Drawing a Straight line

Popular Lecture Series

Arif Zaman

LUMS

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.

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.

James Watt’s Steam Engine

Watt’s Simplified Version

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?”

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.

Inversion

x → 1/xis an inversion that maps (0,1] to [1,∞) and vice versa

x → c2/x similarlymaps (0,c] to [c,∞)

In two dimensions this maps the inside of a circle to the outside

x y = c2 is an inversion

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

E 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.

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

E

B

A C D

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

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

Hart’s 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.

Hart’s 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

Kempe’s Double Rhomboid

Rhomboid has two pairs of adjacent sides equal.

Here we use two similar rhomboids

This solution is not based on any inversion.

Why it works

By parallel lines, the two a’s and the two b’s are equal.

The two rhomboids are similar, hence a=b.

This means that CD is a horizontal line.

But C is fixed so D moves in a straight line.

a

a b

C D

b

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

Enjoy Mathematics