[b]Sub title: How to make Non-slide symmetric umbrella ?[/b][br][br]Line Symmetry Linkage apparatus.[br]Line symmetry (or, line of symmetry)/ rabatment/ line reflection is an elementary function than drawing exact straight line.[br][br]You don't have to know [b]Peaucellier Inversor[/b]. cf. [url=https://en.wikipedia.org/wiki/Peaucellier%E2%80%93Lipkin_linkage]Peaucellier–Lipkin linkage[/url] (wikipedia)[br][br]This tool is basic & very useful in real world. cf. [url=https://www.geogebra.org/m/rzEjssTJ][b]N=4 2R-Virtual Wheel (Vers.-B)[br][/b][/url]cf[sub]2[/sub]. [url=https://www.geogebra.org/m/gU6d63SY][b]N=3 2R-Virtual Wheel[/b][/url]

[b]Remark:[/b] [br]Cross bar must be long than outer edge. Line symmetry line is assigned to outer edge. If outer edge length were long than a cross bar length, please reassign a short length. From my experience, long: short edge ratio is 2:1 is good for easy calculating.[br][br]cf. [url=https://www.geogebra.org/m/crAjr3q4]Chebyshev-like N=3 Polygon Wheel[/url] ---- in this fig. I presented the exact straight line tool sample.[br]( This tool logic/ concept is [b]simpler than[/b] the logic of Peaucellier–Lipkin linkage [on 1864] / Hart's Inversor [on 1874–5]/ Hart's A-frame [on unknown year]. )[br]( I challenged many times to make exact straight line by line symmetry concept from many years ago, but all failed.　I've been abandoned, but I found this [b]by chance[/b]. God is so volatile/ capricious. )[br][br]Above 1,√2,√2,1 ratio Antiparallelogram is one of typical case (i.e. □ form is square), other ratio Antiparallelogram  (i.e. □ form is rectangle) is OK, of course.[br][br][b]■ Is this an invention? ----- NO, it was famous theory "Kempe's reversor".[/b][br]I searched this apparatus on internet web, but I couldn't find it.[br]So, this apparatus is new one, perhaps.[br]So, I named after my name as a memorial to finding.[br][ "[b]Imai[/b]" is pronounced /[b]imai[/b]/ (not /aimai/ etc.) . Japanese Surname. ][br][br]I found the article. about Kempe works.[br]cf. [url=http://111.93.135.171/Volumes/16/03/0220-0237.pdf]Kempe's Linkages and the Universality Theorem[/url] (.pdf)[br]～ Kempe's work of 1876, now known as the Universality Theorem, has a distinctive standing in kinematics. ～[br]-----[br][b]p223 2.2 The Reversor[/b][br] ～～ For this reason,[br]Kempe's reversor is also called the 'angle-doubler'.[br]--- i.e. adding the same angle. ---- multiple repeat is OK. +α, +α, ... [br][br]cf. [url=https://archive.org/details/howtodrawstraigh00kemprich][b]How to draw a straight line ; a lecture on linkages[/b][/url][br]by [b]Kempe, A. B. (Alfred Bray), 1849-1922[/b][br][br]Published 1877[br]Topics Links and link-motion ------- See. Fig. 30.[br]( page moving operation= page dragging from right to left. or, Use/ click "fullscreen view" 「↖↗[br]↙↘」 icon [= nenu shown].)[br][br][b]■ In a nutshell[/b][br]It's 2 Butterflies which a one wing is overlapped. [br][br][b]■ Orange trace shape is interesting.[/b] ----- like a leg/ foot trace.