Chebyshev Wheel[br][br]If finish shape were satisfied, middle form is any.[br]So, Add a bar H₁I₁ +--+, where EH₁=FI₁ [= r₂ ] (No need symmetry.)[br][ H₁I₁ is a good lead very simple bar. ][br][br][b]I recommend [color=#9900ff]Purple color solution[/color].[/b] ------ This is a remarkable method, I think so.[br][br]I think N=3 is very affordable. cf. [url=https://www.geogebra.org/m/J7QQh6nZ]Chebyshev-like N=3 Polygon Wheel (Official ver.)[/url] [[b]2016 version[/b]][br][ But most basic case is N=2, human leg is biped, this is N=2 case. so, no need N>2 (??). [br] Basic question: biped is low energy than N=2 wheel? Which is efficient? ][br]cf. N=4 case: [url=https://www.geogebra.org/m/eekXcx7A]Chebyshev Linkage Wheel[/url] --- typical.[br][br][b]---- Someone, Please convert this apparatus (i.e. N=2 or N=3 Chebyshev wheel) into a commercial reality.[/b] [br]This is no joking.[br][br][color=#ff00ff][b]2018 version[/b] of this N=3 toy is[/color] [url=https://www.geogebra.org/m/UdGbuNpN][color=#0000ff]Polygon-head Chebyshev N3 Wheel[/color][/url][b] [color=#ff00ff].[/color][br][/b][color=#ff00ff]( This is better than above 2016 version. I recommend this N=3 implementation. [br][s]N=4 too[/s]. [/color]---- N=4 case has no merit. why? [color=#ff00ff])[br][br]cf. [b][url=https://www.geogebra.org/m/thEjq3aE]Chebyshev couple N=2[/url] (couple version) 2018/03[/b][br][/color][br][b]■ Analogy :[/b][br]This is virtual [b]r=∞[/b] wheel. [here, r = radius][br]( like as virtual storage/ virtual memory space [we say, [b]vs[/b]] in computer: [br]i.e. in virtual world: memory size =∞, in real world: assigning limited memory size/ disk dynamically. )

[b]■ Line symmetry case was r₂ = 1 case in purple color shape.[/b][br]H₁I₁ = sqrt(4(1/5)^2+(1+3(1/5))^2)=sqrt((4/5)^2+(8/5)^2)=sqrt(80/25)[br]=sqrt(16*5/25)=(4/5)sqrt(5)=0.8√5 (=1.788854382) ----- or = 4/√5[br][br][b]Tip:[/b] By my test, this easy line symmetry simulation by constant length bar exists in not only Chebyshev ratio case, but also in other ratio case . [br]ex. upper base=2, X-bar=√7, lower base=2, height =√3, crank range span is from -60° to +60° (range 120°) case,[br]r₂ = 1.51 , H₁I₁ = 2 constant length bar supports [b]exact line symmetry[/b]. ---- WHY?, very interesting.[br] What natural law exists in this special phenomena?[br]cf. [url=https://www.geogebra.org/m/crAjr3q4]Chebyshev-like N=3 Polygon Wheel[/url] (GeoGebra)[br](upper base = lower base linkage (i.e. [b]Antiparallelogram[/b]) case is exact symmetry. ----- Hart's Inversor.)[br][br][b]My simple Question:[/b][br]Is this N=2 case line symmetry exactly or approximately? which?[br][br][b]Answer:[/b] It's approximately. not exactly. D(0,4), D(2,4), D(4,4) on 3 points, it's equal to exact symmetry shape. --- but, result is very good approximation. [b]--- 99% = exact symmetry.[/b][br]( but, You should be careful of "H₁I₁ ≠ 2 (≒1.79)" property. ) [br][br][b]■ About Other special condition[br][/b]① r[sub]2[/sub] = 1 causes line symmetry.[br]It rotates 2 times angle. 　move at the constant speed　[i.e. nertial law ] add same movement.[br]So, 2 times movement is achieved.[br][br]② r[sub]2[/sub] = 2 causes shrunken [i.e. angle= 0°, initial form mode.] . t[sub]2[/sub] = 2.[br]This special ratio length is useful, perhaps.[br][br]Basically, If foot doesn't touch the ground, its linkage frame form is free.[br]To save energy, I recommend r[sub]2[/sub] = 2 setting. ----- always heel angle = 90°　(⊥). ex. N legs wheel is easy?.[br][br][b]★ r[sub]2[/sub] = 1.95 case is very interesting.[/b][br]Please set r[sub]2[/sub] = 1.95 and check its behavior carefully. [br]Axis D on (0, 4) case: foot is in left below.[br]D on (0.1, 4) case: foot is in left upper. has jumped in a moment. and its shape is right triangle.[br]The heel shape already exists. It looks like a biped lifting the heel highly.