# 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.