[br]This trace is similar to Chebyshev Linkage foot trace (by 3 bars), [br]but above is a exact straight line, and, crank rotates same direction of forward movement (by 7 bars), and, has more flexible length ratio.[br][br]crank range 180° to 360° (in return stroke): After passing the shrunk point, Antiparallelogram switched to Parallelogram form (same as Chebyshev case).[br][br][b]Tip: [/b]Between 180° and 360°, we can realize Antiparallelogram, too.[br]In real world, we can realize Antiparallelogram and Parallelogram, both.[br]In GeoGebra's default specification leads the mixture.[br]Suppose a, b are circles. if so, Intersection a and b have 2 points.[br]Intersect[a, b, 1], Intersect[a, b, 2] ---- default. This does not reflect the real world.[br]Intersect[a, b, if[condition, 1, 2] ] is real world solution. ------ Please try. [br]--------------------------------[br][b]Memo:[/b][br]cf. Related:[br][url=https://en.wikipedia.org/wiki/Origami]Origami[/url] (wikipedia)[br][url=http://www.collinsdictionary.com/dictionary/english/rabatment]rabatment or rabattement /rəˈbætmənt/[/url] [br]--- Definitions[br]Collins English Dictionary[br]noun: geometry the act of rotating a plane to align it with another[br][br][b]■ For educational material[/b][br] See [url=http://whistleralley.com/linkage/linkage.htm]Linkage[/url] (related to "inversion geometry") in [b][url=http://whistleralley.com/]Whistler Alley Mathematics[br][br][/url][/b]Please remember Next. [br][b]Antiparallelogram[/b][br] C◯　　 D◯[br] /| ＼ X. ／ |\[br]　 / | ／ ＼ | \[br]A◯ H･ 　　　 J･ B◯[br]Here, Suppose Antiparallelogram, AC=a, AD=b, CD=cc, AB=dd, CH=h (= vertical)[br]then AH=0.5 (dd-cc), AJ=cc+AH=0.5(dd+cc)[br]So h[sup]2[/sup]=a[sup]2[/sup]-(1/4)(dd-cc)[sup]2[/sup], [br]h[sup]2[/sup]=b[sup]2[/sup]-(1/4)(dd+cc)[sup]2[/sup][br]∴ a[sup]2[/sup]-(1/4)(dd-cc)[sup]2[/sup] = b[sup]2[/sup]-(1/4)(dd+cc)[sup]2[/sup],[br]So, b[sup]2[/sup]-a[sup]2[/sup] =(1/4)[(dd+cc)[sup]2[/sup] -(dd-cc)[sup]2[/sup]) =(1/4)(4 cc×dd)=cc×dd[br][br]i.e. Antiparallelogram upper base CD, lower base AB,[br][b]AB // CD[/b], and [b]AB × CD = b[sup]2[/sup]- a[sup]2 [/sup][= constant][br][br]Tip: [/b]∠DAB=θ, ∠BCD=∠θ, CXA=2θ, [br] (from Parallelogram, AB is rabatment line, from ∠A to ∠A - 2θ, ∠C is the same, ∠X is born +2θ)[b][br][br]Parallelogram[br][/b] C◯---------- D◯[br] / | \ ／ / |[br]　 / | ／ \ / |[br]A◯ H･ -------B◯ J･[br]Suppose, AC=a, CD=b, AD=dd, CD=cc, CH=h (= vertical), AH=α[br]So h[sup]2[/sup]= a[sup]2[/sup]-α[sup]2[/sup],[br]h[sup]2[/sup]= dd[sup]2[/sup]-(b+α)[sup]2[/sup],[br]h[sup]2[/sup]=cc[sup]2[/sup]-(b-α)[sup]2[/sup],[br]So, [br]from last 2 lines, sum, 2 h[sup]2[/sup] = dd[sup]2[/sup] +cc[sup]2[/sup] - 2 b[sup]2[/sup] - 2 α[sup]2[/sup],[br]then 1st image is including in it. So,[br] [b]dd[sup]2[/sup] + cc[sup]2[/sup] [/b]= (2 h[sup]2[/sup] + 2 α[sup]2[/sup] )+2 b[sup]2[/sup] [b]= 2 (a[sup]2[/sup] + b[sup]2[/sup] ) [/b][b][sup] [/sup][= constant][br][/b]---- [b]Diagonal line property:[/b] Sum of square is constant. [b]Not Pythagoras Theorem[/b].