Chebyshev linkage wheel variation[br][br][b]My latest comment: ------ Pending. N= odd number case, this logic is bad (?), angle approach is good. length approach is bad. (?) Please help me![/b][br]This 1st logic is true. but not wise a bit. More elegant/ simple solution exists.[br](ex. only Introduce 2 cramps/ KASUGAI . that's all. [ex. crimson colored [b]WZ ---[/b] constant length restriction] )[br][br][b]■My impression:[/b][br]Brown rigid body interface is beautiful (1 antiparallelogram + 2 parallelograms).[br]DC/ C'C''/ D''D' is Hexagon rigid body diameter.

something rule.[br]1.51/2=0.755[br]2/√7=0.755928946 ---- very near.[br]always triangle similarity ∽ is true.[br]So, 1.51 should be 4/√7=1.511857892 precisely, perhaps.[br][br][b]■ Proof of line symmetry[/b][br]In this case ※, it's easy.[br]Q: How many butterflies in above Fig. ?[br]A: 4 ---- big (Black, Blue, Green butterfly) 3 + small (□DGFC butterfly figure, here DC=GF=2) 1[br]-----[br]4 [url=https://en.wikipedia.org/wiki/Antiparallelogram]Antiparallelogram[/url] (wikipedia) are all similar, [br]so □ABDC ∽ □FCDG (similar ratio is √7 : 2), then, ∠BDC = ∠CDG.[br](if one angle is fixed, other 3 angles are determined automatically. and , this butterfly is symmetry.)[br][b]Remark: ---- There exists recursive structure.[/b][br][br]i.e. We can use Antiparallelogram as a line symmetry making tool. ---- This is Big news !!!! (?!)[br]--- I added the sample picture in above Fig. (1:√2:√2:1, ratio case sample, total 7 bars, very simple) [br]cf. [url=https://en.wikipedia.org/wiki/Peaucellier%E2%80%93Lipkin_linkage]Peaucellier–Lipkin linkage[/url] (wikipedia) same 7 bars implementation. ⊥ direction.[br]　but, above is same direction. somewhat honest (?!). ⊥ direction movement also is easy variation.[br][br]※: Chebyshev linkage 2:5:5:4 bars case, it's not easy like above. ∵ Chebyshev butterfly is not symmetry wing.[br]I can't prove yet. ----- perhaps, different logic.[br][br][b]Tip1:[/b] line symmetry making tool is candidates for making exact straight line tool.[br][url=https://en.wikipedia.org/wiki/Hart%27s_inversor]Hart's inversor[/url] (wikipedia) is so. [br]( But Hart's inversor use the property of Antiparallelogram "OR × PQ = constant". different point of view. )[br][br][b]Tip2:[/b] [br]Exact straight line : Hart's Inversor or Hart's A-frame can create by 5 bars.[br]Line symmetry: Hart's Inversor or Hart's A-frame can create by 7 bars.[br]cf. [url=https://www.geogebra.org/m/ARCcx49T]Chebyshev N=2 Polygon Wheel[/url] -- See left side picture.[br][br]But, above method can support Line symmetry by 5 bars.[br]i.e. Case by case.　They have one's strong （points） and weak points.