# Line Symmetry Linkage (Alias: Imai's Butterfly)

- Author:
- asifsound

- Topic:
- Symmetry

**Sub title: How to make Non-slide symmetric umbrella ?**Line Symmetry Linkage apparatus. Line symmetry (or, line of symmetry)/ rabatment/ line reflection is an elementary function than drawing exact straight line. You don't have to know

**Peaucellier Inversor**. cf. Peaucellier–Lipkin linkage (wikipedia) This tool is basic & very useful in real world. cf.

**N=4 2R-Virtual Wheel (Vers.-B)**cf

_{2}.

**N=3 2R-Virtual Wheel**

**Remark:**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. cf. Chebyshev-like N=3 Polygon Wheel ---- in this fig. I presented the exact straight line tool sample. ( This tool logic/ concept is

**simpler than**the logic of Peaucellier–Lipkin linkage [on 1864] / Hart's Inversor [on 1874–5]/ Hart's A-frame [on unknown year]. ) ( 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

**by chance**. God is so volatile/ capricious. ) 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.

**■ Is this an invention? ----- NO, it was famous theory "Kempe's reversor".**I searched this apparatus on internet web, but I couldn't find it. So, this apparatus is new one, perhaps. So, I named after my name as a memorial to finding. [ "

**Imai**" is pronounced /

**imai**/ (not /aimai/ etc.) . Japanese Surname. ] I found the article. about Kempe works. cf. Kempe's Linkages and the Universality Theorem (.pdf) ～ Kempe's work of 1876, now known as the Universality Theorem, has a distinctive standing in kinematics. ～ -----

**p223 2.2 The Reversor**～～ For this reason, Kempe's reversor is also called the 'angle-doubler'. --- i.e. adding the same angle. ---- multiple repeat is OK. +α, +α, ... cf.

**How to draw a straight line ; a lecture on linkages**by

**Kempe, A. B. (Alfred Bray), 1849-1922**Published 1877 Topics Links and link-motion ------- See. Fig. 30. ( page moving operation= page dragging from right to left. or, Use/ click "fullscreen view" 「↖↗ ↙↘」 icon [= nenu shown].)

**■ In a nutshell**It's 2 Butterflies which a one wing is overlapped.

**■ Orange trace shape is interesting.**----- like a leg/ foot trace. This trace is similar to Chebyshev Linkage foot trace (by 3 bars), but above is a exact straight line, and, crank rotates same direction of forward movement (by 7 bars), and, has more flexible length ratio. crank range 180° to 360° (in return stroke): After passing the shrunk point, Antiparallelogram switched to Parallelogram form (same as Chebyshev case).

**Tip:**Between 180° and 360°, we can realize Antiparallelogram, too. In real world, we can realize Antiparallelogram and Parallelogram, both. In GeoGebra's default specification leads the mixture. Suppose a, b are circles. if so, Intersection a and b have 2 points. Intersect[a, b, 1], Intersect[a, b, 2] ---- default. This does not reflect the real world. Intersect[a, b, if[condition, 1, 2] ] is real world solution. ------ Please try. --------------------------------

**Memo:**cf. Related: Origami (wikipedia) rabatment or rabattement /rəˈbætmənt/ --- Definitions Collins English Dictionary noun: geometry the act of rotating a plane to align it with another

**■ For educational material**See Linkage (related to "inversion geometry") in

**Whistler Alley Mathematics**Please remember Next.

**Antiparallelogram**C◯ D◯ /| ＼ X. ／ |\ / | ／ ＼ | \ A◯ H･ J･ B◯ Here, Suppose Antiparallelogram, AC=a, AD=b, CD=cc, AB=dd, CH=h (= vertical) then AH=0.5 (dd-cc), AJ=cc+AH=0.5(dd+cc) So h

^{2}=a

^{2}-(1/4)(dd-cc)

^{2}, h

^{2}=b

^{2}-(1/4)(dd+cc)

^{2}∴ a

^{2}-(1/4)(dd-cc)

^{2}= b

^{2}-(1/4)(dd+cc)

^{2}, So, b

^{2}-a

^{2}=(1/4)[(dd+cc)

^{2}-(dd-cc)

^{2}) =(1/4)(4 cc×dd)=cc×dd i.e. Antiparallelogram upper base CD, lower base AB,

**AB // CD**, and

**AB × CD = b**∠DAB=θ, ∠BCD=∠θ, CXA=2θ, (from Parallelogram, AB is rabatment line, from ∠A to ∠A - 2θ, ∠C is the same, ∠X is born +2θ)

^{2}- a^{2 }[= constant] Tip:**Parallelogram**C◯---------- D◯ / | \ ／ / | / | ／ \ / | A◯ H･ -------B◯ J･ Suppose, AC=a, CD=b, AD=dd, CD=cc, CH=h (= vertical), AH=α So h

^{2}= a

^{2}-α

^{2}, h

^{2}= dd

^{2}-(b+α)

^{2}, h

^{2}=cc

^{2}-(b-α)

^{2}, So, from last 2 lines, sum, 2 h

^{2}= dd

^{2}+cc

^{2}- 2 b

^{2}- 2 α

^{2}, then 1st image is including in it. So,

**dd**= (2 h

^{2}+ cc^{2}^{2}+ 2 α

^{2})+2 b

^{2}

**= 2 (a**

^{2}+ b^{2})**----**

^{ }[= constant]**Diagonal line property:**Sum of square is constant.

**Not Pythagoras Theorem**.