# N=3 Polygon Wheel

- Author:
- asifsound

Hart's A-frame Application sample
Trigonal wheel
Detail is

**N Polygon Wheel**(GeoGebra)**Pending: in N = odd number !!.**(not easy to control.)Above fig. contains inconsistency.
When N is odd number case, length restriction by rigid bar is bad.
(N-1 bars, another 1 is empty. --- this means , one restriction is missing. --- bad.)
all bars should be replaced into thread or hinge.

**Slack thread is needed. and, N threads should be supported. A・－－－B・ ＼ ／ C・ 3 points A,B,C must be connected by thread (not rigid one bar) or hinge. [ each distance is variable. i.e. constant max. ]**i.e. change method: from Old:**=**restriction, to New:**≦**restriction [ from the bar －－ change to the hinge ／＼ . ] (No bent thread. so, there's no slide friction through Axis point. )**Tip:**in N = 3 Chebyshev wheel, above method was failed by my 1st test. i.e. not feasible. But After more learning, I found there exists above feasible solution in Chebyshev case. ----- above is logic miss. exact rule is " grounding leg's 2 restrict bar is rigid bar, other not grounding leg's bar is thread/ hinge" . i.e. dynamic changing is needed.**it is so difficult for the automatic control.**---- from this, N = 2 case is easy, but N=3 case is not easy. dynamic hinge lock mechanism is needed. (In above fig. bold pink bar F_{1}K_{1}should be removed, and set it as VK_{1}.) ---- Cyan colored leg/ foot is VK_{1}replaced restriction sample. this leg behavior is good. ---- inspiration has occurred. 3D lock mechanism is feasible perhaps. xyz axis space, offset z=0 or not is controllable. See.**Chebyshev N = 2 Polygon Wheel**(GeoGebra).**Chebyshev N = 3 Polygon Wheel**(GeoGebra). Above has yet logic miss. 2 restrictions is enough number of restrict. 3 restriction is odd. grounded foot/ leg is independent, and other 2 legs is dependent by one restriction, so, number of 2 restriction is enough number. Please find the best tuning result. Answer: 3 constant length restriction approach has contradiction. 1 constant length restriction + 2 constant angle movement approach has no contradiction. So, implementation is not so difficult.**■ N=3 Coordinator length (at just 120° span point, another foot has touched on grouned. )**VF_{1}= 9/8 = 1.125 ∵ 0.5=1/2 (3-0.5)*(1/4)=2.5/4=5/8 1/2+5/8=4/8+5/8=9/8 (=1.125) ------ Stride is (3/8）√39=2.341874249 ∵ (3/4)√(4^{2}-2.5^{2})=(3/4）√(16-[5/2]^{2})=(3/4）√([64-25]/4)=(3/4）(1/2)√(39) -------------------- Black, Purple, Cyan are 3 components. Pink is no-needed ( or redundancy). (Bold Purple bar VF_{1}, Bold Cyan bar VP_{1}are needed. Bold Pink bar is no-needed. 3 Points V, F_{1}, P_{1}are completely controlled by 2 bold bars. F1-P1 is not connected directly, but is connected indirectly.) 180° and then -60°, +60° method is needed the thick sandal under the foot in this case. --- why? (result of Chebyshev-like N=3 is different from this result.)