An ellipsograph is a

mechanism Mechanism may refer to: *Mechanism (economics), a set of rules for a game designed to achieve a certain outcome **Mechanism design, the study of such mechanisms *Mechanism (engineering), rigid bodies connected by joints in order to accomplish a ...

that generates the shape of an

ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...

. One common form of ellipsograph is known as the trammel of Archimedes. () It consists of two shuttles which are confined to perpendicular channels or rails and a rod which is attached to the shuttles by pivots at adjustable positions along the rod. As the shuttles move back and forth, each along its channel, all points on the rod move in elliptical paths. The motion of the rod is termed elliptical motion. The semi-axes ''a'' and ''b'' of the ellipses have lengths equal to the distances from the point on the rod to each of the two pivots. The straight lines described by the pivots are special cases of an ellipse, where the length of one axis is twice the distance between the pivots and that of the other is zero. All points on a circle with a diameter defined by the two pivots reciprocate in such straight lines. This circle corresponds to the smaller circle in a

Tusi couple The Tusi couple (also known as Tusi's mechanism) is a mathematical device in which a small circle rotates inside a larger circle twice the diameter of the smaller circle. Rotations of the circles cause a point on the circumference of the smaller ...

. The point midway between the pivots orbits in a circle around the point where the channels cross. This circle is also a special case of an ellipse. Here the axes are of equal length. The diameter of the circle is equal to the distance between the pivots. The direction of travel around the orbit is opposite to the sense of rotation of the trammel. Thus, if a crank centred on the crossing point of the channels is used to engage the trammel at the midway point to drive it, the rotation of the crankpin and the trammel are equal and opposite, which in practical applications results in extra friction and accelerated wear. This is compounded by high forces owing to the short throw of the crank of only one quarter the travel of the pivots. Bullshit grinder

Versions are also made as

toy A toy or plaything is an object that is used primarily to provide entertainment. Simple examples include toy blocks, board games, and dolls. Toys are often designed for use by children, although many are designed specifically for adults and p ...

s or

novelty item A novelty item is an object which is specifically designed to serve no practical purpose, and is sold for its uniqueness, humor, or simply as something new (hence "novelty", or newness). The term also applies to practical items with fanciful or ...

s (sold under the name of Kentucky do-nothings, nothing grinders, do nothing machines, smoke grinders, or bullshit grinders). In these toys the drafting instrument is replaced by a crank handle, and the positions of the sliding shuttles along the rod are usually fixed.

Mathematics

Let be the outer end of the rod, and , be the pivots of the sliders. Let and be the distances from to and to , respectively. Let us assume that sliders and move along the and

coordinate In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...

axes, respectively. When the rod makes an angle with the ''x''-axis, the coordinates of point are given by :

\begin x &= (AB+BC)\cos\theta\, \\ y &= BC\sin\theta\, \end

These are in the form of the standard

parametric equation In mathematics, a parametric equation expresses several quantities, such as the coordinates of a point (mathematics), point, as Function (mathematics), functions of one or several variable (mathematics), variables called parameters. In the case ...

s for an ellipse in canonical position. The further equation :

\frac + \frac= 1

is immediate as well. The trammel of Archimedes is an example of a

four-bar linkage In the study of Mechanism (engineering), mechanisms, a four-bar linkage, also called a four-bar, is the simplest closed-Kinematic chain, chain movable linkage (mechanical), linkage. It consists of four Rigid body, bodies, called ''bars'' or ''link ...

with two sliders and two pivots, and is special case of the more general oblique trammel. The axes constraining the pivots do not have to be perpendicular and the points , and can form a triangle. The resulting locus of is still an ellipse.

Ellipsographs

Ellipse compass-MHS 614-IMG 3827-gradient

An ellipsograph is a trammel of Archimedes intended to draw, cut, or machine ellipses, e.g. in

wood Wood is a structural tissue/material found as xylem in the stems and roots of trees and other woody plants. It is an organic materiala natural composite of cellulosic fibers that are strong in tension and embedded in a matrix of lignin t ...

or other sheet materials. An ellipsograph has the appropriate instrument (pencil, knife, router, etc.) attached to the rod. Usually the

distances Distance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two co ...

''a'' and ''b'' are adjustable, so that the size and shape of the ellipse can be varied. The history of such ellipsographs is not certain, but they are believed to date back to

Proclus Proclus Lycius (; 8 February 412 – 17 April 485), called Proclus the Successor (, ''Próklos ho Diádokhos''), was a Greek Neoplatonist philosopher, one of the last major classical philosophers of late antiquity. He set forth one of th ...

and perhaps even to the time of

Archimedes Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...

Notes

References

* J. W. Downs: ''Practical Conic Sections: The Geometric Properties of Ellipses, Parabolas and Hyperbolas''. Courier Dover 2003, , pp. 4–5 () * I. I. Artobolevskii ''Mechanisms for the Generation of Plane Curves''. Pergamon Press 1964, .

External links

Video of various trammel designs in actionPhoto of a Kentucky Do-NothingVideo
of a Do-Nothing made from

Lego Lego (, ; ; stylised as LEGO) is a line of plastic construction toys manufactured by the Lego Group, a privately held company based in Billund, Denmark. Lego consists of variously coloured interlocking plastic bricks made of acrylonitri ...

bricks
"Wonky Trammel of Archimedes"
{{Webarchive, url=https://web.archive.org/web/20120220152318/http://web.mat.bham.ac.uk/C.J.Sangwin/Publications/WonkyTrammel.pdf , date=2012-02-20 An exploration of a generalized trammel.

* ttps://www.youtube.com/watch?v=7Fn-26Jmi5E ''Secrets of the Nothing Grinder''YouTube video by

Mathologer Burkard Polster (born 26 February 1965 in Würzburg) is a German mathematician who runs and presents the ''Mathologer'' channel on YouTube. He is a professor of mathematics at Monash University in Melbourne, Australia. Education and career P ...

Traditional toys Mechanisms (engineering) Linkages (mechanical) Ellipses Novelty items Educational toys

Mathematics

Ellipsographs

See also

Notes

References

External links