239455, -1.5255925181503076), (2.0554472664936205, -1.4395788203806328), (2.0984541153784577, -1.3750685470533768), (NaN, NaN), (3.3241493085963243, -0.34290417381727906), (2.743556848651019, -1.6976199136896573), (2.7650602730934377, -1.5901027914775638), (2.7865636975358568, -1.504089093707889), (NaN, NaN), (3.646700675232605, -0.32140074937486035), (3.1736253374993932, -1.5901027914775638), (NaN, NaN), (4.829389019565634, 1.3128595082489611), (4.334810257390004, 0.38821225722495684), (4.377817106274842, 0.45272253055221295), (4.442327379602098, 0.5387362283218877), (NaN, NaN), (5.151940386201915, 1.2913560838065423), (4.721871897353541, 0.1946814372431885), (4.786382170680797, 0.23768828612802592), (4.850892444008053, 0.3237019838977007), (4.893899292892891, 0.38821225722495684), (4.936906141777728, 0.45272253055221295), (5.001416415104984, 0.517232803879469), (NaN, NaN), (5.75403627058964, 1.3128595082489611), (5.345471206183683, 0.38821225722495684), (NaN, NaN), (6.119594486110758, 1.1408321127096115), (5.797043119474477, 0.38821225722495684), (NaN, NaN), (6.313125306092526, 1.1193286882671927), (6.012077363898664, 0.10866773947351371), (6.055084212783502, 0.1731780128007698), (NaN, NaN), (6.614173248286388, 0.9473012927278431), (7.452806801540718, -0.17087677827792944), (7.0872485860196, -0.9234966337625841), (7.108752010462019, -0.858986360435328), (NaN, NaN), (7.538820499310392, -0.25689047604760423), (7.173262283789275, -1.3105582737261205), (7.194765708231693, -1.2460480003988645), (7.216269132674112, -1.1600343026291897), (NaN, NaN), (8.09790953481328, -0.25689047604760423), (7.409799952655881, -1.6546130648048198), (7.474310225983136, -1.568599367035145), (7.538820499310392, -1.4610822448230516), (NaN, NaN), (8.506474599219235, -0.25689047604760423), (7.75385474373458, -1.783633611459332), (7.796861592619417, -1.719123338132076), (7.8398684415042545, -1.6546130648048198), (7.882875290389092, -1.568599367035145), (NaN, NaN), (8.91503966362519, -0.4074144471445352), (9.603149245782589, -0.4289178715869539), (8.872032814740352, -1.9771644314411005), (8.807522541413096, -2.020171280325938), (8.74301226808584, -2.0416747047683566), true], 1, (-4.6321177350986, -3.7619486601618526), (15.77463206075676, 5.549034123405446), (0, 433), (949, 0))
Strich2 = AttachCopyToView(PolyLine[(-2.78282323305059, 1.742927997097335), (-2.632299261953659, 1.4418800549034734), (-2.6107958375112403, 1.3773697815762171), (-2.5892924130688217, 1.3128595082489611), (-2.567788988626403, 1.248349234921705), (-2.546285564183984, 1.183838961594449), (-2.546285564183984, 1.097825263824774), (-2.5247821397415655, 1.0333149904975178), (-2.503278715299147, 0.9688047171702618), (-2.4602718664143093, 0.9042944438430057), (-2.4387684419718907, 0.8397841705157496), (-2.395761593087053, 0.7752738971884935), (-2.352754744202216, 0.7107636238612374), (-2.3097478953173782, 0.6462533505339813), (-2.266741046432541, 0.5817430772067251), (-2.2237341975477034, 0.517232803879469), (-2.180727348662866, 0.45272253055221295), (-2.1377204997780286, 0.38821225722495684), (-2.0947136508931914, 0.3237019838977007), (-2.0732102264507724, 0.2591917105704446), (-2.0517068020083538, 0.1946814372431885), (-2.0086999531235166, 0.13017116391593242), (-1.9871965286810978, 0.0656608905886763), (-1.9441896797962603, 0.0011506172614201946), (-1.901182830911423, -0.0633596560658359), (-1.8581759820265855, -0.12786992939309202), (-1.815169133141748, -0.19238020272034811), (-1.7721622842569105, -0.25689047604760423), (-1.729155435372073, -0.32140074937486035), (-1.6861485864872356, -0.38591102270211647), (-1.6431417376023982, -0.45042129602937253), (-1.6001348887175608, -0.5149315693566286), (-1.5786314642751422, -0.5794418426838848), (-1.5356246153903048, -0.6439521160111409), (-1.514121190947886, -0.708462389338397), (-1.4496109176206298, -0.7729726626656531), (-1.3851006442933738, -0.8374829359929092), (-1.3420937954085363, -0.9019932093201654), (-1.2775835220812801, -0.9450000582050028), (-1.2130732487540241, -0.9665034826474214), (-1.1485629754267679, -0.9450000582050028), (-1.062549277657093, -0.9234966337625841), (-0.9765355798874182, -0.9019932093201654), (-0.912025306560162, -0.8804897848777467), (-0.8475150332329059, -0.8374829359929092), (-0.7830047599056498, -0.7944760871080718), (-0.7399979110208124, -0.7299658137808157), (-0.6754876376935562, -0.6654555404535596), (-0.6109773643663001, -0.6224486915687222), (-0.5679705154814626, -0.5579384182414661), (-0.5034602421542065, -0.5149315693566286), (-0.43894996882695037, -0.47192472047179124), (-0.39594311994211295, -0.4074144471445352), (-0.3314328466148568, -0.36440759825969776), (-0.26692257328760066, -0.29989732493244164), (-0.20241229996034452, -0.25689047604760423), (-0.1594054510755071, -0.19238020272034811), (-0.09489517774825096, -0.12786992939309202), (-0.030384904420994823, -0.08486308050825461), (0.03412536890626131, -0.04185623162341721), (0.09863564223351745, 