[br]Totally, this case is near the biped mode. [br][br]That is " biped motion and wheel motion is equal ultimately."[br][b]In that meaning, this N=2 Chebyshev wheel is superior than other biped method apparatus.[br][br][/b][b]★ r[sub]2[/sub] = 2.46 case is very interesting/ honest.[/b][br]r[sub]2[/sub] = 2.46 is near the biped mode, too. [more than 1.95 case (?) ,,, honest foot trace. what happens in next cycle? ] ----- in next cycle, 2 feet touch the ground, and heels take off from the ground. ----- this is not allowable movement. So, r[sub]2[/sub] > 2 value is not allowable. we should inhibit. [b]this is bad pattern.[/b][br]Please check this, too. we can easily step beyond over/ on the high barrier ___|[sup]--- [/sup]by this (?). .[br]heel angle is obtuse angle. this is better.[br][br][b]■　Real implementat[br]　----　Fact: in 3D real world, It doesn't need this arrangement. straight bar is OK/ no conflict.[/b][br]Coordinator bar H₁I₁ conflicts with the red axis D ◯.[br]We can avoid this. In r[sub]2[/sub] ≦ 1 case, point D is inner in H₁I₁ diameter circle (as purple colored circle in above Fig.).[br]So, If you use the bar of H₁I₁ of hollowed-out bar (or circle etc.), you can crank the pedal. [br][br]If you make Leg AE, BF as 凸 (partially convex leg form, upper 1 length part), then, you can get the No-confliction with the axis D. [br][br][b]■ Simple question to YOU from me. [/b] ----- this is educational question.[br]From my experience,[br]Chebyshev linkage and Hart's A-frame are near these forms/ shape.[br]α=∠AEF,　β＝∠BFE[br]then, [br][b]sin(α/2) + sin(β/2) ≒ constant ----- eq. 1[/b][br][br]is true. why?[br](if trace is exact straight line, = is true.[br]Chebyshev trace is not exact straight line, so, ≒ .)[br][br]Please Prove eq. 1[br][br]cf. Hint ----- [url=https://www.geogebra.org/m/RMMUtKGn]N Polygon Wheel (N=2 shape)[/url] Hart's A-frame wheel sample[br]QR ∽ RV (∵ △QSR ∽ △TRV)[br][br][b]■ Noisy roller bag (/ wheeled bag/ luggage trolley etc.)[/b][br]Generally, roller bag is so noisy. It's very annoying. It's all defective product.[br]Please replace it with above caster on the world. [br]--- if necessary, please put on the shoes to feet. (ex. in Venice)[br][br]cf. [url=http://www.flyertalk.com/forum/travel-products/1106450-who-best-quiet-wheel.html]Who is the best "Quiet wheel"? ...[/url][br]cf. [url=http://www.guysgab.com/diy-fixing-that-squeaky-luggage-wheel/]DIY: Fixing That Squeaky Luggage Wheel[/url][br]cf. [url=http://www.independent.co.uk/travel/europe/venice-bans-wheel-suitcases-to-stop-noise-pollution-and-protect-its-historic-streets-9880369.html]Venice isn't banning wheeled suitcases to stop noise pollution and protect its historic streets[br][/url]cf. [url=http://www.fodors.com/community/europe/wheelie-suitcases-ban.cfm]Wheelie Suitcases Ban[/url] (Fodor's Travel)[br] |---- in this. They say that "Rolling bags are one of the great inventions of the 20th century !! "[br] ---- I think " it's somewhat so, but noise of caster is defective." [br] ---- Above walking wheel is one of inventions of the 21th century !!, I think so. At least, it solves the Venice problem.[br] Please enhance/ improve more.[br]cf. [url=https://www.tripadvisor.com/ShowTopic-g187870-i57-k8277838-Wheeled_luggage_ban_in_venice_on_or_off-Venice_Veneto.html]wheeled luggage ban in venice on or off? (on 2015/03)[br][/url]cf. [url=https://margieinitaly.com/2014/11/21/venice-proposes-noisy-luggage-ban-for-tourists/?iframe=true&preview=true]Venice Proposes Noisy Luggage Ban for Tourists (on 2014/11)[/url] [br] ----- I propose, Italy should make a New caster named "[b]Venice Chebyshev wheel[/b]"　as one solution for Venice problem. ---- this is no joking.[br]cf. [url=http://www.dailymotion.com/video/x2as2jc_venice-to-ban-noisy-wheeled-suitcases_travel]Venice To Ban Noisy Wheeled Suitcases (on 2014/11)[/url] ((C) dailymotion) vtr