Chebyshev-like N=3 Polygon Wheel
- Author:
- asifsound
Chebyshev linkage wheel variation
My latest comment: ------ Pending. N= odd number case, this logic is bad (?), angle approach is good. length approach is bad. (?) Please help me!
This 1st logic is true. but not wise a bit. More elegant/ simple solution exists.
(ex. only Introduce 2 cramps/ KASUGAI . that's all. [ex. crimson colored WZ --- constant length restriction] )
■My impression:
Brown rigid body interface is beautiful (1 antiparallelogram + 2 parallelograms).
DC/ C'C''/ D''D' is Hexagon rigid body diameter.
something rule.
1.51/2=0.755
2/√7=0.755928946 ---- very near.
always triangle similarity ∽ is true.
So, 1.51 should be 4/√7=1.511857892 precisely, perhaps.
■ Proof of line symmetry
In this case ※, it's easy.
Q: How many butterflies in above Fig. ?
A: 4 ---- big (Black, Blue, Green butterfly) 3 + small (□DGFC butterfly figure, here DC=GF=2) 1
-----
4 Antiparallelogram (wikipedia) are all similar,
so □ABDC ∽ □FCDG (similar ratio is √7 : 2), then, ∠BDC = ∠CDG.
(if one angle is fixed, other 3 angles are determined automatically. and , this butterfly is symmetry.)
Remark: ---- There exists recursive structure.
i.e. We can use Antiparallelogram as a line symmetry making tool. ---- This is Big news !!!! (?!)
--- I added the sample picture in above Fig. (1:√2:√2:1, ratio case sample, total 7 bars, very simple)
cf. Peaucellier–Lipkin linkage (wikipedia) same 7 bars implementation. ⊥ direction.
but, above is same direction. somewhat honest (?!). ⊥ direction movement also is easy variation.
※: Chebyshev linkage 2:5:5:4 bars case, it's not easy like above. ∵ Chebyshev butterfly is not symmetry wing.
I can't prove yet. ----- perhaps, different logic.
Tip1: line symmetry making tool is candidates for making exact straight line tool.
Hart's inversor (wikipedia) is so.
( But Hart's inversor use the property of Antiparallelogram "OR × PQ = constant". different point of view. )
Tip2:
Exact straight line : Hart's Inversor or Hart's A-frame can create by 5 bars.
Line symmetry: Hart's Inversor or Hart's A-frame can create by 7 bars.
cf. Chebyshev N=2 Polygon Wheel -- See left side picture.
But, above method can support Line symmetry by 5 bars.
i.e. Case by case. They have one's strong （points） and weak points.