The Nothing Grinder(aka. The Trammel of Archimedes)

[image 1_ a nothing grinder] https://commons.wikimedia.org/wiki/File:Trammel_of_Archimedes_Small_White.gif

A nothing grinder, aka the Trammel of Archimedes is an elipsograph; a tool for drawing elipses. In the gif file above, the shaft is moved in such a way that the end part outside the rectangular area trails an elipse.

[image 2_ proof of the end point's trace being an elipse]

Proof)

segment AP= a(long radius), segment BP= b(short radius), line PXY is perpendicular to line BX, AX

when angle(PBX)= t, angle(PAY)= t

    segment PX= b*sin(t)

    segment AY= a*cos(t)

=> P(a*cos(t), b*sin(t))

Since the parametric representation of an elipse is (a*cos(t), b*sin(t)), point P is always a point on an elipse with its center at point 0.

 

Tusi Couple

The Tusi couple is a device in which a circle rotates inside a larger circle twice its radius. It is also a generalization of the Trammel of Archimedes. A point in the smaller circle of the Tusi couple oscilates back and forth trailing the diameter of the larger circle.

[image 3_ Tusi couple] https://en.wikipedia.org/wiki/Tusi_couple

[image 4_ proof of the Tusi couple]

Proof)

point O = center of big circle (radius = 2R)

point A = initial center of small circle (radius = R)

point B = center of small circle after it rolled counterclockwise along the border of the big circle

point P = tangent point of circle O & circle A

point P' = position of point P on circle B

point Q = tangent point of circle O & circle B

point X= tangent point of segment PO and circle B

 

Let angle(POQ) = 2ø

angle(POQ) = angle(XOQ)

                   = angle(XBQ)

                   = ø

sector(PQ) = (2π(2R))*(ø/2π) = 2Rø

sector(XQ) = (2πR)*(2ø/2π) = 2Rø

 

Since sector(XQ) = sector(PQ) and circle B is the later position of circle A, it can be seen that point P' = point X. Therefore, a point in the smaller circle of the Tusi couple oscilates back and forth trailing the diameter of the larger circle.

 

The Trammel of Archimedes can be made by extending the diameter of the inner circle. The tangent points of the small circle( the 'sliders' of the elipsograph) and the extended line will move back and forth in a straight line - the diameter of the bigger circle. Modifications of the Trammel of Archimedes which involve more than two sliders, are just more points on the smaller circle used as sliders.