Drawing a Straight line

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Drawing a Straight line

• Popular Lecture Series
• Arif Zaman
• LUMS

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Need for Straight Line

• Sewing Machine converts rotary motion to up/down
  motion.
• Want to constrain pistons to move only in a
  straight line.
• How do you create the first straight edge in the
  world? (Compass is easy)
• Windshield wipers, some flexible lamps made of
  solid pieces connected by flexible joints.

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James Watts Steam Engine

• Steam engine equipped with Watt's parallelogram.
  
• Five hinges link the rods.
• Two hinges link rods to fixed points
• One hinge links a rod to the piston rod.
• To force the piston rod to move in a straight
  line, to avoid becoming jammed in the cylinder.
• The dashed line shows how this whole assembly can
  be simplified by using three rods and hinge.

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Watts Simplified Version
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Engineer, Mathematician, Accountant

• Engineer 19 20 is approximately 40
• Mathematician 19 20 39
• Accountant Closes the door, and in a low voice,
  whispers What would you like it to be?

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A Straight Line

• The linkage problem attracted designers, and pure
  mathematicians.
• Mathematicians wanted an exact straight line.
• P. L. Tschebyschev (1821-1894) tried
  unsuccessfully.
• Some began to doubt the existence of an exact
  solution.
• Peaucellier (1864) devised a linkage that
  produces straight-line motion, called the
  Peaucellier's cell.
• Soon, a great many solutions of the problem were
  found.
• Solutions were found that would produce various
  curves of which the straight line is only a
  particular case.

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Inversion

• x ? 1/xis an inversion that maps (0,1 to 1,8)
  and vice versa
• x ? c2/x similarlymaps (0,c to c,8)
• In two dimensions this maps the inside of a
  circle to the outside
• x y c2 is an inversion

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Circles that are Tangent to the Center get Mapped
to Straight Lines under Inversion

• ASSUMPTIONS
• Yellow circle is circle of inversion, with center
  A
• Green circle has diameter smaller than the radius
  of the yellow circle, and goes through its center
• C and D are inverses, i.e. AC AD c2
• B is any point on the green circle
• E is the point where AB intersects the
  perpendicular line from D

• TO SHOWE and be are inverses
• ABC and ADE are similar right triangles
• AC/AB AE/AD
• AC AD AB AE
• But AC AD c2
• So AB AE c2
• This shows that E is the inverse of B.
• Since B was an arbitrary point on the circle, we
  have shown that a circle gets mapped to a line.

E
B A C D
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The Peaucellier Inversion

• AE2 BE2 AB2
• EC2 BE2 BC2
• AC AD
• (AE EC)(AE EC)
• AE2 EC2
• AB2 BC2
• is a constant!
• All we need to do is to fix A and move C

B A
C E
D
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The Peaucellier Cell

• Small red circle constrains C to move in a circle
• Red vertical line is image of above circle
• Large red circle is the circle of inversion
• Yellow circle is the limit of movement
• Green circle is just for fun

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Harts Solution

• Linkages are in middle of rods, where the green
  imaginary line intersects the blue rods.
• Green circle is inversion.
• Red circle mapped to red horizontal line
• Centers of yellow and red circles are fixed.

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Harts Inversion

• Points O, P and Q are marked on a line parallel
  to FB and AD
• AOP AFB, FOQ FAD
• OP/FB OA/FA is fixed
• OQ/AD FO/FA is fixed.
• OP OQ FB AD cons
• FB AD AC AD constant (same as
  Peaucellier)

F B O P Q A C E D
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Kempes Double Rhomboid

• Rhomboid has two pairs of adjacent sides equal.
• Here we use two similar rhomboids
• This solution is not based on any inversion.

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Why it works
a a b C D b

• By parallel lines, the two as and the two bs
  are equal.
• The two rhomboids are similar, hence ab.
• This means that CD is a horizontal line.
• But C is fixed so D moves in a straight line.

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What happened next

• Algebraic methods have shown how one can
  approximate any curve using linkages.
• You can design a machine to sign your name!
• This problem is not interesting any more, but
  other similar problems are. Seehttp//www.ams.org
  /new-in-math/cover/linkages1.html

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