0.0011506172614201946), (0.1416424911183549, 0.0656608905886763), (0.206152764445611, 0.10866773947351371), (0.27066303777286715, 0.1731780128007698), (0.31366988665770457, 0.23768828612802592), (0.37818015998496074, 0.302198559455282), (0.42118700886979815, 0.36670883278253813), (0.3351733111001233, 0.36670883278253813), (NaN, NaN), (-0.22391572440276325, -0.34290417381727906), (0.09863564223351745, -0.02035280718099851), (0.1631459155607736, 0.0441574661462576), (0.22765618888802974, 0.10866773947351371), (0.29216646221528586, 0.1731780128007698), (0.35667673554254203, 0.23768828612802592), (0.39968358442737945, 0.302198559455282), (0.44269043331221686, 0.36670883278253813), (0.46419385775463556, 0.43121910610979425), (0.48569728219705427, 0.49572937943705037), (0.5287041310818917, 0.5602396527643064), (0.5502075555243104, 0.6247499260915625), (0.5932144044091479, 0.6892601994188187), (0.6147178288515666, 0.7537704727460748), (0.657724677736404, 0.8397841705157496), (0.7007315266212414, 0.9257978682854244), (0.7437383755060788, 0.9903081416126805), (0.808248648833335, 1.0763218393823553), (0.8297520732757537, 1.1408321127096115), (0.8942623466030099, 1.2268458104792863), (0.9157657710454286, 1.2913560838065423), (0.958772619930266, 1.3558663571337985), (1.0017794688151034, 1.4203766304610546), (1.0447863176999408, 1.4848869037883106), (1.0877931665847782, 1.5709006015579854), (1.1523034399120344, 1.6569142993276602), (1.2168137132392904, 1.742927997097335), (1.2813239865665467, 1.8074382704245913), (1.324330835451384, 1.8719485437518473), (1.3888411087786403, 1.9364588170791035), (1.4533513821058963, 2.0009690904063597), (1.5178616554331525, 2.043975939291197), (1.56086850431799, 2.108486212618453), (1.625378777645246, 2.1514930615032903), (1.711392475414921, 2.2160033348305466), (1.775902748742177, 2.280513608157803), (1.8619164465118518, 2.3450238814850586), (1.926426719839108, 2.388030730369896), (1.9909369931663643, 2.4310375792547334), (2.0554472664936205, 2.474044428139571), (2.1199575398208763, 2.4955478525819896), (2.1844678131481325, 2.5170512770244082), (2.2489780864753888, 2.5385547014668273), (2.3349917842450636, 2.5385547014668273), (2.39950205757232, 2.560058125909246), (2.4855157553419946, 2.560058125909246), (2.5715294531116695, 2.560058125909246), (2.6575431508813443, 2.560058125909246), (2.7220534242086005, 2.5385547014668273), (2.7865636975358568, 2.5170512770244082), (2.8510739708631125, 2.474044428139571), (2.89408081974795, 2.409534154812315), (2.9585910930752064, 2.3450238814850586), (3.0015979419600436, 2.280513608157803), (3.0446047908448812, 2.2160033348305466), (3.1091150641721375, 2.1514930615032903), (3.1521219130569746, 2.0869827881760346), (3.1951287619418123, 2.0224725148487783), (3.2381356108266495, 1.957962241521522), (3.259639035269068, 1.893451968194266), (3.3026458841539057, 1.8289416948670099), (3.3241493085963243, 1.7644314215397539), (3.345652733038743, 1.678417723770079), (3.367156157481162, 1.6139074504428228), (3.3886595819235805, 1.527893752673148), (3.410163006365999, 1.463383479345892), (3.431666430808418, 1.398873206018636), (3.4531698552508368, 1.3343629326913797), (3.4746732796932553, 1.248349234921705), (3.496176704135674, 1.183838961594449), (3.517680128578093, 1.1193286882671927), (3.5391835530205116, 1.0548184149399367), (3.5821904019053488, 0.9903081416126805), (3.603693826347768, 0.9257978682854244), (3.646700675232605, 0.8612875949581683), (3.6897075241174426, 0.7967773216309122), (3.73271437300228, 0.7322670483036561), (3.754217797444699, 0.6677567749764), (3.797224646329536, 0.6032465016491438), (3.8187280707719546, 0.5387362283218877), (3.8402314952143737, 0.47422595499463166), (3.883238344099211, 0.40971568166737554), (3.9047417685416295, 0.3452054083401194), (3.9262451929840485, 0.2806951350128633), (3.947748617426467, 0.21618486168560722), (3.9692520418688857, 0.15167458835835113), (4.012258890753723, 0.08716431503109501), (4.0337623151961415, 0.022654041703838897), (4.0552657396385605, -0.04185623162341721), (4.07676916408098, -0.10636650495067332), (4.098272588523398, -0.17087677827792944), (4.141279437408236, -0.23538705160518553), (4.205789710735492, -0.27839390049002294), (4.248796559620329, -0.34290417381727906), (4.313306832947585, -0.38591102270211647), (4.377817106274842, -0.4289178715869539), (4.442327379602098, -0.49342814491420994), (4.506837652929353, -0.5364349937990474), (4.5713479262566095, -0.5794418426838848), (4.6143547751414475, -0.6439521160111409), (4.678865048468703, -0.6654555404535596), (4.743375321795959, -0.708462389338397), (4.807885595123215, -0.7514692382232344), (4.872395868450472, -0.7729726626656531), (4.936906141777728, -0.7944760871080718), (5.001416415104984, -0.8159795115504905), (5.06592668843224, -0.8374829359929092), (5.1304369617594965, -0.858986360435328), (5.216450659529171, -0.8804897848777467), (5.302464357298846, -0.8804897848777467), (5.3884780550685205, -0.8804897848777467), (5.474491752838196, -0.8804897848777467), (5.539002026165452, -0.858986360435328), (5.625015723935127, -0.858986360435328), (5.6895259972623835, -0.8374829359929092), (5.75403627058964, -0.8159795115504905), (5.818546543916896, -0.7944760871080718), (5.883056817244152, -0.7729726626656531), (5.947567090571408, -0.7299658137808157), (6.012077363898664, -0.708462389338397), (6.07658763722592, -0.6654555404535596), (6.141097910553176, -0.6224486915687222), (6.2056081838804324, -0.6009452671263035), (6.270118457207689, -0.5579384182414661), (6.334628730534945, -0.5364349937990474), (6.399139003862201, -0.49342814491420994), (6.463649277189457, -0.47192472047179124), (6.528159550516714, -0.4074144471445352), (6.592669823843969, -0.36440759825969776), (6.657180097171225, -0.32140074937486035), (6.721690370498481, -0.27839390049002294), (6.786200643825738, -0.25689047604760423), (6.850710917152994, -0.21388362716276682), (6.893717766037831, -0.14937335383551073), (6.958228039365087, -0.08486308050825461), (7.0227383126923435, -0.02035280718099851), (7.0872485860196, 0.022654041703838897), (7.108752010462019, 0.08716431503109501), (7.130255434904437, 0.15167458835835113), (7.151758859346856, 0.21618486168560722), (7.173262283789275, 0.2806951350128633), (7.173262283789275, 0.36670883278253813), (7.194765708231693, 0.43121910610979425), (7.216269132674112, 0.49572937943705037), (7.216269132674112, 0.5817430772067251), (7.23777255711653, 0.6462533505339813), (7.259275981558949, 0.7107636238612374), (7.280779406001368, 0.7752738971884935), (7.280779406001368, 0.8612875949581683), (7.302282830443787, 0.9257978682854244), (7.323786254886206, 1.0333149904975178), (7.323786254886206, 1.1193286882671927), (7.345289679328625, 1.2053423860368675), (7.345289679328625, 1.3128595082489611), (7.366793103771043, 1.398873206018636), (7.388296528213462, 1.4848869037883106), (7.388296528213462, 1.5924040260004042), (7.409799952655881, 1.678417723770079), (7.409799952655881, 1.7859348459821724), (7.431303377098299, 1.8719485437518473), (7.431303377098299, 1.957962241521522), (7.431303377098299, 2.043975939291197), (7.452806801540718, 2.108486212618453), (7.452806801540718, 2.194499910388128), (7.452806801540718, 2.280513608157803), (7.452806801540718, 2.388030730369896), (7.452806801540718, 2.474044428139571), (7.474310225983136, 2.560058125909246), (7.495813650425555, 2.6675752481213393), (7.495813650425555, 2.753588945891014), (7.517317074867974, 2.8180992192182703), (7.517317074867974, 2.904112916987945), (7.538820499310392, 2.99012661475762), (7.5603239237528115, 3.0546368880848758), (7.5818273481952305, 3.1406505858545506), (7.5818273481952305, 3.2266642836242254), (7.603330772637649, 3.2911745569514816), (7.603330772637649, 3.3771882547211565), (7.624834197080068, 3.4416985280484127), (7.624834197080068, 3.5277122258180875), (7.624834197080068, 3.6137259235877623), (7.646337621522486, 3.7212430457998558), (7.646337621522486, 3.8072567435695306), (7.624834197080068, 3.871767016896787), (7.5603239237528115, 3.914773865781624), true], 1, (-4.6321177350986, -3.7619486601618526), (15.77463206075676, 5.549034123405446), (0, 433), (949, 0))
v = false